318.  The  equations  of  continued  motion  of  a set  of  free 
particles  acted  on  by  any  forces,  or  of  a system  connected  in 
any  manner  and  acted  on  by  any  forces,  are  readily  obtained 
in  terms  of  Lagrange’s  Generalized  Co-ordinates  by  the  regular 
and  direct  process  of  analytical  transformation,  from  the  or- 
dinary forms  of  the  equations  of  motion  in  terms  of  Cartesian 
(or  rectilineal  rectangular)  co-ordinates.  It  is  convenient  first 
to  effect  the  transformation  for  a set  of  free  particles  acted 
on  by  any  forces.  The  case  of  any  system  with  invariable 
connexion.s,  or  witli  connexions  varied  in  a given  manner,  is 


Impulsive 
motion  of 
incompres- 
sible liquid. 


Lafrranpe’s 
equations  of 
motion  in 
terms  of 
fienemlizHd 
co-ordinates 


302 


PRELIMINARY. 


[318. 

then  to  be  dealt  with  by  supposing  one  or  more  of  the  gene- 
ralized co-ordinates  to  be  constant : or  to  be  given  functions 
of  the  time.  Thus  the  generalized  equations  of  motion  are 
merely  those  for  the  reduced  number  of  the  co-ordinates  re- 
maining un-given ; and  their  integration  determines  these 
co-ordinates. 


deduced 
direct  by 
transforma- 
tion from 
the  equa- 
tions of 
motion  in 
terms  of 
Cartesian 
co-ordi- 
nates. 


Let  m,,  etc.  be  the  masses,  etc.  be  the  co- 
ordinates of  the  particles;  and  etc.  the  components 

of  the  forces  acting  upon  them.  Let  xf/,  <fi,  etc.  be  other  variables 
equal  in  number  to  the  Cartesian  co-ordinates,  and  let  there  be 
the  same  number  of  relations  given  between  the  two  sets  ot 
variables;  so  that  we  may  either  regard  xf/,  (f>,  etc.  as  known 
functions  of  y^,  etc.,  or  etc.  as  known  functions  of 

xj/,  (f),  etc.  Proceeding  on  the  latter  supposition  we  have  the 
equations  {a),  (1),  of  § 313;  and  we  have  equations  (6),  (6),  of 
the  same  section  for  the  generalized  components  <l>,  etc.  of  the 
force  on  the  system. 


For  the  Cartesian  equations  of  motion  we  have 


cLoo 

Multiplying  the  first  by  the  second  by  and  so  on, 
and  adding  all  the  products,  we  find  by  313  (6) 


’ df  d^f  de  # ■ de  dil 

Now 


^ + wZg  (etc.)  + etc (20). 


d^x-^  dx-^ 

df  dxf/ 


/.  dxj\ 

. d dx^ 

d 

/ , dx  \ 

. dx^ 

V^dxp) 

^ dt  dxj/ 

dt 

_d  \ d (g,’'))  _ , d{x^^) 
dl\^  d^  j ^ dil^ 


(21). 


Using  this  and  similar  expressions  with  reference  to  the  other 
co-ordinates  in  (20),  and  remarking  that 


(iCj*  -f-  -vz^)  -f  (etc.)  -t-  etc.  = T (22), 

if,  as  before,  we  put  T for  the  kinetic  energy  of  the  system;  we 
find 


d<^_dT 

dt  dxj/  d^ 


(23). 


318.] 


DYNAMICAL  LAWS  AND  PRINCIPLES. 


303 


The  substitutions  of  for  and  of 

dip'  dip  dip 


„ d dx,  . 
tor  — — used 
dt  dip 


above,  suppose  to  be  a function  of  the  co-ordinates,  and  of  the 
generalized  velocity-components,  as  shown  in  equations  (1)  of 
§313.  It  is  on  this  supposition  [which  makes  T a quadratic 
function  of  the  generalized  velocity-components  with  functions 
of  the  co-ordinates  as  coefficients  as  shown  in  § 313  (2)]  that  the 

differentiations  and  (23)  are  performed.  Proceeding 

similarly  with  reference  to  cp,  etc.,  we  find  expressions  similar  to 
(23)  for  etc.,  and  thus  we  have  for  the  equations  of  motion  in 
terms  of  the  generalized  co-ordinates 


dip 


% 


d d_T 
dt  dip 
dc^_dT_^ 
dt  dcp  d(p  ’ 
etc. 


(24). 


It  is  to  be  remarked  that  there  is  nothing  in  the  preceding 
transformation  which  would  be  altered  by  supposing  t to  appear 
in  the  relations  between  the  Cartesian  and  the  generalized  co- 
ordinates : thus  if  we  suppose  these  relations  to  be 


2/ij  2^1,  ajg, xp,  e, 

Vxx  ^1,  

etc. 

we  now,  instead  of  § 313  (1),  have 


‘.-(1)*S**S***'i 

etc.  J 


(25), 


(26), 


where 


denotes  what  the  velocity-component  would  be 


if  ip,  (f),  etc.  were  constant;  being  analytically  the  partial  differ- 
ential coefficient  with  reference  to  t of  the  formula  derived  from 
(2G)  to  express  x^  as  a function  of  t,  ip,  (p,  By  etc. 


Using  (26)  in  (22)  we  now  find  instead  of  a homogeneoiis 
quadratic  function  of  ip,  cp,  etc.,  as  in  (2)  of  § 313,  a mixed 


Lagrange’s 
equations  of 
motion  in 
terms  of 
generalized 
co-ordinates 
deduced 
direct  by 
transforma- 
tion from 
the  equa- 
tions of 
motion  in 
terms  of 
Cartesian 
co-ordi- 
nates. 


304 


PRELIMINARY. 


[818. 


Lagrange’s 
equations  of 
motion  in 
terms  of 
generalized 
co-ordinates 
deduced 
direct  by 
transforma- 
tion from 
the  equa- 
tions of 
motion  in 
terms  of 
Cartesian 
co-ordi- 
nates. 


function  of  zero  degree  and  first  and  second  degrees,  for  the 
kinetic  energy,  as  follows  : — 

T =.K+{il/) k}; +{<!>) <j>+  ... 

where 


J 2m 

{(©)■ 

*(©)■• 

(©)’} 

2 m 

( /dx\  dx 
dip 

/dz\  dz) 

ip)  = 

WdxV 

(W/ 

\dilij  Kdij/J 

1 1 , etc. 

{xp-,  cP)  r= 

/dx  dx 

dy  dy  dz  dz\ 

2m 

\dip  d(p 

dxp  dcp  dxp  d<pj  ’ 

etc. 


..(28); 


AT,  (ij/),  (</)),  {if/j  ij/))  {\f/,  </>),  etc.  being  thus  in  general  each  a known 
function  of  t,  {j/,  0,  etc. 


Equations  (24)  above  are  Lagrange’s  celebrated  equations  of 
motion  in  terms  of  generalized  co-ordinates.  It  was  first 
pointed  out  by  Vieille*  that  they  are  applicable  not  only  when 
yjr,  (p,  etc.  are  related  to  x^,  etc.  by  invariable  relations 

as  supposed  in  Lagrange’s  original  demonstration,  but  also 
when  the  relations  involve  t in  the  manner  shown  in  equa- 
tions (25).  Lagrange’s  original  demonstration,  to  be  found 
in  the  Fourth  Section  of  the  Second  Part  of  his  M^canique 
Analytique,  consisted  of  a transformation  from  Cartesian  to 
generalized  co-ordinates  of  the  indeterminate  equation  of 
motion ; and  it  is  the  same  demonstration  with  unessential 
variations  that  has  been  hitherto  given,  so  far  as  we  know, 
by  all  subsequent  writers  including  ourselves  in  our  first  edition 
(§  329).  It  seems  however  an  unnecessary  complication  to 
introduce  the  indeterminate  variations  hx,  Sy^  etc. ; and  we  find 
it  much  simpler  to  deduce  Lagrange’s  generalized  equations 
by  direct  transformation  from  the  equations  of  motion  (19) 
of  a free  particle. 


• Sur  les  equations  differentielles  de  la  dynamique,  Liouville's  Journal, 
1819,  p.  201. 


♦^18.] 


DYNA.MICAL  LAWS  AND  PRINCIPLES. 


.305 


When  the  kinematic  relations  are  invariable,  that  is  to  say  Lagrange’s 

ffcn0i*9;liz6cl 

when  t does  not  appear  in  the  equations  of  condition  (25),  we  form  9f  the 
find  from  (27)  and  (28), 

expanded. 

= 2 1(‘Aj  V")  ’A"  + ^ (‘Aj  + • • • } (29), 

■^^=(‘A;  ‘A)‘A  + ('A.  <A)^+  ••• 

at  dif/ 


•iT^**^’**-)* . 


.(29'), 


and 


2 ^ (<A>  <A)' 

d(f)  dij/ 


Hence  the  i/^-equation  of  motion  expanded  in  this,  the  most 
important  class  of  cases,  is  as  follows  : 

where  | 

(29"').  _ 

Remark  that  Q^{T)  is  a quadratic  function  of  the  velocity-com- 
ponents derived  from  that  which  expresses  the  kinetic  energy 
{T)  by  the  process  indicated  in  the  second  of  these  equations, 
in  which  ij/  appears  singularly,  and  the  other  co-ordinates  sym- 
metrically with  one  another. 

Multiply  the  i/^-equation  by  ij/,  the  (^-equation  by  and  so  Equation  of 
on  ; and  add.  In  what  comes  from  {T)  we  find  terms 


energy. 


which  together  yield 


d(j> 


With  this,  and  the  rest  simply  as  shown  in  (29"'),  we  find 

[(lA,  if/)  {if/,  <f>)  ij)  + . . .]  if/ 

+ [(‘At  <A)  ‘A  + (<A.  <A)  + •••]  'A 


VOL.  I. 


20 


Equation  of 
energy. 


Hamilton’s 

form. 


30G 


PRELIMINARY. 


[318. 


dT  . dT  . 


= ^{}/  + + 


(29"), 


dt 


= ^il/  + ^cj)  + 


,(29'). 


When  the  kinematical  relations  are  invariable,  that  is  to  say, 
when  t does  not  appear  in  the  equations  of  condition  (25),  the 
equations  of  motion  may  be  put  under  a slightly  different  form 
first  given  by  Hamilton,  which  is  often  convenient ; thus  ; — Let 


T,  if;,  be 


in  terms  of  i,  r],...,  the  impulses  re- 


quired to  produce  the  motion  from  rest  at  any  instant  [§  313  (c?)]; 
so  that  T will  now  be  a homogeneous  quadratic  function,  and 
ij/,  ...  each  a linear  function,  of  these  elements,  with  coeffi- 
cients— functions  of  i/r,  <f>,  etc.,  depending  on  the  kinematical 
conditions  of  the  system,  but  not  on  the  particular  motion. 
Thus,  denoting,  as  in  § 322  (29),  by  d,  partial  differentiation  with 
reference  to  rj,  if/,  considered  as  independent  vari- 

ables, we  have  [§  313  (10)] 

dT 


. dT 


d-n  ’ 


(30), 


and,  allowing  d to  denote,  as  in  what  precedes,  the  partial  dif- 
ferentiations with  reference  to  the  system  ij/,  ...,  if/,  ...,  we 
have  [§  313  (8)J 

i = V = (31). 


I4  ' 


The  two  expressions  for  T being,  as  above,  § 313, 

»A)^^^+...+2(i/^,  = ^]^’7+---K32), 

the  second  of  these  is  to  be  obtained  from  the  first  by  substitu- 
ting for  {j/,  <j) , their  expressions  in  terms  of  17,  ...  Hence 


(‘ 


dT  dT 

dk'^'^  d-n' 


dT  dT  dT  dT 

di[/  dij/  dij/  dif/  d(ji  dxj/ 

_d^ 

dif/  d\{/ 

From  this  we  conclude 

dT  dT  , . 1 dT 

Hence  Lagrange’s  equations  become 

di  'bT  ^ ^ 

dt  d^ 


_dT  , d dT 

'~dxf,^^d<i,  di'^^’df  dn'^ 


dT 

= 1-  2 — . 

d\f/  d\(/ 


dT  ^ 
dr^^- 


(33). 


(34). 


DYNAMICAL  LAWS  AND  PRINCIPLES. 


307 


318.] 

In  § 327  below  a purely  analytical  proof  will  be  given  of  Hamilton’s 
Lagrange’s  generalized  equations  of  motion,  establishing  them 
directly  as  a deduction  from  the  principle  of  “ Least  Action,” 
independently  of  any  expression  either  of  this  principle  or  of 
the  equations  of  motion  in  terms  of  Cartesian  co-ordinates.  In 
their  Hamiltonian  form  they  are  also  deduced  in  § 330  (33)  from 
the  principle  of  Least  Action  ultimately,  but  through  the  beau- 
tiful “ Characteristic  Equati(m”  of  Hamilton. 


319.  Hamilton’s  form  of  Lagrange’s  equations  of  motion  in 
terms  of  generalized  co-ordinates  expresses  that  what  is  re- 
quired to  prevent  any  one  of  the  components  of  momentum 
from  varying  is  a corresponding  component  force  equal  in 
amount  to  the  rate  of  change  of  the  kinetic  energy  per  unit 
increase  of  the  corresponding  co-ordinate,  with  all  components 
of  momentum  constant:  and  that  whatever  is  the  amount  of 
the  component  force,  its  excess  above  this  value  measures  the 
rate  of  increase  of  the  component  momentum. 

In  the  case  of  a conservative  system,  the  same  statement 
takes  the  following  form: — The  rate  at  which  any  component 
momentum  increases  per  unit  of  time  is  equal  to  the  rate,  per 
unit  increase  of  the  corresponding  co-ordinate,  at  which  the 
sum  of  the  potential  energy,  and  the  kinetic  energy  for  con- 
stant momen turns,  diminishes.  This  is  the  celebrated  “canonical 
form”  of  the  equations  of  motion  of  a system,  though  why  it 
has  been  so  called  it  would  be  hard  to  say. 


Let  V denote  the  potential  energy,  so  that  293  (3)] 
^Si//  + + . . . = - 8 F, 


and  therefore 


“ Canonical 
form”  of 
Hamilton’s 
general 
equations  of 
motion  of  a 
conserva- 
tive system. 


Let  now  17  denote  the  algebraic  expression  for  the  sum  of  the 
potential  energy,  F,  in  terms  of  the  co-ordinates,  \f/,  cf}...,  and  the 
kinetic  energy,  T,  in  terms  of  the  co-ordinates  and  the  components 
of  momentum,  17,....  Then 


also 


^^_dU 

(it  d\\t 
clil/  _dU 
(It  cl^ 


etc. 


etc. 


(35), 


20—2 


308 


PRELIMINARY. 


[319. 


Examples  of 
the  use  of 
Lagrange’s 
generalized 
equations  of 
motion 
polar  co- 
ordinates. 


the  latter  being  equivalent  to  (30),  since  the  potential  energy  does 
not  contain  r],  etc. 

In  the  following  examples  we  shall  adhere  to  Lagrange’s  form 
(24),  as  the  mobt  convenient  for  such  applications. 

Example  (A). — Motion  of  a single  point  (m)  referred  to  polar 
co-ordinates  (r,  6,  4>).  From  the  well-known  geometry  of  this 
case  we  see  that  8r,  rSO,  and  r sin  are  the  amounts  of  linear 
displacement  corresponding  to  infinitely  small  increments,  Sr,  8^, 
S(fi,  of  the  co-ordinates  : also  that  these  displacements  are  respec- 
tively in  the  direction  of  r,  of  the  arc  r8d  (of  a great  circle) 
in  the  plane  of  r and  the  pole,  and  of  the  arc  rsin^S^  (of  a 
small  circle  in  a plane  perpendicular  to  the  axis) ; and  that  they 
are  therefore  at  right  angles  to  one  another.  Hence  if  F,  G,  H 
denote  the  components  of  the  force  experienced  by  the  point,  in 
these  three  rectangular  directions,  we  have 

F = R,  Gr=®,  and  Hr  sin  ^ ; 

R^  ©,  ^ being  what  the  generalized  components  of  force  (§  313) 
become  for  this  particular  system  of  co-ordinates.  We  also  see 
that  r,  rO,  and  are  three  components  of  the  velocity, 

along  the  same  rectangular  directions.  Hence 

T = + F sin^  Ocp^). 

From  this  we  have 

dT  . dT  dT  , . . 

— = 7tw,  — --  mr  v,  — ; = mF  sm^dd) ; 

dr  ’ de  d4 

ilT  . dT  dT 

- - rz  7nr{6^  + siiF  = mr^sin  $ cos  ^ = 9. 


Hence  the  equations  of  motion  become 


m + sin^ 

{dt  ) 


= Gr, 


7)1- 


d(F  {iud6(j>) 


dt 


Hr  sin  0 


or,  according  to  the  ordinary  notation  of  the  differential  calculus. 


{d^r  (dF 


\ I 


319.] 


DYNAMICAL  LAWS  AND  PRINCIPLES. 


309 


= Or, 


cl 


siii^  Q 


Hr  sin  0. 


""  dt  dtj 

If  the  motion  is  confined  to  one  plane,  that  of  r,  0,  we  have 


Examples  of 
the  use  of 
Lagrange’s 
generalized 
equations  of 
motion 
polar  co- 
ordinates. 


= 0,  and  therefore  11=0,  and  the  two  equations  of  motion 


df 

~di 

which  remain  are 
(dS' 

m -n-r 

\df 


d^ 

df 


F,  m 


d_  ( de\ 

dt\  dt) 


= Gr. 


§ 32,  in  which  it  was  shown  that  and 


These  equations  might  have  been  wiitten  down  at  once  in  terms 
of  the  second  law  of  motion  from  the  kinematical  investigation  of 

df~'  df^  rdt"^^ 
are  the  components  of  acceleration  along  and  perpendicular  to 
the  radius-vector,  when  the  motion  of  a point  in  a plane  is  ex- 
pressed according  to  polar  co-ordinates,  r,  0. 

The  same  equations,  with  instead  of  0,  are  obtained  from  the 
polar  equations  in  three  dimensions  by  putting  ^=l7r,  which 
implies  that  G = 0,  and  confines  the  motion  to  the  plane  (r,  </>). 

Example  (B). — Two  particles  are  connected  by  a string ; one 
of  them,  m,  moves  in  any  way  on  a smooth  horizontal  plane,  and 
the  string,  passing  through  a smooth  infinitely  small  aperture  in 
this  plane,  bears  the  other  particle  m',  hanging  vertically  down- 
wards, and  only  moving  in  this  vertical  line  : (the  string  re- 
maining always  stretched  in  any  practical  illustration,  but,  in 
the  problem,  being  of  course  supposed  capable  of  transmitting 
negative  tension  with  its  two  parts  straight.)  Let  I be  the  whole 
length  of  the  string,  r that  of  the  part  of  it  from  m to  the  aperture 
in  the  plane,  and  let  B be  the  angle  between  the  direction  of  r 
and  a fixed  line  in  the  plane.  We  have 

dT  , dT  . 

dr  ^ dO 


Dynamical 

problem. 


dT 

— =mrb~, 
dr 


dH 

db 


= 0. 


Also,  there  being  no  other  external  force  than  gni,  the  weight 
of  the  second  paHicle, 

II  = - (Jin',  @ — 0. 


810 


PRELIMINARY. 


[319. 


Examples  of 
the  use  of 
Lagrange’s 
generalized 
equations  of 
motion ; 
dynamical 
problem. 


Case  of 
stable  equi- 
librium due 
to  motion. 


Examples 
continued ; 
C (a),  fold- 
ing door. 


Hence  the  equations  of  motion  are 
(m  + 7)1  )r  - mrO^  = — m'g, 


d{r^6) 

dt 


The  motion  of  rn!  is  of  course  that  of  a particle  influenced  cnlj 
by  a force  towards  a fixed  centre;  but  the  law  of  this  force,  F 
(the  tension  of  the  string),  is  remarkable.  To  find  it  we  have 
(§  32),  P = m{-r  But,  by  the  equations  of  the  motion, 

r -r^  ^ — — , {a  + r&),  and  6 = —o , 

m + m mr- 


where  h (according  to  the  usual  notation)  denotes  the  moment 
of  momentum  of  the  motion,  being  an  arbitrary  constant  of  in- 
tegi’ation.  Hence 


pL 


m!)t 
m + 7)1 


The  particular  case  of  projection  which  gives  m a circular  motion 
and  leaves  in'  at  rest  is  interesting,  inasmuch  as  (§  350,  below) 
the  motion  of  7)i  is  stable,  and  therefore  m is  in  stable  equi- 
librium. 


Example  (C). — A rigid  body  m is  supported  on  a fixed  axis, 
and  another  rigid  body  n is  supported  on  the  first,  by  another 
axis ; the  motion  round  each  axis  being  perfectly  free. 

Case  {a). — The  second  axis  parallel  to  the  first.  At  any  time, 
t,  let  ^ and  ij/  be  the  inclinations  of  a fixed  plane  through  tlie 
first  axis  to  the  plane  of  it  and  the  second  axis,  and  to  a 
plane  through  the  second  axis  and  the  centre  of  inertia  of  the 
second  body.  These  two  co-ordinates,  i/f,  it  is  clear,  completely 
specify  the  configuration  of  the  system.  Now  let  a be  the  dis- 
tance of  the  second  axis  from  the  first,  and  b that  of  the  centre 
of  inertia  of  the  second  body  from  the  second  axis.  The  velocity 
of  the  second  axis  will  be  atji ; and  the  velocity  of  the  centre 
of  inertia  of  the  second  body  will  be  the  resultant  of  two  velocities 

a(j>,  and  bif/, 

in  lines  inclined  to  one  another  at  an  angle  equal  to  i//  — and 
its  square  will  therefore  be  equal  to 

+ ^abcjnj/  cos  (ij/  - <f>) 

Hence,  if  m and  n denote  the  masses,  j the  radius  of  gyration 
of  the  first  bo'ly  about  the  fixed  axis,  and  k that  of  the  second 


DYNAMICAL  LAWS  AND  PRINCIPLES. 


311 


.] 


body  about  a parallel  axis  through  its  centre  of  inertia  ; we  have.  Examples 

continued 

according  to  §§  280,  281,  c (a , fold 

ing  door. 

T = \ {mf^^  + n + 2a60i^  cos  (ij/ -<!>)  + + F 

Hence  we  have. 


T 


dT 


9 


- - + nab  cos  {xjj  - cf>)  if/ ; —=  nab  cos  {\f/—<fi)^  + nib'^+k^)\j/‘, 


dT 

dcfi 


clT 

dif/ 


dif/ 

= 7iab  sin  (\f/  — (ft)  ^\j/. 


The  most  general  supposition  we  can  make  as  to  the  applied  forces, 
is  equivalent  to  assuming  a couple,  to  act  on  the  first  body,  and 
a couple,  'k,  on  the  second,  each  in  a plane  perpendicular  to  the 
axes ; and  these  are  obviously  what  the  generalized  components  of 
stress  become  in  this  particular  co-ordinate  system,  if/.  Hence 
the  equations  of  motion  are 


-1-  7ia^)  <f)  -f-  nab 


d [i^  cos  - ^)] 


dt 


nab  sin  (xf/  — cf>)  <f)if/  - 


7^5  + n (b^  h k')  if  -h  nab  sin  [if/  — <ff)  ^if  = 

clt 

If  there  is  no  other  applied  force  than  gravity,  and  if,  as  we  may 
suppose  without  losing  generality,  the  two  axes  are  horizontal,  tho 
potential  energy  of  the  system  will  be 

gmh  (1  - cos  (f>)  + gn  [a  [1  — cos  (<^  + yl)]  -i-  6 [1  — cos  (i/^  -i-  ^)]}, 

the  distance  of  the  centre  of  inertia  of  the  first  body  from  the 
fixed  axis  being  denoted  by  h,  the  inclination  of  the  plane 
through  the  fixed  axis  and  the  centre  of  inertia  of  the  first  body, 
to  the  plane  of  the  two  axes,  being  denoted  by  A,  and  the  fixed 
plane  being  so  taken  that  ^ = 0 when  the  former  plane  is  vertical. 
By  difierentiating  this,  with  reference  to  <f>  and  if/,  we  therefore 
liave 

— ^ = gmh  sin  <f}  + gna  sin  {(f>  + A),  — ^ = gnb  sin  (if/  + A). 

We  shall  examine  this  case  in  some  detail  later,  in  connexion 
with  the  interference  of  vibrations,  a subject  of  much  importance 
in  physical  science. 

When  there  are  no  applied  or  intrinsic  working  forces,  wo 
have  = 0 and  'I'  0 : or,  if  there  are  mutual  forces  between  the 

two  b<Klies,  but  no  forces  applied  from  without,  'I'  = 0.  In 


312 


PRELIMINARY. 


[319. 


Examples 
the  use  of 
Lap^nge’s 
generalized 
equations  of 
motion. 


C(b). 

Motion  of 

governing 

masses  in 

Watt’s 

centrifugal 

governor: 

also  of 

gimballed 

compass- 

bowl. 


either  of  these  cases  we  have  the  following  first  integral : — 

{mf  4-  na^)  (j)  -f-  m'ah  cos  {{j/ - <}>)  {<ji  + i[/)  + n {b^  + k^)  ij/=G ; 

obtained  by  adding  the  two  equations  of  motion  and  integrating. 
This,  which  clearly  expresses  the  constancy  of  the  whole  moment  of 
momentum,  gives  ^ and  xj/  in  terms  of  {ij/  — 0)  and  (if/  — (f>).  Using 
these  in  the  integral  equation  of  energy,  provided  the  mutual  forces 
are  functions  of  \}/ - <f>,  we  have  a single  equation  between 

— ^ , {xj/  — <f>),  and  constants,  and  thus  the  full  solution  of 

(it 

the  problem  is  reduced  to  quadratures.  [It  is  worked  out  fully 
below,  as  Sub-example  G^.] 

Case  (b). — The  second  axis  'perpendicular  to  the  first  For 
simplicity  suppose  the  pivoted  axis  of  the  second  body,  n,  to  be 
a principal  axis  relatively  [§  282  Def.  (2)]  to  the  point,  N,  in 
which  it  is  cut  by  a plane  perpendicular  to  it  through  the  fixed 
axis  of  the  first  body,  m.  Let  NE  and  NF  be  ?^’s  two  other 
])rincipal  axes.  Denote  now  by 

h the  distance  from  N to  m’s  fixed  axis ; 

k,  e,f  the  radii  of  gyration  of  n round  its  three  principal 
axes  through  N ; 

j the  radius  of  gyration  of  m round  its  fixed  axis ; 

6 the  inclination  of  NE  to  m’s  fixed  axis  ; 

\}/  the  inclination  of  the  plane  parallel  to  n^s  pivoted  axis 
through  m’s  fixed  axis,  to  a fixed  plane  through  the 
latter. 

Kemarking  that  the  component  angular  velocities  of  n round 
NE  and  NE  are  xj/  cos  0 and  xj/  sin  0,  we  find  immediately 

^ - h + n{h^  + ^ cos^  B + sin^  ^)]  xf/^  + nk^  6\ 
or,  if  we  put 

mfi  + n ifil  +/")  = n (e"  -/")  = D ; 

T^WiG^D  cos^  Q)  x};^  -t-  nk^  r}. 

The  farther  working  out  of  this  case  we  leave  as  a simple  but 
most  interesting  exercise  for  the  student.  We  may  return 
to  it  later,  as  its  application  to  the  theory  of  centrifugal  chrono- 
raetric  regulators  is  very  important. 


519.] 


DYNAMICAL  LAWS  AND  PRINCIPLES. 


515 


Example  (O').  Take  the  case  C (d)  and  mount  a third  body  1/  Motion  of 
upon  an  axis  00  fixed  relatively  to  n in  any  position  parallel  to  pivoted'^m!' 
JVE.  Suppose  for  simplicity  0 to  be  the  centre  of  inertia  of  M principal 
and  00  one  of  its  principal  axes  ; and  let  OA^  OB  be  its  two 
other  principal  axes  relative  to  0.  The  notation  being  in  other 
respects  the  same  as  in  Example  0 (6),  denote  now  farther  by 
B,  0 the  moments  of  inertia  of  A1  round  OA,  OB^  00  ] </>  the 
angle  between  the  plane  AOO  and  the  plane  through  the  fixed 
axis  of  m perpendicular  to  the  pivoted  axis  oi  n ] 'uy,  p,  a the 
component  angular  velocities  of  M round  OA,  OB,  00. 

In  the  annexed  diagram,  taken  from  § 101  above,  ZOZ'  is  a 


Letter  0 at  cen- 
tre of  sphere 
concealed  by 


\ 


circle  of  unit  radius  having  its  centre  at  0 and  its  plane  parallel 
to  the  fixed  axis  of  m and  perpendicular  to  the  pivoted  axis 
of  n. 

The  component  velocities  of  0 in  the  direction  of  the  arc  ZO 
and  perpendicular  to  it  are  6 and  ij/  sin  6 ; and  the  component 
angular  velocity  of  the  plane  ZOZ'  round  00  is  ij/  cos  0.  Hence 


and 

( C.^omparc  § 101 .] 


SI  4 PRELIMINAKY.  [319. 


Motion  of 
a rigid  body 
pivoted  on 
one  of  its 
principal 
axes  mount- 
ed on  a 
gimballed 
bowl. 

The  kinetic  energy  of  the  motion  of  M relatively  to  0,  its 
centre  of  inertia,  is  (§  281) 

i(Az^^  + Bp^  + Ca^); 

and  (§  280)  its  whole  kinetic  energy  is  obtained  by  adding  the 
kinetic  energy  of  a material  point  equal  to  its  mass  moving  with 
the  velocity  of  its  centre  of  inertia.  This  latter  part  of  the 
kinetic  energy  of  M is  most  simply  taken  into  account  by  sup- 
posing n to  include  a material  point  equal  to  M placed  at  0 ; 
and  using  the  previous  notation  k,  e,  f for  radii  of  gyration  of  n 
on  the  understanding  that  n now  includes  this  addition.  Hence 
for  the  present  example,  with  the  preceding  notation  6^,  Z),  we 
have 

cos^  0)  + nk^e^} 

+ A {6  sincfi-ij/  sin  0 cos  (f>y  + B {0  cos  ^ sin  6 sin 

+ C {xl/ cos  0 + 0)^}. 

Rigid  body 
rotating 
freely;  re- 
ferred to 
the  0 

co-ordinates 
(§  101). 

From  this  the  three  equations  of  motion  are  easily  written  down. 

By  putting  6^  = 0,  D = 0,  and  ^ = 0,  we  have  the  case  of  the 
motion  of  a free  rigid  body  relatively  to  its  centre  of  inertia. 

By  putting  B — A we  fall  on  a case  which  includes  gyroscopes 

Gyroscopes 

and 

gyrostats. 

and  gyrostats  of  every  variety;  and  have  the  following  much 
simplified  formula : 

T=\^G-^A^{B-A)  cos"^]  xj;^  + {nk^  + A)  0^  + C(xj;  cos  6 + ^)"1, 

or 

r = i {(£■  f Fcos^  d)  + {nF‘  + A)6‘  + C (f  cos  6 + 4,)%  > 

if  we  put  B = G + Ay  and  F=D  — A. 

Gyroscopic 

pendulum. 

Example  (D). — Gyroscopic  pendulum. — A ligid  body,  P,  is  ^ 
attached  to  one  axis  of  a universal  flexure  joint  (§  109),  of  which 
the  other  is  held  fixed,  and  a second  body,  Q,  is  supported  on  P by 
a fixed  axis,  in  line  with,  or  parallel  to,  the  first-mentioned  arm  of  i 

the  joint,  l^or  simplicity,  we  shall  suppose  Q to  be  kinetically 
symmetrical  about  its  bearing  axis,  and  OB  to  be  a principal 
axis  of  an  ideal  rigid  body,  PQy  composed  of  P and  a mass  so 
distributed  along  the  bearing  axis  of  the  actual  body  Q as  to 
have  the  same  centre  of  inertia  and  the  same  moments  of  inertia 
round  axes  perpendicular  to  it.  Let  ^0  be  the  fixed  arm,  0 the 
joint,  OB  the  movable  arm  bearing  the  body  P,  and  coinciding 
with,  or  parallel  to,  the  axis  of  Q.  Let  BOA'  = 6 \ let  ^ be  the 

1 


DYNAMICAL  LAWS  AND  PRINCIPLES. 


315 


319.] 


0 


aiio-le  which  the  plane  At  0.S  makes  with  a fixed  plane  of  reference,  Gyroscopic 

° ^ . pendulum. 

through  OA,  chosen  so  as  to  contain  a second 

O ^ >1 

principal  axis  of  the  imagined  rigid  body,  PQ, 
when  OP  is  placed  in  line  with  A 0 ; and  let 
k}/  be  the  angle  between  a plane  of  reference  in 
Q through  its  axis  of  symmetry  and  the  plane 
of  the  two  principal  axes  of  FQ  already  men- 
tioned. These  three  co-ordinates  (^,  \f/) 

clearly  specify  the  configuration  of  the  system  at 
any  time,  t.  Let  the  moments  of  inertia  of  the 
imagined  rigid  body  FQ^  round  its  principal 
axis  OB,  the  other  principal  axis  referred  to  above,  and  the 
remaining  one,  be  denoted  by  © respectively ; and  let 

be  the  moment  of  inertia  of  Q round  its  bearing  axis. 


We  have  seen  (§  109)  that,  with  the  kind  of  joint  we  have  sup- 
posed at  0,  every  possible  motion  of  a body  rigidly  connected  with 
OB,  is  resolvable  into  a rotation  round  01,  the  line  bisecting  the 
angle  AOB,  and  a rotation  round  the  line  through  0 perpen- 
dicular to  the  plane  AOB.  The  angular  velocity  of  the  latter 
is  Q,  according  to  our  present  notation.  The  former  would  give 
to  any  point  in  OB  the  same  absolute  velocity  by  rotation  round 
01,  that  it  has  by  rotation  with  angular  velocity  fj>  round  AA' ; 
and  is  therefore  equal  to 


sin  A’OB  . 

sh^TOB 


sin  6 
cos  ^0 


= 20  sin 


This  may  be  resolved  into  20  sin^  = 0 (1  — cos  round  OB, 
and  20  sin  cos  = 0 sin  0 round  the  perpendicular  to  OB,  in 
plane  AOB.  Again,  in  virtue  of  the  symmetrical  character  of 
the  joint  with  reference  to  the  line  01,  the  angle  0,  as  defined 
above,  will  be  equal  to  the  angle  between  the  plane  of  the  two 
fii-st-mentioned  principal  axes  of  body  F,  and  the  plane  AOB. 
Hence  the  axis  of  the  angular  velocity  0 sin  0,  is  inclined  to  the 
principal  axis  of  moment  at  an  angle  equal  to  0.  Resolving 
therefore  this  angular  velocity,  and  0,  into  components  round  the 
axes  of  H and  (C,  we  find,  for  tlie  whole  component  angular 
velocities  of  the  imagined  rigid  body  FQ,  round  these  axes, 
0 sin  0 cos  <l>-h  6 sin  0,  and  - 0 sin  6 shi  (f>  + 0 cos  0,  respectively. 
The  whole  kinetic  energy,  2\  is  composed  of  that  of  the  imagined 
rigid  body  FQ,  and  tliat  of  Q about  axes  through  its  centre  of 


Gyroscopic 

pendulum. 


Example  of 

varying 

relation 

without 

constraint 

(rotating 

axes). 


SIG 


PRELIMINARY. 


[319. 


inertia  : we  therefore  have 

2T  = ^{\- cos  Of  <j)^ (<j) sin 6 cos (fi  + 6 sin -J- CT ((j^ sin 0 sin ^ - 6 cos (f>y 

+ - cos  0)Y. 

fIT  . . dT 

Hence  — = ^'{\j/-^(l-cosO)},  — =0, 
d\j/ 

dT  . . 

— : = ^ (1  - COS  Oy  ^ + 1^  (0  sin  $ cos  ^ 0 sin  9)  sin  0 cos 

d(j> 

+ (it  sin.  Osin  cf>-0  cos  (ji)  sin0sin</>-^'{i/'  — 0(1  -cos^)}(l  -cos^), 


(tl  • • • • 

— 33  (<^  sin  0 cos  (f)  + 0 sin  0)  (0  sin  0 sin  0 - ^ cos  0) 
dcf> 

+ ©(0sin0sin0-^cos0)(0sin^cos0  + 0sin0), 

dT  . • . . 

— = 33  (0  sin 0 COS0  + ^ sin  0)  sin 0 - © (0  sin  ^ sin  <f>-0  cos  0)  cos  9 
dO 

dT 

and  ^ = ^ ( 1 - cos  0)  sin  d0^  + 13  cos  0 cos  00  (0  sin  d cos  0 + ^ sin  0) 
du 

4 © cos^  sin 00(0  sind  sin0-0  cos0)-^'  sin00{0-(l  — cos^)0}. 


Now  let  a couple,  6^,  act  on  tlie  body  Q,  in  a plane  perpendi- 
cular to  its  axis,  and  let  X,  M,  N act  on  P,  in  the  plane  perpen- 
dicular to  OB,  in  the  plane  A' OB,  and  in  the  plane  through  OB 
perpendicular  to  the  diagram.  If  0 is  kept  constant,  and  0 
varied,  the  couple  G will  do  or  resist  work  in  simple  addition 
with  L.  Hence,  resolving  L G and  N into  components  round 
01,  and  perpendicular  to  it,  rejecting  tJie  latter,  and  remembering 
that  2 sin  1^0  is  the  angular  velocity  round  OT,  we  have 

$=2sinid{-(Z:  + X)sinl(9  + 7V^cos-i-d}  = [-(X  + 6^)(l-cos^)-i-7\^sin^}. 

Also,  obviously 

Using  these  several  expressions  in  Lagrange’s  general  equations 
(24),  we  have  the  equations  of  motion  of  the  system.  They  will 
be  of  great  use  to  us  later,  when  we  shall  consider  several  parti- 
cular cases  of  remarkable  interest  and  of  very  gi’eat  importance. 


Example  (E). — Motion  of  a free  particle  referred  to  rotating  axes. 
Let  X,  y,  z be  the  co-ordinates  of  a moving  particle  referred  to 
axes  rotating  with  a constant  or  varying  angular  velocity  round 
the  axis  OZ.  Let  x^,  y^,  z,  be  its  co-ordinates  referred  to  the 
same  axis,  OZ,  and  two  axes  OX,,  OY^,  fixed  in  the  plane  per- 


319.]  DYNAMICAL  LAWS  AND  PRINCIPLES.  317 

pendicular  to  it.  "We  have 

■=  X cos  a — y sin  a,  y^  = x sin  a.  + y cos  a ; 

£Cj  = X COS  a — y sin  a - (x  sin  a + y cos  a)  d,  y,  = 

where  a,  the  angle  XfiX,  must  be  considered  as  a given  func- 
tion of  t.  Hence 

T = \m  [x^  ->ry^  +2  {xy  — yx)  a + {x^  1/)  d^}, 

dT  , dT  ,,  dT 

dx  ^ ~ Iz  ^ 

^=m(2/a  + xa-),  ~ =ni(~  xo.^  ya),  ^ = 0, 

Also, 

-^=m(x-ya-ya),  - ^ = m (j/ + *a + *a), 

and  hence  the  equations  of  motion  are 

m (x  - 2ya  — xql  - ya)  = X,  m (y  + 2xd  — yiT  -f-  xd)  = F,  mz  — Z, 

X,  Y,  Z denoting  simply  the  components  of  the  force  on  the 
particle,  parallel  to  the  moving  axes  at  any  instant.  In  this 
example  t enters  into  the  relation  between  fixed  rectangular  axes 
and  the  co-ordinate  system  to  which  the  motion  is  referred ; but 
there  is  no  constraint.  The  next  is  given  as  an  example  of  vary- 
ing, or  kinetic,  constraint. 

Example  (F). — A particle,  influenced  hy  any  forces,  and  at- 
tached to  one  end  of  a string  of  which  the  other  is  moved  with  any 
constant  or  varying  velocity  in  a straight  line.  Let  6 be  the 
inclination  of  the  string  at  time  t,  to  the  given  straight  line,  and 
^ the  angle  between  two  planes  through  this  line,  one  containing 
the  string  at  any-  instant,  and  the  other  fixed.  These  two  co- 
ordinates (6,  <f))  specify  the  position,  P,  of  the  particle  at  any 
instant,  the  length  of  the  string  being  a given  constant,  a,  and 
the  distance  OE,  of  its  other  end  E,  from  a fixed  point,  0,  of  the 
line  in  which  it  is  moved,  being  a given  function  of  t,  which  we 
shall  denote  by  u.  Let  x,  y,  z be  the  co-ordinates  of  the  particle 
referred  to  three  fixed  rectangular  axes.  Choosing  OX  as  the  given 
straight  line,  and  YOX  the  fixed  plane  from  which  ^ is  measured, 
we  have 

X — u a cos  6,  y = a sin  6 cos  (f>,  z = a sin  6 sin  (f), 
x = u - a sin  0$  ; 


Examf)Ie  cf 

varying 

relation 

without 

constraint 

(rotating 

axes). 


Example  ol 

varying 

relation 

due  to 

kinetic 

constraint. 


318 


PRBLIMINA.Rr. 


[319. 


Example  of 

varying 

relatiou 

due  to 

kinetic 

constraint. 


and  for  z we  have  the  same  expressions  as  in  Example  (A). 
Hence 

{u^  — 2uda  sin  6) 

where  ® denotes  the  same  as  the  T of  Example  (A),  with 
r 0,  and  r = a.  Hence,  denoting  as  there,  by  G and  H the  two 
components  of  the  force  on  the  particle,  perpendicular  to  EP^ 
respectively  in  the  plane  of  0 and  perpendicular  to  it,  we  find,  for 
the  two  required  equations  of  motion, 

m [a  {6  - sin  6 cos  0j>^)  - sin  0 u]=  G,  and  ma  — = H. 

These  show  that  the  motion  is  the  same  as  if  E were  fixed,  and 
a force  equal  to  — mil  were  applied  to  the  particle  in  a direction 
parallel  to  EX ; a result  that  might  have  been  arrived  at  at  once 
by  superimposing  on  the  whole  system  an  acceleration  equal  and 
opposite  to  that  of  E,  to  effect  which  on  P the  force  — mil  is 
required. 


Example  (F').  Any  case  of  varying  relations  such  that  in 
318  (27)  the  coefficients  {^,^),  (i/^,  ^)...  are  independent  of  t. 
Let  denote  the  quadratic  part,  L the  linear  part,  and  K [as 
in  § 318  (27)]  the  constant  part  of  T in  respect  to  the  velocity 
components,  so  that 

® = i {(‘A?  + + • • •}  I 

L = i («), 

A"=  (lA,  <A,  6*,  ...)  ) 

where  (lA? ’A)?  ("A?  (‘A?  •••  denote  functions  of  the  co-ordi- 

nates without  and  (xj/),  (^),  ...,  (if/,  cf>,  0,  ...)  functions  of  the 
co-ordinates  and,  may  be  also,  of  t ; and 

T = 'E  + L + K (6). 

dK 

We  have  — : = 0. 

dij/ 


Hence  the  contribution  from  K to  the  first  member  of  the  j/r- 
equation  of  motion  is  simply  - . Again  we  have 


d dL  d (xj/)  . d (xj/)  . 

; = -Xy  ^ + -Pu  ^ + etc.  -1- 

dt  dxj/  dxj/  d(f> 


hence 


DYNAMICAL  LAWS  AND  PRINCIPLES. 


819 


819.' 


^ dij/ 


Farther  we  have 

clif/ 

Hence  the  whole  contribution 
motion  is 

^ <4 

Lastly,  the  contribution  from  ^ is  the  same  as  the  whole  from 
T in  § 318  (29"')  ; so  that  we  have 
d d'E  dE  , , 

--  ... 

dt  dip  dip 


(d  iP)  d {<f>)\  . /t 


from  L to  the  i/^-equation  of 

d{f) 


Example  of 

varying 

relation 

due  to 

kinetic 

constraint. 


‘{ 


d iP,  P) 

dp 


P" 


d ip,  p)  . . 


2 d (p,  P)  _ d {p,  p) 
d(p  dp 


...j 


(A 


and  the  completed  j/r-equation  of  motion  is 
d'  dE  dE  (d  (p)  d {p)'\  ; /d  (p) 
dp  / ^ 


dE  ft 


dt  dp  dp  \ dp 


) _ ,^0\ 

c/f  ! 

d {p)\  dK 


0 + 


m 


(e). 


(f)-*'5r’--****** (/>. 


dt  J dp 

It  is  important  to  remark  that  the  coefficient  of  p in  this  p- 
equation  is  equal  but  of  opposite  sign  to  the  coefficient  of  p in 
the  (^-equation.  [Compare  Example  G (19)  below.] 

Proceeding  as  in  § 318  (29*'")  (29''),  we  have  in  respect  to  E Equatii 
precisely  the  same  formulas  as  there  in  respect  to  T.  The  terms 
involving  first  powers  of  the  velocities  simply,  balance  in  the 
sum  : and  we  find  finally 

dE 
dt 

where  denotes  diflferentiation  on  the  supposition  of 

pjp,  ...  variable  ; and  t constant,  where  it  appears  explicitly. 

Now  with  this  notation  we  have 
dL  ^ fdL\ 
dt  \dt)^  dt 

, dK  fdK\  d^j,  A \K 

and  — — = 

dt 

Hence  from  {/)  we  have 
d^  _ d{E  + L+  K) 
dt 


+ {p)p+ {P)p+ 


(f) 


dt 


= 'i'p  + ^p  + ...  + 


dt 


2 


dt 

.)K  [dK 


dt 


■(?)• 


820 


PRELIMINABY. 


[819. 


Exercise  for 
student. 


Ij^noration 
of  co- 
ordinates. 


Take,  for  illustration,  Examples  (E)  and  (F)  from  above;  in 
which  we  have 

[Example  (E)]  \ m + if  + 

L — mix  (xy  — yx), 

K = \mix^  {x^  +y^), 

and  [Example  (F)]  = ^ (sin®0<^®  -f 

L = — mua  sin  09^ 

K—^  mv?. 


Write  out  explicitly  in  each  case  equations  (/)  and  (^),  and 
verify  them  by  direct  work  from  the  equations  of  motion  forming 
the  conclusions  of  the  examples  as  treated  above  (remembering 
that  d and  u are  to  be  regarded  as  given  explicit  functions  of  i). 


Exam, pie  (G). — Preliminary  to  Gyrostatic  connexions  and  to 
Fluid  Motion.  Let  there  be  one  or  more  co-ordinates  x?  Xt  etc. 
which  do  not  appear  in  the  coefficients  of  velocities  in  the 


dT  dT 

expression  for  T ; that  is  to  say  let  — = 0,  = 0,  etc.  The 

equations  corresponding  to  these  co-ordinates  become 


dt  rfx  dt  dx  ^ 


(I). 


Farther  let  us  suppose  that  the  force-components  X,  X',  etc. 
corresponding  to  the  co-ordinates  x>  zero:  we 

shall  have 

dT  dT 

dx  ' dx'  ' ^ 

or,  expanded  according  to  previous  notation  [318  (29)], 

x)  ’A  + x)  ^ + • • • + (x»  x)  X + (x»  x')  x'  + • • • = ^ 1 

(•Aj X ) ‘A  + ("A, X ) + •••  + (x^x)x  + (x'»x)x + — •••(3). 


Hence,  if  we  put 

(’A.  x)«A  + (<A»  x)<A  + --- 
(»A> x)  ‘A^  x)  0 + •• 


(4), 


319.]  DYNAMICAL  LAWS  AND  PRINCIPLES.  321 

we  have 

(x»x)x  + (x>x')x  + •••  =C-P  ^ 

(X^X)X  + (X^X)X  + •••  =0'-F'  ....(5). 

J 

Resolving  these  for  x,  ...  we  find 


(6)» 


(XyX)y  (XyX)y  ••• 

(O-P)^ 

(x",x').  (x".x"),- 

(C'-P')  + . 

(x"»x')»(x^x")v. 

(x"',x'),(x",x"),- 

{X’X)Ax^X)y  (XyX)’  •• 
(XyX)y  (X'»X)»  (x'jX")»  ••• 
{XyX)y  (x'yX)y  {x"yX')y  ••• 


and  symmetrical  expressions  for  ^ ^ or,  as  we  may  write 
them  short, 

X={C,C)(C-P)  + (C,  O')  (O'  - P’)  + . . . ) 

x'  = (O',  0)  {C-P)+  (O',  0')(0'-P')  + ...\ (7), 


where  {C,  C),  (6',  C'),  {G\C'),  ...  denote  functions  of  the  retained 

co-ordinates  if/,  <f>,  0, It  is  to  be  remembered  that,  because 

(x.  x)  = (x.  x).  (x.  x")  = (x">  x)>  "'e  see  from  (6)  that 
(0,  O')  = (O',  C),  (0,  C")  = (C",  0),  (O',  (7")  = (C",  O'),  aad  so  on. ..(8). 

The  following  formulas  for  X)X  » •••»  condensed  in  respect  to 
C,  C',  C"  by  aid  of  the  notation  (14)  below,  and  expanded  in 
respect  to  xf/,  <j),  ...,  by  (4),  will  also  be  useful. 


wliere 


dK  , , _ . . 

x'=§-W  + ^'<^+-) 


Af  = (C,0).(^,x)+(0,0').(^,x:)  + 
X =(0,0).  (^,x)  + (0,0').  (4>,x)  + 


(9). 


/f'  = (0',0).(4,,x)  + {0',0').(^,x’)  + 


(10). 


'I’he  elimination  of  Xt  X i • • I>*cm  T by  tliese  expressions  for 
VOL  I.  21 


Iff  no  ration 
of  co- 
ordinates. 


Ignoration 
of  co- 
ordinates. 


322  PRELIMINARY.  [319. 

them  is  facilitated  by  remarking  that,  as  it  is  a quadratic  func- 
tion of  ...  X)  X>  •••}  have 

^ .(jdT.  (IT  ,dT  „dT  ) 

Hence  by  (3), 

so  that  we  have  now  only  first  powers  of  x,  x^  •••  to  eliminate. 
Gleaning  out  %x^  ...  from  the  first  group  of  terms,  and  denoting 
by  the  part  of  T not  containing  x,  x ? •••» 

^0  + J x)  f + i^’X)  f + •••  + <^]  X 
x)  ‘A  + x')  + • • • x' 



or,  according  to  the  notation  of  (4), 

T=T,  + ^{(0+F)x  + (C'+F')x' 

Eliminating  now  x,  x,  •••  ty  (7)  we  find 
y=  T,  + i {((7,  C)  (C^-  F’)  + 2 {C,  C)  (CC  - FP)  + (C",  C")  ((?'“  - P‘) 

+ •••} (11)- 

It  is  remarkable  that  only  second  powers,  and  products,  ')iot 
first  powers^  of  the  velocity-components  0,  ...  appear  in  this 
expression.  We  may  write  it  thus  : — 

T=Z  + K (12), 

where  ^ denotes  a quadratic  function  of  i^,  0,  . . . , as  follows : — 

^ - 4 0)P^  + 2 {G,  C')  FT'  + {O',  C')  -f- ...}.. ..(13), 

and  K a quantity  independent  of  xj/,  <j>j  ...,  as  follows: — 

j^=  4 {{0,  C)C^  + 2 (G,  C)  CC  + {G\  C)  G'^+...} (14). 

Next,  to  eliminate  x,  X>  •••  from  the  Lagrange’s  equations,  we 
have,  in  virtue  of  (12)  and  of  the  constitutions  of  1\  and 

dT  dTdx  dTdj(  , dZ 

— r 4- ^ -f ^ 4- etc.  = — (15), 

dij/  dx  dij/  dx  dxj/  d\j/ 

where  ^ ^ , etc.  are  to  be  found  by  (7)  or  (9),  and  therefore 
dxj/  dxf/ 

are  simply  the  coefficients  of  in  (9) ; so  that  we  have 



dif/  dxf/ 

where  M^M'  are  functions  of  ...  explicitly  expressed  by 
(10).  Using  (16)  in  (15)  we  find 


319.] 


DYNAMICAL  LAWS  AND  PRINCIPLES. 


823 


^ ^ + G'M^  + etc (17). 


dxp  dij/ 

Again  remarking  that  'SP  + A contains  if/j  both  as  it  appeared 
originally  in  Ty  and  as  farther  introduced  in  the  expressions  (7) 
for  X,  X,  ...,  we  see  that 

dij/  ^ dif/  dx  d\}/  dx  dij/ 

“di/.  # dij/ 


Ignoration 
of  co- 
ordinates. 


And  by  (9)  we  have 

dx  / • dM  . dA 

which,  used  in  the  preceding,  gives 
dT  . dd/ 


^dX 

diJ^dO'^ 


dif/ 


.dM' 

'1'^ 


.dN' 


...) 


etc.  4-2 

dif/ 


Hence 


dij/ 


(M 

d\\f 


dK 


dxj/ 


+ :^c 


dM  . dN 


■■■) (18), 

where  5 denotes  summation  with  regard  to  the  constants  C, 
C'y  etc. 

Using  this  and  (1 7)  in  the  Lagrange’s  i//’-equation,  we  find  finally 
for  the  i/r-equation  of  motion  in  terms  of  the  non-ignored  co- 
ordinates alone,  and  conclude  the  symmetrical  equations  for 
etc.,  as  follows. 


= <^ 


d 

/d®\ 

fdM 

dN\ 

fdM 

dO^ 

dt 

\^/ 

d-j, 

\d<j> 

dif/J 

1 9 + 

\W 

di/^y 

(6^4-...^ 

f # 

d 

fdZ\ 

dZ 

SP) 

/dN 

dM\ 

fdN 

d0\ 

\ dK 

dt 

\d^) 

d<)>^ 

\dx{/ 

d<f>) 

1 lA  + 1 

\Jb' 

dff)/ 

)6^+..v 

I d,!, 

d 

/d^\ 

dZ 

/dO 

dM\ 

'i'^i 

/dO 

dN\ 

dK 

(It 

Kdo) 

\d^ 

~de) 

\d<f> 

de) 

' * dO' 

(19). 


[Compare  Example  F'  (e)  above.  It  is  important  to  remark 
that  in  each  equation  of  motion  the  first  power  of  the  related 
velocity-component  disappears  ; and  the  coeflicient  of  each  of  the 
other  velocity-components  in  this  equation  is  equal  but  of  02:»posite 
sign  to  the  coefficient  of  the  velocity-component  corresponding  to 
this  equation,  in  the  equation  corresponding  to  that  other  velocity- 
(‘omponent,] 

21—2 


324 


PRELIMINARY. 


[819. 


Equation  of 
Ener^. 


The  equation  of  energy,  found  as  above  [§  318  (29^^)  and 
(29^')],  is 

-A_ L = + etc (20). 

The  interpretation,  considering  (12),  is  obvious.  The  contrast 
Avith  Example  F'  {g)  is  most  instructive. 


Sub-Example  (GJ. — Take,  from  above.  Example  C,  case  (a) : 
and  put  (f)  = if/-hd;  also,  for  brevity,  mf+na^=B,n{b^-hk^)  = A, 
and  nab  = c.  We  have* 

T=^  {Axj/^  + 2ci^  (j^  + 0)  cos  0 + ^ (i^  + Oy] ; 

and  from  this  find 

dT  dT  ... 

— = 0,  -— = Aiff  + c {2\l/ + 0)  co^  6 + B l\j/ + 6) ; 
d\{/  d^ 

dT  . . • dT 

= - Clj/  (ij/  + 0)  sin  $,  . = Cx/r  cos  0 -i-  B (ij;  + 0). 

dd  do 

Here  the  co-ordinate  0 alone,  and  not  the  co-ordinate  if/,  appears 
in  the  coefficients.  Suppose  now  ^ = 0 [which  is  the  case  con- 

dT 

sidered  at  the  end  of  C {a)  above].  We  have  — = C,  and 

dip 

deduce 

0 - (c  cos  0 -i-  B)  0 
^ A + B + 2g  cos  0 ’ 

/ dT  . dT\ 

T=\U  -^  + e—.)  = \{if/G  ■¥6[{Gcose  + B)ii/  + BO]} 

\ d\]/  dOJ 


d\f/  dO 
1{^[C+  (c  cos  0-hB)  0]  + B0\ 

, i - {c  cos  0 + By  0^^  , 

A + B + 2ccos0  I 


G^+{AB-c^cos^e)e^ 

^ A + B + 2c  cos  0 


Hence 

and 


AB  - c®  cos*  0 
A + B+2c  cos  6* 
(7* 


^ ^ A + B + 2c  cos  0 


* Remark  that,  according  to  the  alteration  from  \}/,  \p,  0,  0,  to  0,  0,  $,  d, 
as  independent  variables, 

d0“U'/'y  U0y’  dd~\d<t>E 

, dT  fdT\  fdT\  dT  (dT\ 

d0  ~ \d0/  ^ \d0/  ’ dd  ~ \d^)  ’ : 

^here  ( ) indicates  the  original  notation  of  C (a). 


319.]  DYNAMICAL  LAWS  AND  PRINCIPLES.  325 


and  the  one  equation  of  the  motion  becomes 

^ / AB  -c^ cos^  6 _ 1 ^2  cos’  ^ \ ^ dK 

dt  \A  + B -¥20 cos  0 ) ^ dB\A  + B + 2c  cos d)  dd  ’ 

which  is  to  be  fully  integrated  first  by  multiplying  by  dO  and 
integrating  once  ] and  then  solving  for  dt  and  integrating  again 
with  respect  to  6.  The  first  integral,  being  simply  the  equation 
of  energy  integrated,  is  [Example  G (20)] 


and  the  final  integral  is 


% = j®de-K', 


-/V 


AB  — cos’  6 


2{A  + B + 2c  cos  b)  {j®dd  — K) 


In  the  particular  case  in  which  the  motion  commences  from 
rest,  or  is  such  that  it  can  be  brought  to  rest  by  proper  applica- 
tions of  force-components,  etc.  without  any  of  the  force- 

components  X,  X',  etc.,  we  have  (7  = 0,  (7' = 0,  etc.;  and  the 
elimination  of  dc.  by  (3)  renders  T a homogeneous  quad- 

ratic function  of  ij/,  etc.  without  (7,  C',  etc. ; and  the  equations 
of  motion  become 

d dT  dT  , n 

= q/  I 

dt  d\j/  dif/ 

d dT  I 

dt  d<j>  d(f>  y (21)- 

d dT 

dt~de  ' 

etc.  etc. 

We  conclude  that  on  the  suppositions  made,  the  elimination  of 
the  velocity-components  corresponding  to  the  non-appearing  co- 
ordinates gives  an  expression  for  the  kinetic  energy  in  terms 
of  the  remaining  velocity-components  and  corresponding  co- 
ordinates which  may  be  used  in  the  generalised  equations  just 
as  if  these  were  the  sole  co-ordinates.  The  reduced  number  of 
equations  of  motion  thus  found  suffices  for  the  determination 
of  the  co-ordinates  which  they  involve  without  the  necessity 
for  knowing  or  finding  the  other  co-ordinates.  If  the  farther 
question  be  put, — to  determine  the  ignored  co-ordinates,  it  is  to 
be  answered  l)y  a simple  integration  of  equations  (7)  with 
(7=0,  (7'  = 0,  etc. 

One  obvious  case  of  application  for  this  example  is  a system  in 
which  any  number  of  fly  wheels,  that  is  to  say,  bodies  which  are 


Equation 

Energy. 


T^nordtion 
of  co- 
ordinates. 


326 


PRELIMINARY. 


I^noration 
of  co-ordi- 
nates. 


[319. 

kitietically  symmetrical  round  an  axis  (§  285),  are  pivoted  fric- 
tionlessly  on  any  moveable  part  of  the  system.  In  this  case 
with  the  particular  supposition  (7  = 0,  (7'  = 0,  etc.,  the  result  is 
simply  that  the  motion  is  the  same  as  if  each  fly  wheel  were 
deprived  of  moment  of  inertia  round  its  bearing  axis,  that  is  to 
say  reduced  to  a line  of  matter  flxed  in  the  position  of  this  axis 
and  having  unchanged  moment  of  inertia  round  any  axis  per- 
pendicular to  it.  But  if  (7,  C\  etc.  be  not  each  zero  we  have  a 
case  embracing  a very  interesting  class  of  dynamical  problems 
in  which  the  motion  of  a system  having  what  we  may  call 
gyrostatic  links  or  connexions  is  the  subject.  Example  (D) 
above  is  an  example,  in  which  there  is  just  one  fly  wheel  and  one 
moveable  body  on  which  it  is  pivoted.  The  ignored  co-ordinate 
is  i}/ ; and  supposing  now  ^ to  be  zero,  we  have 

ij/-  (f>  {I  - cob6)=-- G (a). 

If  we  suppose  (7  = 0 all  the  terms  having  W for  a factor  vanish 
and  the  motion  is  the  same  as  if  the  fly  wheel  were  deprived  of 
inertia  round  its  bearing  axis,  and  we  had  simply  the  motion  of 
the  “ideal  rigid  body  FQ'’  to  consider.  But  when  (7  does  not 
vanish  we  eliminate  ij/  from  the  equations  by  means  of  (a).  It 
is  important  to  remark  that  in  every  case  of  Example  (G)  in 
which  (7  = 0,  (7'  = 0,  etc.  the  motion  at  each  instant  possesses  the 
property  (§  312  above)  of  having  less  kinetic  energy  than  any 
other  motion  for  which  the  velocity-components  of  the  non-ignored 
co-ordinates  have  the  same  values. 


Take  for  another  example  the  final  form  of  Example  C'  above, 
putting  B for  (7,  and  A for  nk^  + A.  We  have 

T=i{{E  + Fcos^  0)iP^  + B (,/r  cos  6 + + AO^}  ...(22). 

Here  neither  ij/  nor  ^ appears  in  the  coefficients.  Let  us  suppose 
^>  = 0,  and  eliminate  to  let  us  ignore  We  have 


d(ji 

C 

Hence  ^ = -^-^cob6 (23), 

Z = \\{E-¥F  cos^  6)  + AO^] (24), 

and  K-\^ (25). 


The  place  of  x in  (9)  above  is  now  taken  by  and  comparing 
with  (23)  we  find 

J/'=cos0,  iV^=0,  0 = 0. 


DYNAMICAL  LAWS  AND  PRINCIPLES. 


327 


319.] 


Hence,  and  as  K is  constant,  the  equations  of  motion  (19) 
become  ordinates. 


and 


dt  dij/ 
d d^ 
dt  dO 


_^_(7sin  ee=^ 
# [ 
_ + C sin  6\j/  = @\ 

dQ  J 


(26); 


and,  using  (24)  and  expanding, 

_Csine0-4-| 

dt  i (27). 

A 6 + i^sin  6 cos  OiJ/^  + C sin  6^  = ®] 


A most  important  case  for  the  “ ignoration  of  co-ordinates”  is 
presented  by  a large  class  of  problems  regarding  the  motion  of 
frictionless  incompressible  fluid  in  which  we  can  ignore  the 
infinite  number  of  co-ordinates  of  individual  portions  of  the  fluid 
and  take  into  account  only  the  co-ordinates  which  suffice  to 
specify  the  whole  boundary  of  the  fluid,  including  the  bounding 
surfaces  of  any  rigid  or  flexible  solids  immersed  in  the  fluid. 
The  analytical  working  out  of  Example  (G)  shows  in  fact  that  when 
the  motion  is  such  as  could  be  produced  from  rest  by  merely 
moving  the  boundary  of  the  fluid  without  applying  force  to  its 
individual  particles  otherwise  than  by  the  transmitted  fluid 
pressure  we  have  exactly  the  case  of  (7  = 0,  C = 0,  etc. : and 
Lagrange’s  generalized  equations  with  the  kinetic  energy  expressed 
in  terms  of  velocity-components  completely  specifying  the  motion 
of  the  boundary  are  available.  Thus, 


320.  Problems  in  fluid  motion  of  remarkable  interest  and  Kinetics  of 

a perfect 

importance,  not  hitherto  attacked,  are  very  readily  solved  by 
the  aid  of  Lagrange’s  generalized  equations  of  motion.  For 
brevity  we  shall  designate  a mass  which  is  absolutely  incom- 
pressible, and  absolutely  devoid  of  resistance  to  change  of  shape, 
by  the  simple  appellation  of  a liquid.  We  need  scarcely  say 
that  matter  perfectly  satisfying  this  definition  does  not  exist 
in  nature : but  we  shall  see  (under  properties  of  matter)  how 
nearly  it  is  approached  by  water  and  other  common  real 
liquids.  And  we  shall  find  that  much  practical  and  interesting 
information  regarding  their  true  motions  is  obtained  by  deduc- 
tions from  the  principles  of  abstract  dynamics  applied  to  the 
ideal  perfect  liquid  of  our  defiuition.  It  follows  from  Example 


Kinetics  of 
a perfect 
liquid. 


328  PRELIMINARY.  [320 

(G)  above  (and  several  other  proofs,  some  of  them  more 
synthetical  in  character,  will  be  given  in  our  Second  Volume,) 
that  the  motion  of  a homogeneous  liquid,  whether  of  infinite 
extent,  or  contained  in  a finite  closed  vessel  of  any  form,  with 
any  rigid  or  flexible  bodies  moving  through  it,  if  it  has  ever 
been  at  rest,  is  the  same  at  each  instant  as  that  determinate 
motion  (fulfilling,  § 312,  the  condition  of  having  the  least 
possible  kinetic  energy)  which  would  be  impulsively  produced 
from  rest  by  giving  instantaneously  to  every  part  of  the 
bounding  surface,  and  of  the  surface  of  each  of  the  solids 
within  it,  its  actual  velocity  at  that  instant.  So  that,  for 
example,  however  long  it  may  have  been  moving,  if  all  these 
surfaces  were  suddenly  or  gradually  brought  to  rest,  the  whole 
fluid  mass  would  come  to  rest  at  the  same  time.  Hence,  if 
none  of  the  surfaces  is  flexible,  but  we  have  one  or  more  rigid 
bodies  moving  in  any  way  through  the  liquid,  under  the  in- 
fluence of  any  forces,  the  kinetic  energy  of  the  whole  motion 
at  any  instant  will  depend  solely  on  the  finite  number  of  co- 
ordinates and  component  velocities,  specifying  the  position  and 
motion  of  those  bodies,  whatever  may  be  the  positions  reached 
by  particles  of  the  fluid  (expressible  only  by  an  infinite  number 
of  co-ordinates).  And  an  expression  for  the  whole  kinetic 
energy  in  terms  of  such  elements,  finite  in  number,  is  precisely 
what  is  wanted,  as  we  have  seen,  as  the  foundation  of  Lagrange’s 
equations  in  any  particular  case. 

It  will  clearly,  in  the  hydrodynamical,  as  in  all  other  cases, 
be  a homogeneous  quadratic  function  of  the  components  of  velo- 
city, if  referred  to  an  invariable  co-ordinate  system;  and  the 
coefficients  of  the  several  terms  will  in  general  be  functions  of 
the  co-ordinates,  the  determination  of  which  follows  immediately 
from  the  solution  of  the  minimum  problem  of  Example  (3)  § 317, 
in  each  particular  case. 

Exam, pie  (1). — A hall  set  in  motion  through  a mass  of  incom- 
pressible fluid  extending  inflnitely  in  all  directions  on  one  side  of 
an  infinite  plane^  and  originally  at  rest.  Let  £c,  y,  z be  the  co- 
ordinates of  the  centre  of  the  ball  at  time  t,  with  reference  to 
rectangular  axes  through  a fixed  point  0 of  the  bounding  plane, 
with  OX  perpendicular  to  this  plane.  If  at  any  instant  either 


DYNAMICAL  LAWS  AND  PKINCIPLES. 


329 


320.] 


component  y or  z of  the  velocity  be  reversed,  the  kinetic  energy  Kinetics  of 
will  clearly  be  unchanged,  and  hence  no  terms  yz,  zx^  or  xy  can  liquid, 
appear  in  the  expression  for  the  kinetic  energy  : which,  on  this 
account,  and  because  of  the  symmetry  of  circumstances  with 
reference  to  y and  z,  is 


T=\{Px^^Q{f^z^)}. 

Also,  we  see  that  P and  Q are  functions  of  x simply,  since  the 
circumstances  are  similar  for  all  values  of  y and  z.  Hence,  by 
differentiation. 


dT  p.  dT 

-^P^,-=Qy, 


dT 

dz 


Q^, 


d(dT\ 
dt  \dx  ) 


dy 


dQ  ..  , 

+ ^2/x,etc., 
= 0,  etc., 


and  the  equations  of  motion  are 

QyP^Jx^-Y,  Qz  + f^i±  = Z. 


Principles  sufficient  for  a practical  solution  of  the  problem  of 
determining  P and  Q will  be  given  later.  In  the  meantime,  it 
is  obvious  that  each  decreases  as  x increases.  Hence  the  equa- 
tions of  motion  show  that 


321.  A ball  projected  through  a liquid  perpendicularly  Effect  of  a 
from  an  infinite  plane  boundary,  and  influenced  by  no  other  Sh?mo- 

1 1 p n • 1 • 11  tionofaball 

forces  than  those  oi  fluid  pressure,  experiences  a gradual  ac- through  a 
celeration,  quickly  approximating  to  a limiting  velocity  which 
it  sensibly  reaches  when  its  distance  from  the  plane  is  many 
times  its  diameter.  But  if  projected  parallel  to  the  plane,  it 
experiences,  as  the  resultant  of  fluid  pressure,  a resultant  attrac- 
tion towards  the  plane.  The  former  of  these  results  is  easily 
proved  by  first  considering  projection  towards  the  plane  (in 
which  case  the  motion  of  the  ball  will  obviously  be  retarded), 
and  by  taking  into  account  the  general  principle  of  reversibility 
(§  272)  which  has  perfect  application  in  the  ideal  case  of  a per- 
fect liquid.  The  second  result  is  le.ss  easily  foreseen  without 


Seeming 
attruction 
b^^tween 
two  ships 
moving  side 
by  side  in 
the  same 
direction. 


Hydro- 

dynamical 

examples 

continued. 


“ Centre  of 

reaction” 

defined. 


330  PRELIMINARY.  [321 

the  aid  of  Lagrange’s  analysis ; but  it  is  an  obvious  consequence 
of  the  Hamiltonian  form  of  his  equations,  as  stated  in  words 
in  § 319  above.  In  the  precisely  equivalent  case,  of  a 
liquid  extending  infinitely  in  all  directions,  and  given  at  rest ; 
and  two  equal  balls  projected  through  it  with  equal  velocities 
perpendicular  to  the  line  joining  their  centres — the  result  that 
the  two  balls  will  seem  to  attract  one  another  is  most  re- 
markable, and  very  suggestive. 


Eocample  (2). — A solid  symmetrical  round  an  axis,  moving 
through  a liquid  so  as  to  keep  its  axis  always  in  one  plane. 
Let  o)  be  the  angular  velocity  of  the  body  at  any  instant  about 
any  axis  perpendicular  to  the  fixed  plane,  and  let  u and  q be  the 
component  velocities  along  and  perpendicular  to  the  axis  of 
figure,  of  any  chosen  point,  C,  of  the  body  in  this  line.  By  the 
general  principle  stated  in  § 320  (since  changing  the  sign  of 
u cannot  alter  the  kinetic  energy),  we  have 

(Au^  + Eq^  + pW  -h  2Eioq) (a), 

where  A,  B,  p,  and  E are  constants  depending  on  the  figure  of 

the  body,  its  mass,  and  the  density  of  the  liquid.  Now  let  v 

denote  the  velocity,  perpendicular  to  the  axis,  of  a point  which 

we  shall  call  the  centre  of  reaction,  being  a point  in  the  axis  and 

E E 

at  a distance  ^ from  G,  so  that  (§  87)  q = v--^ei.  Then, 

E^ 

denoting  p - ^ by  p,  we  have  T—  J (Au^  + Bv^  + pu)^) (a). 

Let  X and  y be  the  co  ordinates  of  the  centre  of  reaction  relatively 
to  any  fixed  rectangular  axes  in  the  plane  of  motion  of  the  axis 
of  figure,  and  let  Q be  the  angle  between  this  line  and  OX,  at 
any  instant,  so  that 

CD  = u = x cos  B -vy  sin  B,  v = - x sin  B -\-y  cos  B (b). 


Substituting  in  T,  differentiating,  and  retaining  the  notation 
u,  V where  convenient  for  brevity,  we  have 


dT 


= = Au  COB  B - Bv  sin  B,  ^ = Au  sin  B + Bv  cos  B, 

^ dx  dij 


dT 


dB 


dy 


dT  ..  dT  . dT  . 

T0  = ^A-B)uv,  ^ = 0,  -^  = 0, 


dy 


321.J 


DYNAMICAL  LAWS  AND  PRINCIPLES. 


331 


Hence  the  equations  of  motion  are 


fxd  — (A  — uv  = 7/, 

d(Auco^6  — Bvsui6)  ^ 

r- — A, 


d (Au  sin  0 + Bv  cos  0) 
dt 


(d), 


Hydro- 

dynamical 

examples 

continued. 


where  X,  Y are  the  component  forces  in  lines  through  C parallel 
to  OX  and  0 Y,  and  L the  couple,  applied  to  the  body. 


Denoting  by  A,  rj  the  impulsive  couple,  and  the  components 
of  impulsive  force  through  C,  required  to  produce  the  motion  at 
any  instant,  we  have  of  course  [§313  (c)], 


A 


dO 


dT  dT 
dx^  ^ dy 


and  therefore  by  (c),  and  (6), 

u = cos  0 + rj  sin  6),  -y  = ^ (-  ^ sin  ^ cos  ^),  0 = -. 
. /cos®0  sin'^^X  /I  1\  . ^ ^ ' 

X = j f + (^2  - ^ 


A 

. /I  In  . . . /sin^6^  cos^0\  ; ’ 

2^ = (a  - 5 j ^ ^ {-A- 

and  the  equations  of  motion  become 
" § - cos  2^}  = A I = J=r,  (4). 


•(/)> 

•(?). 


The  simple  case  of  X=  0,  T=  0,  Z = 0,  is  particularly  interesting. 
In  it  ^ and  rj  are  each  constant;  and  we  may  therefore  choose  the 
axes  OX,  OY,  so  that  rj  shall  vanish.  Thus  we  have,  in  (g),  two 
first  integrals  of  the  equations  of  motion;  and  they  become 


+ 


sin^  A — B. 

-B~)’  y=-^^ 


sin  W 


and  the  first  of  equations  (4)  becomes 


d:Q 


(0- 


In  this  let,  for  a moment,  20  = <4,  and 


A-B 

AB 


e=9hlV. 


It  becomes 


de 


+ gh  W sin  <4  = 


which  is  the  equation  of  motion  of  a common  pendulum,  of 
mass  TI',  moment  of  inertia  fx  round  its  fixed  axis,  and  lengtli 


332 


PKELIMINARY. 


[321. 

h from  axis  to  centre  of  gravity;  if  </>  be  the  angle  from 
the  position  of  equilibrium  to  the  position  at  time  t.  As  we 
shall  see,  under  kinetics,  the  final  integral  of  this  equation 
expresses  in  terms  of  t by  means  of  an  elliptic  function. 
By  using  the  value  thus  found  for  0 or  in  (k),  we  have 
equations  giving  x and  y in  terms  of  t by  common  integration ; 
and  thus  the  full  solution  of  our  present  problem  is  reduced  to 
quadratures.  The  detailed  working  out  to  exhibit  both  the  actual 
curve  described  by  the  centre  of  reaction,  and  the  position  of 
the  axis  of  the  body  at  any  instant,  is  highly  interesting.  It  is 
very  easily  done  approximately  for  the  case  of  very  small  angular 
vibrations;  that  is  to  say,  when  either  A — ^ is  positive,  and 
always  very  small,  or  A — ^ negative,  and  ^ very  nearly 
equal  to  Jtt.  But  without  attending  at  present  to  the  final 
integrals,  rigorous  or  approximate,  we  see  from  {Ic)  and  (1)  that 

322.  If  a solid  of  revolution  in  an  infinite  liquid,  be  set  in 
motion  round  any  axis  perpendicular  to  its  axis  of  figure,  or 
simply  projected  in  any  direction  without  rotation,  it  will  move 
with  its  axis  always  in  one  plane,  and  every  point  of  it  moving- 
only  parallel  to  this  plane;  and  the  strange  evolutions  which 
it  will,  in  genera],  perform,  are  perfectly  defined  by  comparison 
with  the  common  pendulum  thus.  First,  for  brevity,  we  shall 
Quadrantai  Call  by  the  name  of  quadrantal  pendulum  (which  will  be  further 
dSued?*^  exemplified  in  various  cases  described  later,  under  electricity 
and  magnetism;  for  instance,  an  elongated  mass  of  soft  iron 
pivoted  on  a vertical  axis,  in  a “uniform  field  of  magnetic 
force”),  a body  moving  about  an  axis,  according  to  the  same 
law  with  reference  to  a quadrant  on  each  side  of  its  position  of 
equilibrium,  as  the  common  pendulum  with  reference  to  a half 
circle  on  each  side. 

Let  now  the  body  in  question  be  set  in  motion  by  an  im- 
pulse, f,  in  any  line  through  the  centre  of  reaction,  and  an 
impulsive  couple  \ in  the  plane  of  that  line  and  the  axis.  This 
will  (as  will  be  proved  later  in  the  theory  of  statical  couples) 
have  the  same  effect  as  a simple  impulse  f (applied  to  a point, 
if  not  of  the  real  body,  connected  with  it  by  an  imaginary  in- 
finitely light  framework)  in  a certain  fixed  line,  which  we  shall 
call  the  line  of  resultant  impulse,  or  of  resultant  momentum. 


Hydro- 

dynamical 

examples 

continued. 


DYNAMICAL  LAWS  AND  PRINCIPLES. 


883 


822.' 


beiug  parallel  to  the  former  line,  and  at  a distance  from  it  equal  to 

“ . The  whole  momentum  of  the  motion  generated  is  of  course 

(§  295)  equal  to  The  body  will  move  ever  afterwards 
according  to  the  following  conditions  : — (1.)  The  angular  velo- 
city follows  the  law  of  the  quad  ran  tal  pendulum.  (2.)  The 
distance  of  the  centre  of  reaction  from  the  line  of  resultant 
impulse  varies  simply  as  the  angular  velocity.  (8.)  The 
velocity  of  the  centre  of  reaction  parallel  to  the  line  of 
impulse  is  found  by  dividing  the  excess  of  the  whole  con- 
stant energy  of  the  motion  above  the  part  of  it  due  to  the 
angular  velocity  round  the  centre  of  reaction,  by  half  the 
momentum.  (4)  If  A,  B,  and  ya  denote  constants,  depending 
on  the  mass  of  the  solid  and  its  distribution,  the  density  of  the 
liquid,  and  the  form  and  dimensions  of  the  solid,  such  that 

I ? ^ 

’ H' 

respectively  produced  by  an  impulse  f along  the  axis,  an  im- 
pulse f in  a line  through  the  centre  of  reaction  perpendicular 
to  the  axis,  and  an  impulsive  couple  \ in  a plane  through  the 
axis ; the  length  of  the  simple  gravitation  pendulum,  whose 
motion  would  keep  time  with  the  periodic  motion  in  question, 

is  when  the  angular  motion  is  vibratory,  the 

vibrations  will,  according  as  ^ or  A < B,  be  of  the 

axis,  or  of  a line  perpendicular  to  the  axis,  vibrating  on 
each  side  of  the  line  of  impulse.  The  angular  motion  will 
in  fact  be  vibratory  if  the  distance  of  the  line  of  resultant 
impulse  from  the  centre  of  reaction  is  anything  less  than 
(A  ~B)  /JL  cos  2x 


^ are  the  linear  velocities,  and  the  angular  velocity. 


AB 


where  a denotes  the  inclination  of  the 


im- 


pulse to  the  initial  position  of  the  axis.  In  this  case  the  path 
of  the  centre  of  reaction  will  be  a sinuous  curve  symmetrical  on 
the  two  sides  of  the  line  of  impulse ; every  time  it  cuts  this  line, 
the  angular  motion  will  reverse,  and  the  maximum  inclination 
will  be  attained ; and  every  time  the  centre  of  reaction  is  at  its 
greatest  distance  on  either  side,  the  angular  velocity  will  be  at 
its  greatest,  positive  or  negative,  value,  and  the  linear  velocity  of 


Motion  of 
a solid  of 
revolution 
through  a 
liquid. 


334 


PRELIMINARY. 


[322. 


Motion  of 
a solid  of 
revolution 
through  a 
liquid. 


the  centre  of  reaction  will  be  at  its  least.  If,  on  the  other  hand, 
the  line  of  the  resultant  impulse  be  at  a greater  distance  than 
'(^~5)^cos2a 
~AB 


from  the  centre  of  reaction,  the  angular  motion 


will  be  always  in  one  direction,  but  will  increase  and  diminish 
periodically,  and  the  centre  of  reaction  will  describe  a sinuous 
curve  on  one  side  of  that  line ; being  at  its  greatest  and  least 
deviations  when  the  angular  velocity  is  greatest  and  least.  At 
the  same  points  the  curvature  of  the  path  will  be  greatest  and 
least  respectively,  and  the  linear  velocity  of  the  describing 
point  will  be  least  and  greatest. 


323.  At  any  instant  the  component  linear  velocities  along 


f cos  Q 


and  perpendicular  to  the  axis  of  the  solid  will  be  ^ 
^sin  6 


and 


B 


respectively,  if  d be  its  inclination  to  the  line  of  re- 

Py  . 

sultant  impulse ; and  the  angular  velocity  will  be  — if  y be  the 

distance  of  the  centre  of  reaction  from  that  line.  The  whole 
kinetic  energy  of  the  motion  will  be 


f ^ cos^  Q ^ sin^  0 

2B  ■'■"V  ’ 


and  the  last  term  is  what  we  have  referred  to  above  as  the 
part  due  to  rotation  round  the  centre  of  reaction  (defined  in 
§ 321).  To  stop  the  whole  motion  at  any  instant,  a simple 
impulse  equal  and  opposite  to  f in  the  fixed  “ line  of  resultant 
impulse”  will  suffice  (or  an  equal  and  parallel  impulse  in  any 
line  through  the  body,  with  the  proper  impulsive  couple,  accord- 
ing to  the  principle  already  referred  to). 


324.  From  Lagrange’s  equations  applied  as  above  to  the  case 
of  a solid  of  revolution  moving  through  a liquid,  the  couple 
which  must  be  kept  applied  to  it  to  prevent  it  from  turning  is 
immediately  found  to  be 


uv  (A  — B) 


DYNAMICAL  LAWS  AND  PRINCIPLES. 


335 


324.] 

if  u and  v be  the  component  velocities  along  and  perpendicular  MoUmi^of 

to  the  axis,  or  TS  321  (f)]  revolution 

' through  a 

_ . ..  liquid. 

— B)  Sin  20 
^ 2AB  ’ 

if,  as  before,  f be  the  generating  impulse,  and  0 the  angle  be- 
tween its  line  and  the  axis.  The  direction  of  this  couple  must 
be  such  as  to  prevent  0 from  diminishing  or  from  increasing, 
according  as  ^4  or  ^ is  the  greater.  The  former  will  clearly 
be  the  case  of  a flat  disc,  or  oblate  spheroid ; the  latter  that  of 
an  elongated,  or  oval-shaped  body.  The  actual  values  of  A 
and  B we  shall  learn  how  to  calculate  (hydrodynamics)  for 
several  cases,  including  a body  bounded  by  two  spherical  sur- 
faces cutting  one  another  at  any  angle  a submultiple  of  two 
right  angles;  two  complete  spheres  rigidly  connected;  and  an 
oblate  or  a prolate  spheroid. 

325.  The  tendency  of  a body  to  turn  its  flat  side,  or  its  Observed 
length  (as  the  case  may  be),  across  the  direction  of  its  motion 
through  a liquid,  to  which  the  accelerations  and  retardations  of 
rotatory  motion  described  in  § 322  are  due,  and  of  which  we 
have  now  obtained  the  statical  measure,  is  a remarkable  illus- 
tration of  the  statement  of  § 319  ; and  is  closely  connected 
with  the  dynamical  explanation  of  many  curious  observations 
well  known  in  practical  mechanics,  among  which  may  be  men- 
tioned : — 

(1)  That  the  course  of  a symmetrical  square-rigged  ship 
sailing  in  the  direction  of  the  wind  with  rudder  amidships  is 
unstable,  and  can  only  be  kept  by  manipulating  the  rudder  to 
check  infinitesimal  deviations ; — and  that  a child’s  toy-boat, 
whether  “square-rigged”  or  “fore-and-aft  rigged*,”  cannot  be 

* “Fore-and-aft”  rig  is  any  rig  in  which  (as  in  “ cutters  ” and  “ schooners  ”) 
the  chief  sails  come  into  the  plane  of  mast  or  masts  and  keel,  hy  the  action  of 
the  wind  upon  the  sails  when  the  vessel’s  head  is  to  wind.  This  position 
of  the  sails  is  unstable  when  the  wind  is  right  astern.  Accordingly,  in 
“wearing”  a fore-and-aft  rigged  vessel  (that  is  to  say  turning  her  round 
stern  to  wind,  from  sailing  with  the  wind  on  one  side  to  sailing  with  the 
wind  on  the  other  side)  the  mainsail  must  be  hauled  in  as  closely  as  may  be 
towards  the  middle  position  before  the  wind  is  allowed  to  get  on  the  other  side 
of  the  sail  from  that  on  which  it  had  been  pressing,  so  that  when  the  wind 


33G 


PRELIMINARY. 


[325. 


AjjpWcat^ns  got  to  Sail  permanently  before  the  wind  by  any  permanent  ad- 
dyuamics  justment  of  rudder  and  sails,  and  that  (without  a wind  vane,  or 
a weighted  tiller,  acting  on  the  rudder  to  do  the  part  of 
steersman)  it  always,  after  running  a few  yards  before  the  wind, 
turns  round  till  nearly  in  a direction  perpendicular  to  the 
wind  (either  gibing”  first,  or  “luffing”  without  gibing  if  it 
is  a cutter  or  schooner)  : — 

(2)  That  the  towing  rope  of  a canal  boat,  when  the  rudder 
is  left  straight,  takes  a position  in  a vertical  plane  cutting  the 
axis  before  its  middle  point : — 

(3)  That  a boat  sculled  rapidly  across  the  direction  of  the 
wind,  always  (unless  it  is  extraordinarily  unsymmetrical  in 
its  draught  of  water,  and  in  the  amounts  of  surface  exposed 
to  the  wind,  towards  its  two  ends)  requires  the  weather  oar 
to  be  worked  hardest  to  prevent  it  from  running  up  on  the 
wind,  and  that  for  the  same  reason  a sailing  vessel  generally 
“carries  a weather  helm*”  or  “gripes;”  and  that  still  more  does 
so  a steamer  with  sail  even  if  only  in  the  forward  half  of  her 
length — ^griping  so  badly  with  any  after  canvass *f’  that  it  is  often 
impossible  to  steer  : — 

(4)  That  in  a heavy  gale  it  is  exceedingly  difficult,  and 
often  found  impossible,  to  get  a ship  out  of  “the  trough  of  the 
sea,”  and  that  it  cannot  be  done  at  all  without  rapid  motion 
ahead,  whether  by  steam  or  sails  : — 

(5)  That  in  a smooth  sea  with  moderate  wind  blowing 
parallel  to  the  shore,  a sailing  vessel  heading  towards  the  shore 
with  not  enough  of  sail  set  can  only  be  saved  from  creeping 
ashore  by  setting  more  sail,  and  sailing  rapidly  towards  the 
shore,  or  the  danger  that  is  to  be  avoided,  so  as  to  allow  her  to 
be  steered  away  from  it.  The  risk  of  going  ashore  in  fulfilment 

does  get  on  the  other  side,  and  when  therefore  the  sail  dashes  across  through 
the  mid-ship  position  to  the  other  side,  carrying  massive  boom  and  ga£E  with  it, 
the  range  of  this  sudden  motion,  which  is  called  “gibing,”  shall  be  as  small 
as  may  be. 

* The  weather  side  of  any  object  is  the  side  of  it  towards  the  wind.  A ship 
is  said  to  “carry  a weather  helm”  when  it  is  necessary  to  hold  the  “helm”  or 
“tiller”  permanently  on  the  weather  side  of  its  middle  position  (by  which  the 
rudder  is  held  towards  the  lee  side)  to  keep  the  ship  on  her  course. 

t Hence  mizen  masts  are  altogether  condemned  in  modern  war-ships  by 
many  competent  nautical  authorities. 


DYNAMICAL  LAWS  AND  PRINCIPLES. 


337 


325.] 

of  Lagrange’s  equations  is  a frequent  incident  of  “getting 
under  way”  while  lifting  anchor,  or  even  after  slipping  from 
moorings  : — 

(6)  That  an  elongated  rifle-bullet  requires  rapid  rotation  and  gun- 
about  its  axis  to  keep  its  point  foremost. 

(7)  The  curious  motions  of  a flat  disc,  oyster-shell,  or  the 
like,  when  dropped  obliquely  into  water,  resemble,  no  doubt,  to 
some  extent  those  described  in  § 322.  But  it  must  be  re- 
membered that  the  real  circumstances  differ  greatly,  because 
of  fluid  friction,  from  those  of  the  abstract  problem,  of  which 
we  take  leave  for  the  present. 

326.  Maupertuis’  celebrated  principle  of  Least  Action  has  Least 

1 . Ill  . action 

been,  even  up  to  the  present  time,  regarded  rather  as  a curious 
and  somewhat  perplexing  property  of  motion,  than  as  a useful 
guide  in  kinetic  investigations.  We  are  strongly  impressed 
with  the  conviction  that  a much  more  profound  importance 
will  be  attached  to  it,  not  only  in  abstract  dynamics,  but  in  the 
theory  of  the  several  branches  of  physical  science  now  beginning 
to  receive  dynamic  explanations.  As  an  extension  of  it.  Sir 
W.  R.  Hamilton*  has  evolved  his  method  of  Varying  Action, 
which  undoubtedly  must  become  a most  valuable  aid  in  future 
generalizations. 

What  is  meant  by  “ Action  ” in  these  expressions  is,  unfor-  Action, 
tunately,  something  very  different  from  the  Actio  Agentis  de- 
fined by  Newton  "f,  and,  it  must  be  admitted,  is  a much  less 
judiciously  chosen  word.  Taking  it,  however,  as  we  find  it, 
now  universally  used  by  writers  on  dynamics,  we  define  the  energy. 
Action  of  a Moving  System  as  proportional  to  the  average 
kinetic  energy,  which  the  system  has  possessed  during  the  time 
from  any  convenient  epoch  of  reckoning,  multiplied  by  the  time. 
According  to  the  unit  generally  adopted,  the  action  of  a system 
which  has  not  varied  in  its  kinetic  energy,  is  twice  the  amount 
of  the  energy  multiplied  by  the  time  from  the  epoch.  Or  if 
the  energy  has  been  sometimes  greater  and  sometimes  less, 


• Phil.  Tram.  18.34—18.35. 

+ Which,  however  (§  203),  we  have  translated  “ activity”  to  avoid  confusion. 

22 


VOL.  1. 


Time  aver- 
age of 
energy 
doubled. 


Space  aver- 
age of  mo- 
mentums. 


Least 

action. 


General 
l^roblera  of 
aim. 

High  aim 
and  low  aim 
for  same 
target. 
Target  too 
far  off  to  be 
reached. 


338  PRELIMINARY.  [326. 

the  action  at  time  t is  the  double  of  what  we  may  call  the 
time-integral  of  the  energy,  that  is  to  say,  it  is  what  is  de- 
noted in  the  integral  calculus  by 

2 [Vjr, 

where  T denotes  the  kinetic  energy  at  any  time  r,  between 
the  epoch  and  t 

Let  m be  the  mass,  and  v the  velocity  at  time  t,  of  any  one  of 
the  material  points  of  which  the  system  is  composed.  We  have 

(1), 

and  therefore,  if  A denote  the  action  at  time  t, 

A = r ^mv^dr (2). 

Jo 

This  may  be  put  otherwise  by  taking  ds  to  denote  the  space  de- 
scribed by  a particle  in  time  dr,  so  that  vdr  = ds,  and  therefore 

A =f  Xmvds (3), 

or,  if  X,  y,  z be  the  rectangular  co-ordinates  of  m at  any  time, 

A — ^ %m  {xdx  + ydy  -f  zdz) (4). 

Hence  we  might,  as  many  writers  in  fact  have  virtually  done, 
define  action  thus  : — 

The  action  of  a system  is  equal  to  the  sum  of  the  average 
momentu7ns  for  the  spaces  described  by  the  particles  from  any 
era  each  multiplied  by  the  length  of  its  path. 

327.  The  principle  of  Least  Action  is  this : — Of  all  the 
different  sets  of  paths  along  which  a conservative  system  may 
be  guided  to  move  from  one  configuration  to  another,  with  the 
sum  of  its  potential  and  kinetic  energies  equal  to  a given  con- 
stant, that  one  for  which  the  action  is  the  least  is  such  that 
the  system  will  require  only  to  be  started  with  the  proper 
velocities,  to  move  along  it  unguided.  Consider  the  Problem  : — 
Given  the  whole  initial  kinetic  energy ; find  the  initial  velocities 
through  one  given  configuration,  which  shall  send  the  system 
unguided  to  another  specified  configuration.  This  problem  is 
essentially  determinate,  but  generally  has  multiple  solutions 
(§  363  below);  (or  only  imaginary  solutions.) 


DYNAMICAL  LAWS  AND  PRINCIPLES. 


339 


327.] 

If  there  are  any  real  solutions,  there  is  one  of  them  for  which  Least 
the  action  is  less  than  for  any  other  real  solution,  and  less  than 
for  any  constrainedly  guided  motion  with  proper  sum  of  po- 
tential and  kinetic  energies.  Compare  §§  346 — 366  below. 

Let  Xj  y,  z be  the  co-ordinates  of  a particle,  of  the  system, 
at  time  t,  and  V the  potential  energy  of  the  system  in  its  pai-ti- 
cular  configuration  at  this  instant ; and  let  it  be  required  to  find 
the  way  to  pass  from  one  given  coiiHguration  to  another  with 
velocities  at  each  instant  satisfying  the  condition 

2 \ (ic"  4-  y"  + + V = E,  ^ constant (5), 

so  that  d,  or 

j ^ni  {xdx  yd y zdz) 
may  be  the  least  possible. 

By  the  method  of  variations  we  must  have  = 0,  where 

SA  = f %m  (xdSx  + ydSy  + MBz  + oxdx  + ^ydy  + Szdz) (6). 

Taking  in  this  dx  — xdr,  dy  = ydr,  dz  = zdr^  and  remarking  that 
'Zm  {xSx  + y?)y  + zBz)  = hT (7), 

we  liave 

/ 2m  [hxdx  + hjdy  + hzdz)  ~ f hTdr (8). 

. Jo 

Also  by  integration  by  parts, 

/ 'S,m{xdhx  + ...)  = + ...)}  - [2m(ic8a;  -l*...)]— j%m{xhx+ ...)dT, 

where  [...]  and  {...}  denote  the  values  of  the  quantities  enclosed, 
at  the  beginning  and  end  of  the  motion  considered,  and  where, 
further,  it  must  be  remembered  that  dx  = xdr^  etc.  Hence, 
from  above, 

8d  = {2m  {xhx  + y8y  4-  — [2m  (xSx  4-  yhy  4-  zhz)^ 

-h  f dr  [ST-  2m  (xSx  + ijZy  4-  ^S;2:)] (9). 

Jo 

This,  it  may  be  observed,  is  a perfectly  general  kineniatical  expres- 
sion, unrestricted  by  any  terminal  or  kinetic  conditions.  Now 
in  the  present  problem  we  suppose  the  initial  and  final  positions 
to  b(‘,  invariable.  Hence  the  terminal  variations,  hx,  etc.,  must 
all  vanish,  and  therefore  the  integrated  expressions  [••.]  dis- 
appear. Also,  in  the  present  })roblem  ST  = — 8F,  by  the  equation 
of  energy  (5).  Hence,  to  make  8d  = 0,  since  the  intermediate 
variations,  Sx,  etc.,  are  quite  arbitrary,  subject  only  to  the  con- 

22—2 


340 


PRELIMINARY. 


[327. 


Least 

actioTi. 


Principle  of 
Least  Action 
applied 
to  find 
Laf^ranpe’e 
>?eneralized 
equations 
of  motion. 


ditions  of  the  system,  we  must  have 

%m  (xSx  + ijhy  + z^z)  + 8 F—  0 (10), 

which  [(4),  § 293  above]  is  the  general  variational  equation  of 
motion  of  a conservative  system.  This  proves  the  proposition. 

It  is  interesting  and  instructive  as  an  illustration  of  the  prin- 
ciple of  least  action,  to  derive  directly  from  it,  without  any  use 
of  Cartesian  co-ordinates,  Lagrange’s  equations  in  generalized 
co-ordinates,  of  the  motion  of  a conservative  system  [§  318  (24)]. 
We  have 

A ^j2Tdt, 

where  T denotes  the  formula  of  § 313  (2).  If  now  we  put 

^df’ 


so  that 

ds^  = (xj/,  xf/)  dx{/^  + 2 (xf/,  ^)  dx(/d(f)  -1-  etc., 

we  have 

I HO. 

II 

Hence 

J \ dt  dt  J J dt  dt  j dt 

^ jdtBT  + 

1 x{/)dx(/+  {x(/,(fi)d<f>+  etc 

I 


{\f/  (f>)  clij/  -f-  -f  etc. 

dt 


Sdcf>  + etc.  + fdtS(^^^^  etc.)  T, 


where  ^(4,,<i,,etc.)  denotes  variation  dependent  on  the  explicit  a{)- 
pearance  of  if/,  cf>,  etc.  in  the  coefficients  of  the  quadratic  func- 
tion T.  The  second  chief  term  in  the  formula  for  is  clearly 
idT 

equal  to  I — r and  this,  integrated  by  parts,  becomes 

j dxL 


di}/ 


d\p 


~ — 8x1/, 


d dT 

dt  d^ 


where  [ ] denotes  tlie  difference  of  the  values  of  the  bracketed 
expression,  at  the  beginning  and  end  of  the  time  Jdt.  Thus  we 
have  finally 


327.] 


DYNAMICAL  LAWS  AND  PRINCIPLES. 


341 


So  far  we  have  a purely  kinematical  formula.  Now  introduce  Principlo  of 
the  dynamical  condition  [§  293  (7)] 

T=^G- V (10)". 

From  it  we  find 

'dV^.  dV 


8T 





LeastAction 

applied 

to  find 

Lagraniire’s 

generalized 

equations 

of  motion. 


Again,  we  have 


Hence  (10)'  becomes 
SJ 


. rp  dT  ^ dT 

6(^,</»,etc.)  ^ ^9  + etc. 


,(10)‘\ 


+ — 60  + etc. 
dcf> 


=r^  ^ 


,(ior, 


To  make  this  a minimum  we  have 

d dT  dT  dV  ^ , 

- -r  — + -TT  + -,T  = 0,  etc 

dt  dij/  dij/ 

which  are  the  required  equations  [§318  (24)]. 

From  the  proposition  that  8A  = 0 implies  the  equations  of 
motion,  it  follows  that 

328.  In  any  unguided  motion  whatever,  of  a conservative  why  called 
system,  the  Action  from  any  one  stated  position  to  any  other,  a£n^”  b7 
though  not  necessarily  a minimum,  fulfils  the  stationary  condi- 
tion,  that  is  to  say,  the  condition  that  the  variation  vanishes, 
which  secures  either  a minimum  or  maximum,  or  maximum- 
minimum. 

This  can  scarcely  be  made  intelligible  without  mathematical  Stationary 
language.  Let  (jc^,  y^,  z^),  etc.,  be  the  co  ordinates 

of  particles,  m^,  etc.,  composing  the  system ; at  any  time  t of 
the  actual  motion.  Let  V be  the  potential  energy  of  the  system, 
in  this  configuration ; and  let  B denote  the  given  value  of  the 
sum  of  the  potential  and  kinetic  energies.  The  equation  of 
energy  is — 

i {"^1  (^1*  + 2/1"  + + ^2  (^2*  + + ^2)  + r=  ^. . . (5)  bis. 

Choosing  any  part  of  the  motion,  for  instance  that  from  time  0 
to  time  we  have,  for  the  action  during  it, 


A = [‘(A’-  >')*  = - f 


(11). 


Si2 


PRELIMINARY. 


Stationary 

action. 


[328. 

Let  now  the  system  be  guided  to  move  in  any  other  way  possible 
for  it,  with  any  other  velocities,  from  the  same  initial  to  the  same 
final  configuration  as  in  the  given  motion,  subject  only  to  the 
condition,  that  the  sum  of  the  kinetic  and  potential  energies  shall 
still  be  E.  Let  z^'),  etc.,  be  the  co-ordinates,  and  V 

the  corresponding  potential  energy ; aiid  let  (ic/,  i/),  etc., 

be  the  compoient  velocities,  at  time  r in  this  arbitrary  motion; 
equation  (2)  still  holding,  for  the  accented  letters,  with  only  E 
unchanged.  For  the  action  we  shall  have 


where  t'  is  the  time  occupied  by  this  supposed  motion.  Let  now 
0 denote  a small  numerical  quantity,  and  let  etc.,  be  finite 

lines  such  that 


<11^  . ytUL  = ^ etc.  = 6^. 

V'  - V 

The  “principle  of  stationary  action”  is,  that  — - — vanishes 

when  0 is  made  infinitely  small,  for  every  possible  deviation 
7)^0,  etc.)  from  the  natural  way  and  velocities,  subject  only 
to  the  equation  of  energy  and  to  the  condition  of  passing  through 
the  stated  initial  and  final  configurations  : and  conversely,  that  if 
V'  - V . 

— ^ — vanishes  with  0 for  every  possible  such  deviation  from  a 
u 

certain  way  and  velocities,  specified  by  {x^,  y^,  z^,  etc.,  as  the 
co-ordinates  at  t,  this  way  and  these  velocities  are  such  that  the 
system  unguided  will  move  accordingly  if  only  started  with 
proper  velocities  from  the  initial  configuration. 


Varying  329.  From  this  principle  of  stationary  action,  founded,  as 
action.  have  seen,  on  a comparison  between  a natural  motion,  and 

any  other  motion,  arbitrarily  guided  and  subject  only  to  the 
law  of  energy,  the  initial  and  final  configurations  of  the 
system  being  the  same  in  each  case,  Hamilton  passes  to  the 
consideration  of  the  variation  of  the  action  in  a natural  or 
unguided  motion  of  the  system  produced  by  varying  the  initial 
and  final  configurations,  and  the  sum  df  the  potential  and 
kinetic  energies.  The  result  is,  that 


DYNAMICAL  LAWS  AND  PRINCIPLES. 


330.] 


SiS 


330.  The  rate  of  decrease  of  the  action  per  unit  of  increase  Varyin?? 
of  any  one  of  the  free  (generalized)  co-ordinates  (§  204)  speci- 
fying the  initial  configuration,  is  equal  to  the  correspond- 
ing (generalized)  component  momentum  [§  313,  (c)]  of  the 
actual  motion  from  that  configuration : the  rate  of  increase  of 
the  action  per  unit  increase  of  any  one  of  the  free  co-ordi- 
nates specifying  the  final  configuration,  is  equal  to  the  corre- 
sponding component  momentum  of  the  actual  motion  towards 
this  second  configuration  : and  the  rate  of  increase  of  the  action 
per  unit  increase  of  the  constant  sum  of  the  potential  and  kinetic 
energies,  is  equal  to  the  time  occupied  by  the  motion  of  which 
the  action  is  reckoned. 

To  prove  this  we  must,  in  our  previous  expression  (9)  for  hA, 
now  suppose  the  terminal  co-ordinates  to  vary;  hT  to  become 
8^  - 8 F,  in  which  '^E  is  a constant  during  the  motion ; and  each 
set  of  paths  and  velocities  to  belong  to  an  unguided  motion  of 
the  system,  which  requires  (10)  to  hold.  Hence 

8d  - { 2m  {xhx  + yZy  + ^8.2;)}  - [2m  (x'^x  + yhy  -h  ;s8;2;)]  + thE . ..(13). 

If,  now,  in  the  first  place,  we  supjiose  the  particles  constituting 
the  system  to  be  all  free  from  constraint,  and  therefore  (x,  y,  z) 
for  each  to  be  three  independent  variables,  and  if,  for  distinctness, 
we  denote  by  (a?/,  y/,  z^')  and  (x^,  y^,  z^)  the  co-ordinates  of 
in  its  initial  and  final  positions,  and  by  {x^,  y^^  z^'),  (cc^,  z^) 

the  components  of  the  velocity  it  has  at  those  points,  we  have, 
from  the  preceding,  according  to  the  ordinary  notation  of  partial 
differential  coefficients. 


Action 


as  a func- 
tion of 
initial  and 
final  co- 
ordinates 
and  the 
energy; 


dA 

dA 

dA  , 1 

w: 

= - 

= - mi,  , etc. 
dz^  ‘ ‘ 

dA 

dA 

dA  . , 

--  =m,a:,, 
dx  * * 

dy, 

and 


dA_ 


v,..(14). 


In  these  equations  we  must  suppose  A to  be  expressed  as  a func- 
tion of  the  initial  and  final  co-ordinates,  in  all  six  times  as  many 
independent  variables  as  there  are  of  particles  ; and  E,  one  more 
variable,  the  sum  of  the  potential  and  kinetic  energies. 

If  the  system  consist  not  of  free  particles,  but  of  particles  con- 
nected in  any  way  forming  cither  one  rigid  body  or  any  number 


its  diffe- 
rential co- 
efficients 
equal  re- 
spectively 
to  initial 
and  final 
momen- 
tums,  and 
to  the  time 
from  be- 
ginning to 
end. 


344 


PRELIMINARY. 


[330. 


Varying 

action. 


Same  pro- 
positions 
for  gene- 
ralized co- 
ordinates. 


of  rigid  bodies  connected  with  one  another  or  not,  we  might,  it  is 
true,  be  contented  to  regard  it  still  as  a system  of  free  particles, 
by  taking  into  account  among  the  impressed  forces,  the  forces 
necessary  to  compel  the  satisfaction  of  the  conditions  of  con- 
nexion. But  although  this  method  of  dealing  with  a system  of 
connected  particles  is  very  simple,  so  far  as  the  law  of  energy 
merely  is  concerned,  Lagrange’s  methods,  whether  that  of  “equa- 
tions of  condition,”  or,  what  for  our  present  purposes  is  much 
more  convenient,  his  “generalized  co-ordinates,”  relieve  us  from 
very  troublesome  interpretations  when  we  have  to  consider  the 
displacements  of  particles  due  to  arbitrary  variations  in  the  con- 
figuration of  a system. 

Let  us  suppose  then,  for  any  particular  configuration  [x^,  y^,  z^) 
(^2>  ^2)  • • •>  expression 


{xfix^  -f  -I-  etc.,  to  become  + etc.  (15), 


when  transformed  into  terms  of  {}/,  0...,  generalized  co-ordi- 

nates, as  many  in  number  as  there  are  of  degrees  of  freedom  for 
the  system  to  move  [§  313,  (c)]. 

The  same  transformation  applied  to  the  kinetic  energy  of  the 
system  would  obviously  give 

J {x^  + -f-  z^)  -f-  etc.  = J {^ij/  + rj(j>  + ^0  + etc.) (16), 

and  hence  r],  etc.,  are  those  linear  functions  of  the  generalized 
velocities  which,  in  § 313  (e),  we  have  designated  as  “gene- 
ralized components  of  momentum ; ” and  which,  when  the 
kinetic  energy,  is  expressed  as  a quadratic  function  of  the  velo- 
cities (of  course  with,  in  general,  functions  of  the  co-ordinates 
if/,  <f>,  6,  etc.,  for  the  coefficients)  are  derivable  from  it  thus  : 


etc, 


(17). 


Hence,  taking  as  before  non-accented  letters  for  the  second,  and 
accented  letters  for  the  initial,  configurations  of  tlie  system  re- 
spectively, we  have 


dA 

dA 

dA  ,,  , 1 

• d^'  — ^’ 

drff=-^’^^-  j 

dA  ^ 

dA 

dA  5.  , 1 

dif/  ■’ 

dA 

before. 

J 

(18). 


DYNAMICAL  LAWS  AND  PKINCIPLES. 


345 


830.] 


These  equations  (18),  including  of  course  (14)  as  a particular  case,  Varying 
express  in  mathematical  terms  the  proposition  stated  in  words 
above,  as  the  Principle  of  Varying  Action. 

The  values  of  the  momentums,  thus,  (14)  and  (18),  expressed 
in  terms  of  differential  coefficients  of  ff,  must  of  course  satisfy 
the  equation  of  energy.  Hence,  for  the  case  of  free  particles. 


1 dAf 

m\dx^  dy‘^  d'^ , 


f (dAf  dA^  dA\ 
m \dx"‘  ^ dy'^  ^ dz'') 


2(^-  V) 
V) 


(19) , 

(20) . 


Hamilton’s 
“ character- 
istic equa- 
tion” of 
motion  in 
Cartesian 
co-ordi- 
nates. 


Or,  in  general,  for  a system  of  particles  or  rigid  bodies  connected 
in  any  way,  we  have,  (16)  and  (18), 


. dA  .dA^dA  ^ ^ --- 

dA  .,dA  A,  dA 


Hamilton’s 
(21)  character- 

' •'*  istic  equa- 

tion of 
motion  in 
generalized 


( dA.  • ^ dA  dA  \ c\  / TA  IT '\  /oo\  generalii 

-(^  = ref- 


where  \j/,  <j>,  etc.,  are  expressible  as  linear  functions  of 
etc.,  by  the  solution  of  the  equations 

dA  'I 

(‘A,  »A) ‘A  + (‘A,  <A) 0 + ('A,  + = ^ I 

dA 

(<A,  »A)  ‘A  + (<A,  <A)  <A  + (<A,  ^ ^ 


dA  M 
dxj;  ’ d<f>  ’ 


(23), 


etc. 


etc. 


and  ij/',  cj>\  etc.,  as  similar  functions  of  — 


dA 

d\f 


dA 


etc , by 


(i/'',  \p')  if/'  + {if/',  (f>)  <j)'  + (if/',  O')  6'  + etc.  = 
(<A',  ‘A')  4^'  + (^',  'A')  <A'  + (<A',  ^')  + etc.  =7}' 


(U' 

d\j/' 

dA  '^..(24), 
dcf>  I 


etc. 


etc. 


j 


where  it  must  be  remembered  that  (if/,  if/),  {\f/,  cf>),  etc.,  are  func- 
tions of  the  specifying  elements,  ifr,  <f>,  0,  etc.,  depending  on  the 
kinematical  nature  of  the  co-ordinate  system  alone,  and  quite 
independent  of  the  dynamical  problem  with  which  we  are  now 
concerned  ; being  the  coefficients  of  the  half  squares  and  the 
])roducts  of  the  generalized  velocities  in  the  expression  for  the 


346 


PRELIMINARY. 


Varying 

action. 


Proof  that 
the  charac- 
teristic 
equation 
defines  the 
motion,  for 
free 

particles. 


[330. 

kinetic  energy  of  any  motion  of  the  system ; and  that  (i}/\ 

{if/',  etc.,  are  the  same  functions  with  ij/',  <f>',  etc.,  written  for 
ij/,  cf>,  0,  etc. ; but,  on  the  other  hand,  that  .4  is  a function  of  all  the 
elements  xf/,  <^,  etc.,  xj/',  etc.  Thus  the  first  member  of  (21) 

is  a quadratic  function  of  ^ , etc.,  with  coeflBcients, 

dxf/  aqi 

known  functions  of  xf/,  cf),  etc.,  depending  merely  on  the  kine- 
matical  relations  of  the  system,  and  the  masses  of  its  parts,  but 
not  at  all  on  the  actual  forces  or  motions ; while  the  second 
member  is  a function  of  the  co-ordinates  xf/,  <f>,  etc.,  depending 
on  the  forces  in  the  dynamical  problem,  and  a constant  expressing 
the  particular  value  given  to  the  sum  of  the  potential  and  kinetic 
energies  in  the  actual  motion ; and  so  for  (22),  and  xf/',  (f>,  etc. 

It  is  remarkable  that  the  single  linear  partial  differential  equa- 
tion (19)  of  the  first  order  and  second  degree,  for  the  case  of 
free  particles,  or  its  equivalent  (21),  is  sufficient  to  determine  a 
function  A,  such  that  the  equations  (14)  or  (18)  express  the  mo- 
mentums  in  an  actual  motion  of  the  system,  subject  to  the  given 
forces.  For,  taking  the  case  of  free  particles  first,  and  differen- 
tiating (19)  still  on  the  Hamiltonian  understanding  that  A is 
expressed  merely  as  a function  of  initial  and  final  co-ordinates, 
and  of  E,  the  sum  of  the  potential  and  kinetic  energies,  we  have 


2:s 


1 /dA  d\i  dA  d^A 


\dx  dx^dx  ^ dy  dx^dy  ^ dz  dx^dz) 


dV 
^ dx^  ' 


But,  by  (14), 


and  therefore 
d^A  dx, 
dx^ 


J_dA 
m,  dx^ 

d^A 


X. 


1 dA 


= 3/u 


dx^dy^  "^^dx^ 


■’  dy^ 

dy^  dx,  d?A 


dz. 


dx. 


d^A 


= m, 


dx^ 

dx. 


’ dy^  ’ dx^dz^ 
dx. 


dx^  dz^ 


dx^dx^  ^ dx^  ^ dx^ 


etc. 


Using  these  properly  in  the  preceding  and  taking  half ; and 
writing  out  for  two  particles  to  avoid  confusion  as  to  the  mean- 
ing of  we  have 

dx,  dx,  . dE  . dx.  , dx.  ,dx  \ dV  . . 

Now  if  we  multiply  the  first  member  by  dt,  we  have  clearly  the 
change  of  the  value  of  due  to  varying,  still  on  the  Hamil- 


DYNAMICAL  LAWS  AND  PRINCIPLES. 


347 


380.' 


tonian  supposition,  the  co-ordinates  of  all  the  points,  that  is  to  say, 
the  configuration  of  the  system,  from  what  it  is  at  any  moment  to 
what  it  becomes  at  a time  dt  later ; and  it  is  therefore  the  actual 
change  in  the  value  of  in  the  natural  motion,  from  the  time, 
t,  when  the  configuration  is  {x^,  x^^  ...,  E),  to  the  time 

t -h  dt.  It  is  therefore  equal  to  m^x^dt,  and  hence  (25)  becomes 
dV 

simply  m^x^  = - — . Similarly  we  find 
(too 


Varying 

aciion. 

Proof  that, 
the  charac- 
teristic 
equation 
defines  the 
motion,  for 
free 

particles. 


dV  dV  dV  ^ 


dx„ 


But  these  are  [§  293,  (4)]  the  elementary  differential  equations 
of  the  motions  of  a conservative  system  composed  of  free  mutually 
influencing  particles. 

If  next  we  regard  x^,  z^,  x^,  etc.,  as  constant,  and  go 

through  precisely  the  same  process  with  reference  to  x^',  y^,  x^, 

etc.,  we  have  exactly  the  same  equations  among  the  accented 
letters,  with  only  the  diffei  ence  that  - A appears  in  place  of  A ; 
, dV' 


and  end  with  m^x^ 


dx/ 


from  which  we  infer  that,  if  (20) 


is  satisfied,  the  motion  represented  by  (14)  is  a natural  motion 
through  the  configuration  [x^',  y^',  2;/,  xj,  etc.). 

Hence  if  both  (19)  and  (20)  are  satisfied,  and  if  when  x^  = x^', 

dA 


2/1=2/,. 


etc.,  we 


1.  dA 

have  -r-  = 
dx^ 


dxy 


etc.,  the 


motion  represented  by  (14)  is  a natural  motion  through  the 
two  configurations  [x^,  y^,  x^,  etc.),  and  {x^,  y^,  z^,  x^^ 

etc.).  Although  the  signs  in  the  preceding  expressions  have  been 
fixed  on  tlie  supposition  that  the  motion  is  from  the  former,  to  the 
latter  configuration,  it  may  clearly  be  from  either  towards  the 
other,  since  whichever  way  it  is,  the  reverse  is  also  a natural 
motion  (§  271),  according  to  the  general  property  of  a conserva- 
tive system. 

To  [irove  the  same  thing  for  a conservative  system  of  particles  samepro- 
or  rigid  bodies  connected  in  any  way,  we  have,  in  the  first  place, 
from  (18) 

dif/  d<f>  ’ dif/  dO  ’ 


etc. 


connected 
system,  and 
generalized 
/9C\  co-ordi- 
nate3. 


where,  on  the  Hamiltonian  principle,  we  suppose  ij/,  <^,  etc.,  and 
rf,  etc.,  to  be  expressed  as  functions  of  xj/,  (f>,  etc.,  xj/',  <f)',  etc., 


MS 


PRELIMINARY. 


[830. 


Vaiyhig 

actiuu. 


and  the  sum  of  the  potential  and  kinetic  energies.  On  the  same 
supposition,  differentiating  (21),  we  have 


. . dri  . ddf  dcL  ^dO  , 


But,  by  (26),  and  by  the  considerations  above,  we  have 

di 


. d$  . dn  A di  ^ 


d^  dO 


^;^  + etc.  = ^....(28), 


where  ^ denotes  the  rate  of  variation  of  i per  unit  of  time  in  the 
actual  motion. 

Again,  we  have 
dij/ 


__ 

dij^  d^  dij;  drjdxl,  dilf  I 

d<^  00  d^  00  dr]  00 

di{/  d$  dxj/  ^ dr]  di}/  ^ ^ dif/ 

etc.  etc. 


(29\ 


if,  as  in  Hamilton’s  system  of  canonical  equations  of  motion,  we 
suppose  0,  0,  etc.,  to  be  expressed  as  linear  functions  of  77,  etc., 
with  coefficients  involving  0,  0,  etc.,  and  if  we  take  0 to  denote 
the  partial  differentiation  of  these  functions  with  reference  to  the 
system  r],...\l/,  0,...,  regarded  as  independent  variables.  Let 
the  coefficients  be  denoted  by  [0,  0],  etc.,  according  to  the  plan 
followed  above;  so  that,  if  the  formula  for  the  kinetic  energy  be 

^=i  {[‘Aj  0]  f + [<A.  <A]  ’y"  + • • • + 2 [0,  0]  it]  + etc.} (30), 

we  have 

. dT  1 

i = j 

0 = ^ »A]  ^ + [<^5  <A]  ’y  + [<f>,  + etc.  ^ 

etc.  etc. 

where  of  course  [0,  0],  and  [0,  0],  mean  the  same. 


Hence 


and  therefore,  by  (29), 


. dxj/  d^  y dO 


330.]  DYNAMICAL  LAWS  AND  PRINCIPLES.  349 

whence,  by  (28),  we  see  that 

.dxj/  d<f>  ^dO  : . 

+ + ^-j-  + etc.  = $ + 2 — (32). 

dij/  ' dij/  dij/  dij/  ' ' 

pj  yr 

This,  and  (28),  reduce  the  first  member  of  (27)  to  2^+2  — , 
and  therefore,  halving,  we  conclude 

^dT  dv  ......  ar  c/r  ^ 


These,  in  all  as  many  difierential  equations  as  there  are  of  vari- 
ables, xj/,  cf>,  etc.,  suffice  for  determining  them  in  terms  of  t and 
twice  as  many  arbitrary  constants.  But  every  solution  of  the 
dynamical  problem,  as  has  been  demonstrated  above,  satisfies 
(21)  and  (23) ; and  therefore  it  must  satisfy  these  (33),  which  we 
liave  derived  from  them.  These  (33)  are  therefore  the  equations 
of  motion,  of  the  system  referred  to  generalized  co-ordinates,  as 
many  in  number  as  it  has  of  degrees  of  freedom.  They  are  the 
Hamiltonian  explicit  equations  of  motion,  of  which  a direct  de- 
monstration was  given  in  § 318  above.  Just  as  above,  it  appears 
therefore,  that  if  (21)  and  (22)  are  satisfied,  (18)  expresses  a 
natural  motion  of  the  system  from  one  to  another  of  the  two  con- 
figurations (»//,  cfi',  6',...).  Hence