VIII. 
THE  PRINCIPLE  OF    LEAST  ACTION. 

Theorig-  I.    Maupertuis  enunciated,   in   1747,   a  principle 

scurefortn  which  he  Called  **  le  principe  de  la  moindre  quantity  d'ac- 

dpie*o?""  iion,^^  the  principle  of  least  action.     He  declared  this 

'  principle  to  be  one  which  eminently  accorded  with  the 

wisdom  of  the  Creator.     He  took  as  the  measure  of 

the  "action"  the  product  of  the  mass,  the  velocity, 

and  the  space  described,  or  m%)s,      Why^  it  must  be 

confessed,  is  not  clear.    By  mass  and  velocity  definite 

quantities  may  be  understood ;  not  so,  however,  by 


THE  EXTENSION  OF  THE  PRINCIPLES.         365 

space,  when  the  time  is  not  stated  in  which  the  space 
is  described.  If,  however,  unit  of  time  be  meant,  the 
distinction  of  space  and  velocity  in  the  examples  treated 
by  Maupertuis  is,  to  say  the  least,  peculiar.  It  appears 
that  Maupertuis  reached  this  obscure  expression  by  an 
unclear  mingling  of  his  ideas  of  vis  viva  and  the  prin- 
ciple of  virtual  velocities.  Its  indistinctness  will  be 
more  saliently  displayed  by  the  details. 

2.  Let  us  see  how  Maupertuis  applies  his  principle.  Determina- 

If  My  m  be  two  inelastic  masses,  Cand  c  their  velocities  laws  of  im- 
pact by  this 

before  impact,  and  u  their  common  velocity  after  im-  principle, 
pact,  Maupertuis  requires,  (putting  here  velocities  for 
spaces,)  that  the  "action"  expended  in  the  change  of 
the  velocities  in  impact  shall  be  a  minimum.  Hence, 
M{C  —  u)^  '\-  m{c  —  «)  2  is  a  minimum \  that  is, 
m\c—  »)  +  w  (r  —  «)  ==  0 ;  or 

MC  -\-  mc 

For  the  impact  of  elastic  masses,  retaining  the  same 
designations,  only  substituting  Kand  v  for  the  two  ve- 
locities after  impact,  the  expression  M{C —  ^)^  + 
in{c — ^^7^)2  is  a  minimum;  that  is  to  say, 

M{^C—V)dV^m{c  —  v^dv  =  ^ (1) 

In  consideration  of  the  fact  that  the  velocity  of  ap- 
proach before  impact  is  equal  to  the  velocity  of  reces- 
sion after  impact,  we  have 

C—c  =  —  (^V—v)  or 

C-^  r-(r  +  rO  =  0 (2) 

and 

d  V—  dv  =  0 (3) 

The  combination  of  equations  (i),  (2),  and  (3) 
readily  gives  the  familiar  expressions  for  ^  and  v. 
These  two  cases  may,  as  we  see,  be  viewed  as  pro- 


366  THE  SCIENCE  OF  MECHANICS, 

cesses  in  which  the  least  change  of  vis  viva  by  reaction 
takes  place,  that  is,  in  which  the  Uast  counter-work  is 
done.     They  fall,   therefore,   under  the  principle  of 
Gauss. 
Maupcr-  3.   Peculiar  is  Maupertuis's  deduction  of  the  law  of 

tuis;sde-  •'  .  _. 

ductionof  the  lever.     Two  masses  M  and  m  (Fig.  188)  rest  on  a 

the  law  of  .  .  . 

the  lever  by  bar  <z,  which  the  fulcrum  divides  into  the  portions 

this  pnn-  1  1       • 

cipie.  X  and  a  —  jc.  If  the  bar  be  set  in  rotation^  the  veloci- 
ties and  the  spaces  described  will  be  proportional  to 
the  lengths  of  the  lever-arms,  and  Mx^  •\-  m{a  —  x)'^ 
is  the  quantity  to  be  made  a  minimum,  that  is  Mx  — 
m{a  —  jc)  =  0  ;  whence  x  =  majM  +  /«, — a  condition 

that  in  the  case  of  equilib- 

, rium  is  actually  fulfilled.  In 

^' • -w       criticism  of  this,  it  is  to  be 

^  ^_^  remarked,  first,  that  masses 

Fig.  188.  not    subject  to  gravity   or 

other  forces,  as  Maupertuis 
here  tacitly  assumes,  are  always  in  equilibrium,  and, 
secondly,  that  the  inference  from  Maupertuis*s  deduc- 
tion is  that  the  principle  of  least  action  is  fulfilled 
only  in  the  case  of  equilibrium,  a  conclusion  which  it 
was  certainly  not  the  author's  intention  to  demonstrate. 
The  correc-  If  it  Were  sought  to  bring  this  treatment  into  ap- 
pertuis's  proximatc  accord  with  the  preceding,  we  should  have 
to  assume  that  the  heavy  masses  M  and  m  constantly 
produced  in  each  other  during  the  process  the  least 
possible  change  of  vis  viva.  On  that  supposition,  we 
should  get,  designating  the  arms  of  the  lever  briefly  by 
tf,  b^  the  velocities  acquired  in  unit  of  time  by  u^  v,  and 
the  acceleration  of  gravity  by  g,  as  our  minimum  ex- 
pression, M{g —  «)2  -|-  m{g —  v)^\  whence  Af{g —  «) 
du  +  m(^g —  v)dv  =  0.  But  in  view  of  the  connection 
of  the  masses  as  lever, 


deduction. 


THE  EXTENSION  OF  THE  PRINCIPLES. 


367 


u 
a 

du 


7' 


whence  these  equations  correctly  follow 

Afa  —  md  Afa  —  mfi 

and  for  the  case  of  equilibrium,  where  u  =  v  =  0, 

Ma  —  mb  =  0. 

Thus,  this  deduction  also,  when  we  come  to  rectify 
it,  leads  to  Gauss's  principle. 

4.   Following  the  precedent  of  Fermat  and  Leib-  Trcatmeni 

of  the  mo- 

nitz,  Maupertuis  also  treats  by  his  method  the  motion rAonoiw^x 

by  the  pnn- 

of  light.     Here  again,  however,  cfpie  of 

•  he  employs  the  notion  *'  least  ac- 
tion "  in  a  totally  different  sense. 
The   expression   which   for    the 
case  of  refraction  shall  be  a  min- 
imum,    is    m  ,  AR    +    «  •  RB^ 
where  AR  and  RB  denote  the 
paths  described  by  the  light  in 
the  first  and  second  media  re- 
spectively,   and    m    and   n   the    corresponding   velo- 
cities.    True,  we  really  do  obtain  here,  if  R  be  de- 
termined in  conformity  with  the  minimum  condition, 
the  result  sma /sin  ft  =  n/m  =^  const.     But  before,  the 
* '  action  "  consisted  in  the  change  of  the  expressions 
mass  X  velocity  X  distance ;  now,  however,  it  is  con- 
stituted of  the  sum  of  these  expressions.     Before,  the 
spaces  described  in  unit  of  time  were  considered  ;  in 
the  present  case  the  total  spaces  traversed  are  taken. 
Should  not  m,  AR  —  n.  RB  or  [m  —  n){AR  —  RB) 
be  taken  as  a  minimum,  and  if  not,  why  not?     But 


Fig.  189. 


368  THE  SCIENCE  OF  MECHANICS. 

even  if  we  accept  Maupertuis's  conception,  the  recip- 
rocal values  of  the  velocities  of  the  light  are  obtained, 
and  not  the  actual  values, 
characteri-        It  wiU  thus  be  Seen  that  Maupertuis  really  had  no 
Manper-     principle,  properly  speaking,  but  only  a  vague  form- 
cipie.         ula,  which  was  forced  to  do  duty  as  the  expression  of 
different  familiar  phenomena  not  really  brought  under 
one  conception.     I  have  found  it  necessary  to  enter 
into  some  detail  in  this  matter,  since  Maupertuis's  per- 
formance, though  it  has  been  unfavorably  criticised  by 
all  mathematicians,  is,  nevertheless,  still  invested  with 
a  sort  of  historical  halo.     It  would  seem  almost  as  if 
something  of  the  pious  faith  of  the  church  had  crept 
into  mechanics.     However,  the  mere  endeavor  to  gain 
a  more  extensive  view,  although  beyond  the  powers  of 
the  author,  was  not  altogether  without  results.    Euler, 
at  least,  if  not  also  Gauss,  was  stimulated  by  the  at- 
tempt of  Maupertuis. 
Euier'scon-        5.   Euler's  vicw  is,   that  the  purposes  of  the  phe- 

tributions  tt       ■%  j  •  r 

to  this  sub-  nomena  of  nature  afford  as  good  a  basis  of  explana- 

ject.  , 

tion  as  their  causes.  If  this  position  be  taken,  it  will 
be  presumed  a  priori  that  all  natural  phenomena  pre- 
sent a  maximum  or  minimum.  Of  what  character  this 
maximum  or  minimum  is,  can  hardly  be  ascertained 
by  metaphysical  speculations.  But  in  the  solution  of 
mechanical  problems  by  the  ordinary  methods,  it  is 
possible,  if  the  requisite  attention  be  bestowed  on  the 
matter,  to  find  the  expression  which  in  all  cases  is 
made  a  maximum  or  a  minimum.  Euler  is  thus  not 
led  astray  by  any  metaphysical  propensities,  and  pro- 
ceeds much  more  scientifically  than  Maupertuis.  He 
seeks  an  expression  whose  variation  put  =  0  gives  the 
ordinary  equations  of  mechanics. 

For  a  single  body  moving  under  the  action  of  forces 


THE  EXTENSION  OF  THE  PRINCIPLES.  369 

Euler  finds  the  requisite  expression  in  the  formula  The  form 

which  the 

Cv  dsy  where  ds  denotes  the  element  of  the  path  and  principle 
V  the  corresponding  velocity.  This  expression  is  sm  aller  Euier's 
for  the  path  actually  taken  than  for  any  other  infinitely 
adjacent  neighboring  path  between  the  same  initial 
and  terminal  points,  which  the  body  may  be  constrained 
to  take.  Conversely,  therefore,  by  seeking  the  path  that 
makes  Cv  ds  a  minimum,  we  can  also  determine  the 
path.  The  problem  of  minimising  Cv  d's  is,  of  course, 
as  Euler  assumed,  a  permissible  one,  only  when  v  de- 
pends on  the  position  of  the  elements  ds,  that  is  to 
say,  when  the  principle  of  vis  viva  holds  for  the  forces, 
or  a  force-function  exists,  or  what  is  the  same  thing, 
when  Z7  is  a  simple  function  of  coordinates.  For  a  mo- 
tion in  a  plane  the  expression  would  accordingly  as- 
sume the  form 


^  .  ,  c^y 


In  the  simplest  cases  Euler 's  principle  is  easily  veri- 
fied.    If  no  forces  act,  v  is  constant,  and  the  curve  of 
motion  becomes  a  straight  line,  for  which     Cvds^=^ 
V  C  ds  is  unquestionably  shorter  than  for  any  other 
curve  between  the  same  terminal  points. 
Also,  a  body  moving  on  a  curved  surface 
without  the  action  of  forces  or  friction, 
preserves  its  velocity,  and  describes  on 
the  surface  a  shortest  line. 

The  consideration  of  the  motion  of  a       |  \^      euIct's 

projectile  in  a  parabola  ^^C  (Fig.  190)  n^     SppulS^to 

will  also  show  that  the  quantity  Jv  ds    d  AC  S^Trofec- 

is  smaller  for  the  parabola  than  for  any  „.  ****' 

Other  neighboring  curve ;  smaller,  even, 
than  for  the  straight  line  ABC  between  the  same  ter- 
minal points.  The  velocity,  here,  depends  solely  on  the 


370  THE  SCIENCE  OF  MECHANICS, 

Mathemat-  vertical  space  described  by  the  body,  and  is  therefore 

ical  dcvel-  j  -j  * 

opmentof  the  Same  for  all  curves  whose  altitude  above  OC  is  the 
same.  If  we  divide  the  curves  by  a  system  of  horizontal 
straight  lines  into  elements  which  severally  correspond, 
the  elements  to  be  multiplied*  by  the  same  t^'s,  though 
in  the  upper  portions  smaller  for  the  straight  line  AD 
than  for  AB,  are  in  the  lower  portions  just  the  reverse ; 
and  as  it  is  here  that  the  larger  v^s  come  into  play,  the 
sum  upon  the  whole  is  smaller  iox  ABC  than  for  the 
straight  line. 

Putting  the  origin  of  the  codrdinates  at  Ay  reckon- 
ing the  abscissas  x  vertically  downwards  as  positive, 
and  calling  the  ordinates  perpendicular  thereto  y^  we 
obtain  for  the  expression  to  be  minimised 

0 

where  g  denotes  the  acceleration  of  gravity  and  a  the 
distance  of  descent  corresponding  to  the  initial  velocity. 
As  the  condition  of  minimum  the  calculus  of  variations 
gives 


V 


=  C  or 


4 


' + m 


dy 


dx  ^  —^  -' °' 


Vlg\a  +  *)  -  C» 

/Cdx 
l/2g(a  +  x)-^^C^ 
and,  ultimately, 


y  = 


y=-\/2g(a-^x)—C^^C\ 


THE  EXTENSION  OF  THE  PRINCIPLES,         371 

where  C  and  C*  denote  constants  of  integration  that 
pass  into  C=^V  7,ga  and  C=  0,  if  for  jt  =  0,  dx/dy  =  0 
and  j^  =  0  be  taken.  Therefore,  y  =  21/ ax.  By  this 
method,  accordingly,  the  path  of  a  projectile  is  shown 
to  be  of  parabolic  form. 

6.-  Subsequently,  Lagrange  drew  express  attention  The  addi- 
to  the  fact  that  Euler's  principle  is  applicable  only  in  Rrange  and 
cases  in  which  the  principle  of  vis  viva  holds.  Jacobi 
pointed  out  that  we  cannot  assert  that  Cv  ds  for  the  ac- 
tual motion  is  a  minimum^  but  simply  that  the  variation  of 
this  expression,  in  its  passage  fo  an  infinitely  adjacent 
neighboring  path,  is  =  0.  Generally,  indeed,  this  con- 
dition coincides  with  a  maximum  or  minimum,  but  it 
is  possible  that  it  should  occur  without  such ;  and  the 
minimum  property  in  particular  is  subject  to  certain 
limitations.  For  example,  if  a  body,  constrained  to 
move  on  a  spherical  surface,  is  set  in  motion  by  some 
impulse,  it  will  describe  a  great  circle,  generally  a 
shortest  line.  But  if  the  length  of  the  arc  described 
exceeds  1 80°,  it  is  easily  demonstrated  that  there  exist 
shorter  infinitely  adjacent  neighboring  paths  between 
the  terminal  points. 

7.   So  far,  then,  this  fact  only  has  been  pointed  out,  Baler's, 
that  the  ordinary  equations  of  motion  are  obtained  by  but  one  of 
equating  the  variation  of   Cv  ds  to  zero.    But  since  the  give^he  * 
properties  of  the  motion  of  bodies  or  of  their  paths  may  oJmoUon. 
always  be  defined  by  differential  expressions  equated 
to  zero,  and  since  furthermore  the  condition  that  the 
variation  of  an  integral  expression  shall  be  equal  to 
zero  is  likewise  given  by  differential  expressions  equated 
to  zero,  unquestionably  various  other  integral  expres- 
sions may  be  devised  that  give  by  variation  the  ordi- 
nary equations  of  motion,  without  its  following  that  the 


372  THE  SCIENCE  OF  MECHANICS, 

integral  expressions  in  question  must  possess  on  that 
account  any  particular  physical  significance. 
Yet  the  ex-         8.  The  Striking  fact  remains,  however,  that  so  simple 
must  pos-    an  expression  as  fr  ds  does  possess  the  property  men- 
icai  import,  tioued,  and  we  will  now  endeavor  to  ascertain  its  phys- 
ical import.     To  this  end  the  analogies  that  exist  be- 
tween the  motion  of  masses  and  the  motion  of  light,  as 
well  as  between  the  motion  of  masses  and  the  equilib- 
rium of  strings — analogies  noted  by  John  Bernoulli 
and  by  Mobius — will  stand  us  in  stead. 

A  body  on  which  'ho  forces  act,  and  which  there- 
fore preserves  its  velocity  and  direction  constant,  de- 
scribes a  straight  line.  A  ray  of  light  passing  through 
a  homogeneous  medium  (one  having  everywhere  the 
same  index  of  refraction)  describes  a  straight  line.  A 
string,  acted  on  by  forces  at  its  extremities  only,  as- 
sumes the  shape  of  a  straight  line. 
Elucidation       A  body  that  moves  in  a  curved  path  from  a  point 

Of  this  im-  •  r»  «         1  1        • 

port  by  the  ^  to  a  point  B  and  whose  velocity  7f  =^  fl>(^,  y,  z)  is  a 

motion  of  a  .  ,  , 

mass,  the    function  of  Coordinates,  describes  between  A  and  B  a 

motion  of  a  ,  •    i  n        /•       ,     •  •     •  a 

ray  of  light,  curve  f or  which  generally   H'  as  is  a  minimum.    A  ray 
eauiiibrium  of  light  passing  from  A  to  B  describes  the  same  curve, 
if  the  refractive  index  of  its  medium,  «  =  ^(jc,  y,  z), 
is  the  same  function  of  coordinates ;  and  in  this  case 
fnds  is  a  minimum.     Finally,  a  string  passing  from 
A  to  B  will  assume  this  curve,   if  its  tension  S  = 
<P  (p^f  y*  ^)  is  the  same  above-mentioned  function  of  co- 
ordinates ;  and  for  this  case,  also,    CSds  is  a  minimum. 
The  motion  of  a  mass  may  be  readily  deduced  from 
the  equilibrium  of  a  stringy  as  follows.     On  an  element 
ds  oidL.  string,  at  its  two  extremities,  the  tensions  S,  S' 
act,  and  supposing  the  force  on  unit  of  length  to  be  P^ 
in  addition  a  force  P.  ds.     These  three  forces,  which 
we  shall  represent  in  magnitude  and  direction  by  BA, 


THE  EXTENSION  OF  THE  PRINCIPLES. 


373 


BC^  BD  (Fig.  191),  are  in  equilibrium.  If  now,  a  body,  The  moUon 
with  a  velocity  ly  represented  in  magnitude  and  diiec-  deduced 
tion  by  AB^  enter  the  element  of  the  path  ds^  and  re-  eauiiibriun 
ceive  within  the  same  the  velocity  component  BF  = 
—  BD,  the  body  will  proceed  on- 
ward with  the  velocity  v'  =  BC. 
Let   Q  be   an  accelerating  force 
whose  action  is  directly  opposite 
to  that  of  B;  then  for  unit  of  time 
the  acceleration  of  this  force  ^11 
be  Q,  for  unit  of  length  of  the 
string  Q/v,  and  for  the  element 
of  the  string  (Q/v)ds,  The  body  will  move,  therefore, 
in  the  curzfe  of  the  string,  if  we  establish  between  the 
forces  P  and  the  tensions  S,  in  the  case  of  the  string, 
and  the  accelerating  forces  Q  and  the  velocity  v  in  the 
case  of  the  mass,  the  relation 


Fig.  Z9Z. 


V 

The  minus  sign  indicates  that  the  directions  of  P  and 
Q  are  opposite. 

A  closed  circular  string  is  in  equilibrium  when  be-  The  eqm- 

,  librinm  ox 

tween  the  tension  S  of  the  strmg,  everywhere  constant,  closed 

strings. 

and  the  force  P  falling  radially  outwards  on  unit  of 
length,  the  relation  P  =  Sjr  obtains,  where  r  is  the 
radius  of  the  circle.  A  body  will  move  with  the  con- 
stant velocity  v  in  a  circle,  when  between  the  velocity 
and  the  accelerating  force  Q  acting  radially  inwards 
the  relation 

Q 


7f 


71 


a 


=   -  or  ^  =   -   obtains. 
V        r  r 

A  body  will  move  with  constant  velocity  v  in  any  curve 
when  an  accelerating  force  Qz=v^/r  constantly  acts 


374 


THE  SCIENCE  OF  MECHANICS. 


on  it  in  the  direction  of  the  centre  of  curvature  of  each 
element.  A  string  will  lie  under  a  constant  tension  S 
in  any  curve  if  a  force  P  =  Sjr  acting  outwardly  from 
the  centre  of  curvature  of  the  element  is  impressed  on 
unit  of  length  of  the  string. 
The  deduc*  No  concept  aualogous  to  that  of  force  is  applicable 
motion  of    to  the  tnotioTi  of  light.     Consequently,  the  deduction  of 

light  froni 

the  motions  the  motion  of  light  from  the  equilibrium  of  a  string  or 

of  masses  , 

and  the,     the  motion  of  a  mass  must  be  differently  effected.     A 

equilibnnm  ,  ,  , 

of  strings,    mass,  let  us  say,  is  moving  with  the  velocity  AB  =  v. 

(Fig.  192.)  A  force  in  the  direction 
B£>  is  impressed  on  the  mass  which 
produces  an  increase  of  velocity  B£, 
so  that  by  the  composition  of  the  ve- 
locities BC  =  AB  and  B£  the  new 
velocity  BP  =  v'  is  produced.  If  we 
resolve  the  velocities  v,  v*  into  com- 
ponents parallel  and  perpendicular  to 
the  force  in  question,  we  shall  per- 
ceive that  the  parallel  components  alone 
are  changed  by  the  action  of  the  force. 
This  being  the  case,  we  get,  denoting 
by  k  the  perpendicular  component,  and  by  a  and  c^ 
the  angles  v  and  v*  make  with  the  direction  of  the 


Fig.  193. 


force. 


k^=v  sin  a 
k  =  7/  sin  a'  or 


sm^ 
sin  a* 


V 
V 


If,  now,  we  picture  to  ourselves  a  ray  of  light  that 
penetrates  in  the  direction  of  v  a  refracting  plane  at 
right  angles  to  the  direction  of  action  of  the  force,  and 
thus  passes  from  a  medium  having  the  index  of  refrac- 


THE  EXTENSION  OF  THE  PRINCIPLES,         375 
tion  n  into  a  medium  having  the  index  of  refraction  ri^  Develop- 

,  mentor  this 

where  «/«'  =  7^7^',  this  ray  of  light  will  describe  the  illustration, 
same  path  as  the  body  in  the  case  above.  If,  there- 
fore, we  wish  to  imitate  the  motion  of  a  mass  by  the 
motion  of  a  ray  of  light  (in  the  same  curve),  we  must 
everywhere  put  the  indices  of  refraction,  «,  proportional 
to  the  velocities.  To  deduce  the  indices  of  refraction 
from  the  forces,  we  obtain  for  the  velocity 


a 


//  I -^  1  =  Pdq,  and 
for  the  index  of  refraction,  by  analogy. 


where  P  denotes  the  force  and  dg  a  distance-element 
in  the  direction  of  the  force.  If  ds  is  the  element  of 
the  path  and  a  the  angle  made  by  it  with  the  direction 
of  the  force,  we  have  then 

d[  (,-]  =  Pcosa.  ds 

(n^\ 
d\      ]  =  Pcosa.  ds. 

For  the  path  of  a  projectile,  under  the  conditions  above 
assumed,  we  obtained  the  expression y  =  2yax.  This 
same  parabolic  path  will  be  described  by  a  ray  of  light, 
if  the  law  n  ==  \/2g(a  -}-  x)  be  taken  as  the  index  of 
refraction  of  the  medium  in  which  it  travels. 

9.  We  will  now  more  accurately  investigate  the  Relation  of 
manner  in  which  this  minimum  property  is  related  to  mum  prop- 
the/^r»f  of  the  curve.     Let  us  take,  first,  (Fig.  193)  afoimof 
broken  straight  line  ABC,  which  intersects  the  straight 
line  MJV,  put  AB  =  s,  BC  =  s',  and  seek  the  condition 
that  makes  vs  -f  7'V  a  minimum  for  the  line  that  passes 


376 


7'HE  SCIENCE  OF  MECHANICS, 


First,  de- 
duction of 
the  mini- 
mum condi' 
tion. 


through  the  fixed  points  A  and  B^  where  v  and  v'  are 
supposed  to  have  different,  though  constant,  values 
above  and  below  MN,  If  we  displace  the  point  B  an 
infinitely  small  distance  to  D^  the  new  line  through  A 
and  C  will  remain  parallel  to  the  original  one,  as  the 
drawing  symbolically  shows.  The  expression  vs  -\-  v's' 
is  increased  hereby  by  an  amount 

—  timsma  +  v'ms\na\ 

where  m  ^=  DBj  or  by  an  amount  —  v  sin  a  +  7/  sin  or'. 
The  condition  of  the  minimum,  consequently,  is  that 

—  V  sin  a  +  v'  sin  or'  =  0 


s\na 

or  -T      ,= 

smar 


7f 
V 


R       N 


FiR.  193. 


ri«.  19,. 


If  the  expression  sjv  -{■  s  ji^'  is  to  be  made  a  minimum, 
we  have,  in  a  similar  way, 


/• 


smo:  V 

sin «'       V* 

Second,  the        If,  next,  we  Consider  the  case  of  a  string  stretched 

SfoiU^con*?  in  the  direction  ABC,  the  tensions  of  which  .Sand  ^S" 

eauiiibrium  are  different  above  and  below  MN,  in  this  case  it  is 

the  minimum  of  Ss  -f  S*s*  that  is  to  be  dealt  with.   To 

obtain  a  distinct  idea  of  this  case,  we  may  imagine  the 


THE  EXTENSION  OF  THE  PRINCIPLES. 


yn 


motion  of  a 
ray  of  light. 


String  stretched  once  between  A  and  B  and  thrice  be- 
tween B  and  C,  and  finally  a  weight  P  attached.  Then 
S=  P  and  5'  =  3 -P.  If  we  displace  the  point  B  a  dis- 
tance m,  any  diminution  of  the  expression  Ss -{- S's' 
thus  effected,  will  express  the  increase  of  work  which 
the  attached  weight  P  performs.  If  — Sm  sin  a -^ 
S'm  sin  a'  =  0,  no  work  is  performed.  Hence,  the  mini- 
mum of  Ss  -\-  S's'  corresponds  to  a  maximum  of  work. 
In  the  present  case  the  principle  of  least  action  is  sim- 
ply a  different  form  of  the  principle  of  virtual  displace- 
ments. 

Now  suppose  that  ABC  is  a  ray  of  light,  whose  ve-  Third,  the 

application 

locities  V  and  v'  above  and  below  MN  are  to  each  other  of  thi«  con- 
dition to  the 
as  3  to  I.     The  motion  of  light  be- 
tween two  points  A  and  B  is  such 
that  the  light  reaches  ^  in  a  mini- 
mum of  time.     The  physical  reason 
of  this  is  simple.     The  light  travels 
from  A  to  B^   in  the  form    of  ele- 
mentary waves,  by  different  routes. 
Owing  to  the  periodicity  of  the  light, 
the    waves    generally  destroy  each 
other,  and  only  those  that  reach  the 
designated  point  in  equal  times,  that  is,  in  equal  phases, 
produce  a  result.     But  this  is  true  only  of  the  waves 
that  arrive  by  the  minimum  path  and  its  adjacent  neigh- 
boring paths.     Hence,  for  the  path  actually  taken  by 
the  light  s/v  +  j'/p'  is  a  minimum.     And  since  the  in- 
dices of  refraction  n  are  inversely  proportional  to  the 
velocities  v  of  the  light,  therefore  also  ns  ■\-  n*s*  is  a 
minimum. 

In  the  consideration  of  the  motion  of  a  mass  the  con- 
dition that  vs  -\-  v's*  shall  be  a  minimum,  strikes  us  as 
something  novel.   (Fig.  195.)  If  a  mass,  in  its  passage 


Fig-  195- 


378  THE  SCIENCE  OF  MECHANICS, 

Fourth,  it:!  through  a  plane  MN^  receive,  as  the  result  of  the  action 
fo  the  m<^"  of  a  force  impressed  in  the  direction  DB,  an  increase  of 
mass.         velocity,  by  which  v,  its  original  velocity,  is  made  v' y  we 
have  for  the  path  actually  taken  by  the  mass  the  equa- 
tion 7/  sin  a  =  z/'  sin  a'  =  k.    This  equation^  which  is  also 
the  condition  of  minimum,  simply  states  that  only  the  ve- 
locity-component parallel  to  the  direction  of  the  force  is 
altered,  but  that  the  component  k  at  right  angles  thereto  re- 
mains unchanged.    Thus,  in  this  case  also,  Euler's  prin- 
ciple simply  states  a  familiar  fact  in  a  new  form. 
Formofthe        lo.  The  minimum  condition  —  t^sinar+ e^'sina'sTrO 
condition    may  also  be  written,  if  we  pass  from  a  finite  broken 
to  curves.    Straight  line  to  the  elements  of  curves,  in  the  form 

—  V  sin  a  -\-  (y  -\-  dv)  sin(a  +  dd)  =  0 
or 

d(v  sin  «)  =  0 

or,  finally, 

tf  sin  a  =  const. 

In  agreement  with  this,  we  obtain  for  the  motion 

of  light 

d  {n  sin  «)  =  0,  n  sin  a  =  const, 

\      ^'      J  V 

and  for  the  equilibrium  of  a  string 

d{Ssmd)  =  0,  ^sina  =  const. 

To  illustrate  the  preceding  remarks  by  an  ex- 
ample,  let  us  take  the  parabolic  path  of  a  projectile, 
where  a  always  denotes  the  angle  that  the  element  of 
the  path  makes  with  the  perpendicular.  Let  the  ve- 
locity he  7f  =  \/2g(^a  -\-  x),  and  let  the  axis  of  the_y-or- 
dinates  be  horizontal.  The  condition  v.  sin  a  =  const, 
or  V  2g{a  -f-  A*)  .  dy/ds  =  const,  is  identical  with  that 
which  the  calculus  of  variation  gives,  and  we  now  know 


THE  EXTENSION  OF  THE  PRINCIPLES, 


yi^ 


Fig.  xgC 


its  xww^/^/^w/Va/ significance.  If  we  picture  to  ourselves  niustration 

.  ,  T        .  / >-  of  the  three 

a  string  whose  tension  is  5  =  k  2r  («  +  x\  an  arrange-  typical 

...  caaea  by 

ment  which  might  be  effected  by  fixing  frictionless  curvilinear 

.  .        motiona. 

pulleys  on  horizontal  parallel  rods  placed  in  a  vertical 
plane,  then  passing  the  string  through  these  a  sufficient 
number  of  times,  and  finally  attaching 
a  weight  to  the  extremity  of  the  string, 
we  shall  obtain  again,  for  equilibrium, 
the  preceding  condition,  the  phys- 
ical significance  of  which  is  now  ob- 
vious. When  the  distances  between 
the  rods  are  made  infinitely  small  the 
string  assumes  the  parabolic  form. 
In  a  medium,  the  refractive  index  of 
which  varies  in  the  vertical  direction 
by  the  law  n  =  \^2g{a  +  x),  or  the  velocity  of  light  in 
which  similarly  varies  by  the  law  v  =  \/\/'2g{a  +  x)^ 
a  ray  of  light  will  describe  a  path  which  is  a  parabola. 
If  we  should  make  the  velocity  in  such  a  medium 
V  =  \/2g{a-\-x),  the  ray  would  describe  a  cycloidal  path, 
for  which,  not  CV2g{a  +  x).  ds,  but  the  expression 
Cds/\/2g{a  +  x)  would  be  a  minimum. 

II.   In  comparing  the  equilibrium  of  a  string  with 
the  motion  of  a  mass,  we  may  employ  in  place  of  a 
string  wound  round  pulleys, 
a  simple  homogeneous  cord, 
provided  we  subject  the  cord 
to  an  appropriate  system   of 
forces.     We  readily    observe 
that    the    systems    of    forces 
that   make    the    tension,    or, 
as  the  case  may  be,   the  ve- 
locity,  the  same  function    of   coordinates,   are   differ- 
ent.   If  we  consider,  for  example,  the  force  of  gravity. 


Fig.  197. 


38o  THE  SCIENCE  OF  MECHANICS. 


The  condi-  V  =  V igia  +  x\     A  String,  howcver,  subjected  to  the 

tionsand  ...  -  -x  •  t 

conse-        action  of  gravity,  forms  a  catenary,  the  tension   ot 
Oiepreced-  which  IS  given  by  the  formula  S  =  m  —  nx^  where  m 

ing  analo- 
gies- and  n  are  constants.     The  analogy  subsisting  between 

the  equilibrium  of  a  string  and  the  motion  of  a  mass  is 
substantially  conditioned  by  the  fact  that  for  a  string 
subjected  to  the  action  of  forces  possessing  a  force- 
function  U,  there  obtains  in  the  case  of  equilibrium 
the  easily  demonstrable  equation  U  -\-  S=  const.  This 
physical  interpretation  of  the  principle  of  least  action 
is  here  illustrated  only  for  simple  cases ;  but  it  may 
also  be  applied  to  cases  of  greater  complexity,  by 
imagining  groups  of  surfaces  of  equal  tension,  of  equal 
velocity,  or  equally  refractive  indices  constructed  which 
divide  the  string,  the  path  of  the  motion,  or  the  path 
of  the  light  into  elements,  and  by  making  a  in  such  a 
case  represent  the  angle  which  these  elements  make 
with  the  respective  surface- normals.  The  principle  of 
least  action  was  extended  to  systems  of  masses  by  La- 
grange, who  presented  it  in  the  form 

62m  Czfds  =  0. 

If  we  reflect  that  the  principle  of  vis  viva,  which  is  the 
real  foundation  of  the  principle  of  least  action,  is  not 
annulled  by  the  connection  of  the  masses,  we  shall 
comprehend  that  the  latter  principle  is  in  this  case  also 
valid  and  physically  intelligible. 

IX. 

Hamilton's  principle. 

I.  It  was  above  remarked  that  various  expressions 
can  be  devised  whose  variations  equated  to  zero  give 
the  ordinary  equations  of  motion.  An  expression  of 
this  kind  is  contained  in  Hamilton's  principle 


THE  EXTENSION  OF  THE  PRINCIPLES.  381 

S  f(C/'+  T)  dt  =  0,  or  The  points 

J  ^         '  ^  '  of  identity 

'0  of  Hamil- 

tx  ton's  and 

/q  ciples. 

where  <^C/'and  c^T' denote  the  variations  of  the  work 
and  the  vis  viva^  vanishing  for  the  initial  and  terminal 
epochs.  Hamilton's  principle  is  easily  deduced  from 
D'Alembert's,  and,  conversely,  D'Alembert's  from 
Hamilton's ;  the  two  are  in  fact  identical,  their  differ- 
ence being  merely  that  of  form.  * 

2.  We  shall  not  enter  here  into  any  extended  in-  Hamilton's 

.    .  pnncipie 

vestigation  of  this  subject,  but  simply  exhibit  the  iden-  applied  to 

°  .       .     •*        '  '^  •*  the  motion 

tity  of  the  two  principles  by  an  example —  of  a  wheel 

the  same  that  served  to  illustrate  the  prin- 
ciple  of  D'Alembert :  the  motion  of  a  wheel 
and  axle  by  the  over-weight  of  one  of  its 
parts.    In  place  of  the  actual  motion,  we 
may  imagine,  performed  in  the  same  inter- 
val of  time,  a  different  motion,  varying  in- 
finitely little  from  the  actual  motion,  but         p.    ^^ 
coinciding  exactly  with  it  at  the  beginning 
and  end.     There  are  thus  produced  in  every  element 
of  time  dtf  variations  of  the  work  (SU)  and  of  the  vis 
viva  (^ST);  variations,  that  is,  of  the  values  C/'and  T 
realised  in  the  actual  motion.     But  for  the  actual  mo- 
tion, the  integral  expression,  above  stated,  is  =  0,  and 
may  be  employed,  therefore,  to  determine  the  actual 
motion.     If  the  angle  of  rotation  performed  varies  in 
the  element  of  time  di  an  amount  a  from  the  angle  of 
the  actual  motion,  the  variation  of  the  work  corre- 
sponding to  such  an  alteration  will  be 

6C/=  {PR  —Qr)a  =  Ma. 

*  Compare,  for  example,  Kirchhoff,  VarUtungtn  Mber  matkematitche  Pky- 
siky  Meekanik,  p.  aj  et  teqg.^  and  Jacobi,  VorUsungtn  Mhtr  DynamiA,  p.  58. 


382  THE  SCIENCE  OF  MECHANICS. 

Mathemat-  The  vis  viva,  for  any  given  angular  velocity  a?,  is 

ical  devel- 
opment of  ,^         1    X  ^  ^^  ^  00)^ 
tfiiscase.                                          T=^    -  {PR'^  +  ^/-a)  --, 

and  for  a  variation  dco  of  this  velocity  the  variation  of 
the  vis  viva  is 

But  if  the  angle  of  rotation  varies  in  the  element  <//  an 
amount  a, 

dcD:^--  and 

a/ 

The  form  of  the  integral  expression,  accordingly,  is 


fUa  +  N'^'' 


<//  =  0. 


/( 


But  as 

d  ,j.^   .       dN  da 

therefore, 

'o 

The  second  term  of  the  left-hand  member,  though, 
drops  out,  because,  by  hypothesis,  at  the  beginning 
and  end  of  the  motion  a  =  0.     Accordingly,  we  have 

'o 

an  expression  which,  since  a  in  every  element  of  time 
is  arbitrary,  cannot  subsist  unless  generally 

--  '1= »■ 


THE  EXTENSION  OF  THE  PRINCIPLES.  383 

Substituting  for  the  symbols  the  values  they  represent, 
we  obtain  the  familiar  equation 

//«_    PR—Qr 
d't  '^  PR^  ^r  Qr^^' 
D'Alembert*s  principle  gives  the  equation  The  same 

results  ob- 
'^       (iN\  .  Uinedby 

/  bert's  prin- 

which  holds  for  every  possible  displacement.  We  might,  *^*^*®' 
in  the  converse  order,. have  started  from  this  equation, 
have  thence  passed  to  the  expression 


/(^_^)«^,  =  0. 


dt 

to 

and,  finally,  from  the  latter  proceeded  to  the  same  re- 
sult 


/( 


f(Ma  +  JSr^]  di  =  0. 


d^ 

to 

3.  As  a  second  and  more  simple  example  let  us  ninstration 
consider  the  motion  of  vertical  descent.  .For  every  Sy  the  rnc^ 
infinitely  small  displacement  s  the  equation  subsists  ticai  de- 

SCCDt 

[ntg — m(^dv/diy]s  =  0f  in  which  the  letters  retain 
their  conventional  significance.  Consequently,  this 
equation  obtains 

tt 

w  VT  ]s .di  =  0, 

to 

which,  as  the  result  of  the  relations 


'1 
f[mg  - 


.(mvs)  dv     ,         ds       ^ 


384  THE  SCIENCE  OF  MECHANICS. 


J.  fit  =  (m  7'  s)    =  0, 


provided  ^  vanishes  at  both  limits,  passes  into  the  form 

mgs  +  mv  ,\tf/=  0, 


to     ^ 
that  is,  into  the  form  of  Hamilton's  principle. 

Thus,  through  all  the  apparent  differences  of  the 
mechanical  principles  a  common  fundamental  same- 
ness is  seen.  These  principles  are  not  the  expression 
of  different  facts,  but,  in  a  measure,  are  simply  views 
of  different  aspects  of  the  same  fact.