This note is a companion to the cornerstone manuscript. It expands the content of Section 5 there into a self-contained treatment with sharper hypotheses and a worked model.

1. Motivation

The cornerstone manuscript (Section 5) introduces weak stationarity, mollifier probing, and corner/impulse conditions as Propositions P3.1–P3.4. Those statements are sufficient for the structural chain developed there, but they compress the hypotheses and omit worked computations. This satellite note serves three purposes:

State the mollifier localization result as a formal theorem with explicit, numbered hypotheses (Section 2).
Work through a complete model — the delta-kick free particle — showing trajectory, momentum jump, and action evaluation in full detail (Section 4).
Separate two superficially similar but logically distinct uses of the Dirac delta in variational mechanics (Section 5).

2. Mollifier Localization Theorem

We work on a time interval $[t_i,t_f]$ with Lagrangian $\mathcal{L}(q,\dot{q},t)$ and candidate trajectory $q:[t_i,t_f]\to\mathbb{R}^d$.

Theorem 2.1 (Mollifier localization of the Euler–Lagrange equation). Assume:

(H1) $q\in C^1([t_i,t_f];\mathbb{R}^d)$ and $\mathcal{L}$ is $C^2$ in $(q,\dot{q})$ and $C^0$ in $t$.

(H2) The first variation satisfies $\delta S[q;\eta]=0$ for every $\eta\in C_c^\infty((t_i,t_f);\mathbb{R}^d)$.

(H3) The Euler–Lagrange expression $F[q](t):=\frac{\partial\mathcal{L}}{\partial q}(q,\dot{q},t) -\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial\dot{q}}(q,\dot{q},t)$ is continuous at a point $t_0\in(t_i,t_f)$.

Then $F[q](t_0)=0$.

Proof. Fix a nonnegative mollifier $\rho\in C_c^\infty(\mathbb{R})$ with $\int\rho=1$ and set $\rho_\varepsilon(s)=\varepsilon^{-1}\rho(s/\varepsilon)$. For any unit vector $u\in\mathbb{R}^d$, the test variation $\eta_\varepsilon(t)=\rho_\varepsilon(t-t_0)\,u$ is in $C_c^\infty$ for $\varepsilon$ small enough. By (H2): $0=\delta S[q;\eta_\varepsilon]=\int_{t_i}^{t_f}F[q](t)\cdot\rho_\varepsilon(t-t_0)\,u\,dt =u\cdot\int_{t_i}^{t_f}\rho_\varepsilon(t-t_0)\,F[q](t)\,dt.$ By (H3) the convolution converges to $F[q](t_0)$ as $\varepsilon\to0^+$. Since $u$ is arbitrary, $F[q](t_0)=0$. $\square$

Remark 2.2 (Role of each hypothesis). (H1) ensures $F[q]$ is locally integrable so the distributional pairing makes sense [Hormander2003]. (H2) is the global stationarity input. (H3) is the local regularity gate: without it, mollifier limits may fail to converge or may converge to an averaged value rather than a pointwise one. If $F[q]$ is continuous on all of $(t_i,t_f)$, iteration of Theorem 2.1 recovers the classical Euler–Lagrange equation everywhere.

Remark 2.2a (Regularity chain: from a.e. to pointwise to smooth). Theorem 2.1 is the mollifier-specific form of the du Bois-Reymond lemma (1879): for $f\in L^1_{\mathrm{loc}}$, vanishing against all $\eta\in C_c^\infty$ gives $f=0$ only a.e. Hypothesis (H3) upgrades the conclusion to pointwise: continuity at $t_0$ is the minimal gate. If $\mathcal{L}$ is smooth, Hilbert’s differentiability theorem further bootstraps $q\in C^1$ to $q\in C^\infty$ by iterating the Euler–Lagrange equation — so (H3) is genuinely the weakest hypothesis for pointwise recovery.

Remark 2.3 (Structural parallel with polygonal refinement). The mollifier-localization argument of Theorem 2.1 shares its logical structure with Newton’s polygonal construction (cornerstone, Section 3): both begin with an invariant that holds exactly at finite resolution (stationarity against every test function / equal areas at every polygon step), introduce a refinement parameter ($\varepsilon$ / $\Delta t$), and extract a continuous or pointwise statement in the limit under a local regularity hypothesis (continuity of $F[q]$ / controlled vertex convergence; see [Nauenberg2003KeplerArea] for the polygon limit). The equal-area law is algebraically exact at every polygon step; the stationarity integral vanishes exactly at every mollifier width. In the time-slicing bridge of Section 4.5, the same pattern recurs a third time: each segment’s normalization must be chosen so that the $N\to\infty$ limit yields a well-defined composition law.

3. Corners and Impulses: Formal Statements

When hypothesis (H3) fails — because $\dot{q}$ or external forcing is discontinuous — two distinct situations arise.

3.1 Corners (unforced velocity jump)

Theorem 3.1 (Corner condition / Weierstrass–Erdmann). Assume $q$ is piecewise $C^2$ with a single velocity discontinuity at $t_0$, satisfying the unforced Euler–Lagrange equation on $(t_i,t_0)$ and $(t_0,t_f)$ separately. Then the canonical momentum is continuous at $t_0$: $\left[\frac{\partial\mathcal{L}}{\partial\dot{q}}\right]_{t_0^-}^{t_0^+}=0.$

Proof. Integrate the Euler–Lagrange equation over $[t_0-\varepsilon,t_0+\varepsilon]$. The integral of $\partial_q\mathcal{L}$ vanishes as $\varepsilon\to0$ by boundedness; the derivative term yields the momentum jump. $\square$

Remark 3.1b (Energy condition — Weierstrass–Erdmann second condition). Theorem 3.1 follows from test variations with fixed support. Allowing variations that shift the corner time $t_0$ yields a second condition: the Hamiltonian $H=(\partial_{\dot{q}}\mathcal{L})\cdot\dot{q}-\mathcal{L}$ is continuous at the corner, $\left[\frac{\partial\mathcal{L}}{\partial\dot{q}}\cdot\dot{q}-\mathcal{L}\right]_{t_0^-}^{t_0^+}=0.$ Together, Theorem 3.1 (momentum continuity) and this energy condition constitute the classical Weierstrass–Erdmann corner conditions. For autonomous $\mathcal{L}$, the second condition reduces to energy conservation across the junction.

Example 3.1a (No corner for a free particle). For a free particle ($\mathcal{L}=\tfrac{m}{2}\dot{q}^2$) with a piecewise-linear trajectory and velocities $v_-$, $v_+$ on either side of $t_0$, Theorem 3.1 demands $mv_+=mv_-$, so $v_+=v_-$ — the trajectory is a single straight line and no genuine corner exists. In contrast, the impulse condition of Section 3.2 permits $v_+\neq v_-$ with jump $v_+-v_-=J/m$. Corners are the homogeneous case of the impulse condition: the momentum is continuous (soft junction), while an impulse breaks that continuity (hard junction).

3.2 Impulses (delta forcing)

Theorem 3.2 (Impulse jump condition). Consider the forced distributional equation $\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial\dot{q}} -\frac{\partial\mathcal{L}}{\partial q} =J\,\delta(t-t_0), \quad J\in\mathbb{R}^d.$ If $\partial_{\dot{q}}\mathcal{L}$ has one-sided limits at $t_0$, then $\frac{\partial\mathcal{L}}{\partial\dot{q}}(t_0^+) -\frac{\partial\mathcal{L}}{\partial\dot{q}}(t_0^-) =J.$

Proof. Same integration argument: the delta integrates to $J$, the smooth remainder vanishes. $\square$

The distinction is structural: corners arise from variational boundary conditions (matching at a junction), while impulses arise from external forcing (a source term in the equation of motion).

Remark 3.2a (Energy jump across an impulse — the impulse-work formula). For the free particle $\mathcal{L}=\frac{m}{2}\dot{q}^2$, the Hamiltonian $H=\frac{m}{2}\dot{q}^2$ jumps by $\Delta H=\frac{m}{2}(v_+^2-v_-^2)=J\cdot\bar{v}$, where $\bar{v}=(v_++v_-)/2$ is the average velocity at the impulse. This is the discrete analog of the power relation $P=F\cdot v$: the impulse does work $J\cdot\bar{v}$ at the mean velocity. Contrast with Remark 3.1b: at an unforced corner $J=0$, so the energy is automatically continuous. The corner/impulse dichotomy thus has matched momentum and energy statements — corners preserve both $p$ and $H$; impulses shift $p$ by $J$ and $H$ by $J\cdot\bar{v}$.

4. Worked Model: Free Particle with a Single Delta-Kick

We give a complete computation that illustrates both Theorem 3.2 and the evaluation of action on a kinked trajectory.

4.1 Setup

Consider a particle of mass $m$ in one dimension with Lagrangian $\mathcal{L}=\frac{m}{2}\dot{q}^2$ and an external impulsive force $J\,\delta(t-t_0)$ applied at time $t_0\in(0,T)$. The equation of motion is $m\ddot{q}=J\,\delta(t-t_0).$

4.2 Solution

The trajectory is piecewise linear: $q(t)=\begin{cases} q_i+v_-\,t & 0\le t<t_0,\\ q_i+v_-\,t_0+v_+\,(t-t_0) & t_0\le t\le T, \end{cases}$ with the velocity jump $v_+-v_-=J/m$ from Theorem 3.2.

Boundary conditions $q(0)=q_i$, $q(T)=q_f$ fix the velocities. Writing $\Delta v=J/m$: $v_-=\frac{q_f-q_i-\Delta v\,(T-t_0)}{T}, \qquad v_+=v_-+\Delta v.$

4.3 Action evaluation

The action splits across the kink: $S=\frac{m}{2}\bigl(v_-^2\,t_0+v_+^2\,(T-t_0)\bigr).$ In the unforced limit ($J=0$, so $\Delta v=0$): $S_0=\frac{m}{2}\frac{(q_f-q_i)^2}{T},$ the standard free-particle result. The impulse adds a positive-definite kinetic energy cost: $S-S_0=\frac{m}{2}\frac{t_0(T-t_0)}{T}\,(\Delta v)^2>0\quad(J\neq0).$ This confirms that the delta-kick raises the action above the free minimum — the impulsive trajectory is not an extremum of the unforced problem.

Remark 4.3a (Midpoint kick maximizes the action excess). The factor $t_0(T-t_0)/T$ achieves its maximum value $T/4$ at the midpoint $t_0=T/2$ and vanishes at the endpoints $t_0\to0$ or $t_0\to T$. Physically, a midpoint kick divides the trajectory into two equal segments, maximizing the “leverage” for kinetic-energy cost; an endpoint kick leaves no time for the velocity perturbation to accumulate path deviation.

Remark 4.3b (Strict minimality of the free path — Jacobi sufficiency). The strong Legendre condition $\partial^2\mathcal{L}/\partial\dot{q}^2=m>0$ holds everywhere. For $\mathcal{L}=\frac{m}{2}\dot{q}^2$, the Jacobi equation is $\ddot{q}_J=0$, whose non-trivial solutions $q_J(t)=At$ (with $q_J(0)=0$) have no further zeros in $(0,T]$ — no conjugate points. Together these give Jacobi’s sufficiency theorem: the straight-line path is a strict weak local minimum of the action. The action excess $S-S_0>0$ of Section 4.3 is a direct manifestation of this sufficiency: every kinked trajectory (an admissible weak competitor) has strictly higher action.

Remark 4.3c (Weierstrass excess function — strong minimality). The Weierstrass excess function for $\mathcal{L}=\frac{m}{2}\dot{q}^2$ is $\mathcal{E}(\dot{q},\dot{q}')=\frac{m}{2}(\dot{q}'-\dot{q})^2\ge 0$, with equality only when $\dot{q}'=\dot{q}$. Combined with the no-conjugate-point condition of Remark 4.3b, this gives Weierstrass’s sufficiency theorem: the straight-line path is a strong local minimum — competitors need only be $C^0$-close, not $C^1$-close. The kinked trajectory of Section 4.3, with its velocity discontinuity, is precisely a strong competitor: $C^0$-close to the straight path but not $C^1$-close. Its positive action excess $S-S_0>0$ is thus a concrete manifestation of Weierstrass sufficiency.

4.4 Angular momentum preservation under central impulses

For a central force in the plane, the impulse is radial: $J=J_r\,\hat{r}$. Decomposing the velocity into radial and transverse components, $\mathbf{v}=v_r\,\hat{r}+r\dot{\theta}\,\hat{\theta}$, the impulse jump of Theorem 3.2 reads $v_r^+-v_r^-=J_r/m,\qquad \dot{\theta}^+=\dot{\theta}^-,$ so the angular momentum $L=mr^2\dot{\theta}$ is unchanged across the kick.

This is precisely the mechanism behind Newton’s proof of the equal-area law (Principia, Proposition 1). A particle moves freely from $A$ to $B$ in time $\Delta t$, sweeping out triangle $SAB$. At $B$, a central impulse directed toward the force center $S$ adds a radial velocity component, deflecting the inertial continuation $Bc$ to $C$ with $cC\parallel BS$ (the parallelogram construction). Since $c$ and $C$ are equidistant from line $SB$, the triangles $SBc$ and $SBC$ have the same area, so $\text{Area}(SAB)=\text{Area}(SBC)$. The construction repeats at every vertex: the swept-area equality is an algebraic identity at each polygon step, exact for any finite number of impulses.

In the language of Section 4.5, a sequence of $N$ central impulses produces a polygon whose area-sweep rate is constant at every step. As $N\to\infty$ and $\Delta t\to0$ (Newton’s Lemma 3 on “ultimate ratios”), the polygon converges to a smooth orbit under a continuous central force, recovering Kepler’s second law [Nauenberg2003KeplerArea]. The distributional impulse-matching of Theorem 3.2 thus provides the modern functional-analytic underpinning of Newton’s original polygonal argument.

Remark 4.4a (Noether charges across impulses: the general criterion). For a Lagrangian admitting a continuous symmetry $\delta q=\varepsilon\,\xi(q)$, the Noether charge $Q=(\partial\mathcal{L}/\partial\dot{q})\cdot\xi$ is conserved on each smooth segment. Across an impulse $J\,\delta(t-t_0)$, Theorem 3.2 gives $\Delta(\partial\mathcal{L}/\partial\dot{q})=J$, and since $q$ is continuous at $t_0$, the charge jump is $\Delta Q=J\cdot\xi(q(t_0))$. The charge is therefore conserved if and only if $J\cdot\xi=0$: the impulse must be orthogonal to the symmetry generator in configuration space. Section 4.4’s angular-momentum preservation under central impulses is the special case $\xi=r\,\hat\theta$, $J=J_r\hat r$, where radial-tangential orthogonality gives $J\cdot\xi=0$ identically.

4.5 From N impulses to the time-sliced path integral

The single-impulse model extends naturally to a sequence of $N$ impulses. This extension bridges the distributional mechanics of Sections 3–4 to the path-integral composition framework of the cornerstone manuscript (Section 6 there).

Partition $[0,T]$ into $N+1$ equal intervals of length $\Delta t=T/(N+1)$, with junction times $t_k=k\,\Delta t$ for $k=1,\ldots,N$. Fix the endpoints $q_0=q_i$, $q_{N+1}=q_f$, and let $q_1,\ldots,q_N$ be free intermediate positions. On each segment the particle is free, so the trajectory is piecewise linear with velocities $v_k=\frac{q_{k+1}-q_k}{\Delta t},\qquad k=0,\ldots,N.$ The discrete action is $S_N[\{q_k\}]=\sum_{k=0}^{N}\frac{m}{2}\frac{(q_{k+1}-q_k)^2}{\Delta t}.$ At each junction $t_k$, the velocity jumps from $v_{k-1}$ to $v_k$. By Theorem 3.2, each jump requires an impulse $J_k=m(v_k-v_{k-1})$. The classical stationary condition $\partial S_N/\partial q_k=0$ imposes $v_k=v_{k-1}$ for all $k$ — that is, Theorem 3.1’s corner condition (momentum continuity) at every junction — and the path collapses to a single straight line.

In the quantum theory, one instead sums over all intermediate configurations with amplitude weights: $K(q_f,q_i;T)=\lim_{N\to\infty}\left(\frac{m}{2\pi i\hbar\,\Delta t}\right)^{(N+1)/2}\int\prod_{k=1}^{N}dq_k\; \exp\!\left(\frac{i}{\hbar}\,S_N[\{q_k\}]\right).$ There are $N+1$ segments and $N$ intermediate integrations; each segment contributes one factor of $\sqrt{m/(2\pi i\hbar\,\Delta t)}$, giving the exponent $(N+1)/2$. This is precisely the half-density normalization required for the composition law to hold at each intermediate integration [BatesWeinstein1997] — a point treated systematically in the cornerstone’s half-density framework. The distributional impulse-matching of Theorem 3.2 thus connects, through this $N\to\infty$ limit [FeynmanHibbs1965], to the composition law for transition amplitudes: the classical limit of the composition law recovers Theorem 3.2’s momentum-continuity condition at each junction.

Remark 4.5a (Exact evaluation for the free particle). For $\mathcal{L}=\frac{m}{2}\dot{q}^2$, each of the $N$ intermediate Gaussian integrals can be performed in closed form. The result is $K(q_f,q_i;T)=\sqrt{m/(2\pi i\hbar T)}\,\exp(iS_0/\hbar)$ with $S_0=m(q_f-q_i)^2/(2T)$, independent of the slicing parameter $N$: each intermediate integration reproduces the same functional form with the segment endpoints merged (the composition law). The prefactor $\sqrt{m/T}$ is the Van Vleck determinant $\sqrt{\det(-\partial^2 S_0/\partial q_f\,\partial q_i)}$, confirming that the half-density normalization of each segment (Section 4.5 above) encodes precisely the information needed for the composition to produce the correct semiclassical prefactor without any extraneous measure choice.

Remark 4.5b (Multi-dimensional propagator and the d/2 exponent). In $d$ spatial dimensions, the free-particle propagator generalizes to $K(\mathbf{q}_f,\mathbf{q}_i;T)=(m/(2\pi i\hbar T))^{d/2}\exp(iS_0/\hbar)$ with $S_0=m|\mathbf{q}_f-\mathbf{q}_i|^2/(2T)$. The Van Vleck determinant is now $\det(-\partial^2 S_0/\partial q_f^i\,\partial q_i^j)=(m/T)^d$, and the prefactor $(m/T)^{d/2}$ is its square root. The exponent $d/2$ is the same one that controls the convergence of the diagonal Green function $G_0(0,0;E)\propto\int_0^\infty t^{-d/2}e^{-\kappa^2 t}\,dt$ (Remark 5.2b): the path-integral normalization and the renormalization threshold for delta potentials share a common analytical root in the $d$-dimensional heat kernel.

Remark 4.5c (Trotter product formula: the operator-theoretic backbone). The $N$-impulse time-slicing is the configuration-space face of the Trotter product formula: each composition step alternates $e^{-iH_0\Delta t/\hbar}$ (free segment) with the impulse operator $e^{-iV\Delta t/\hbar}$ (junction phase kick). The Trotter–Kato theorem provides the strong-operator convergence guarantee for the $N\to\infty$ limit, complementing the pointwise semiclassical argument with a rigorous $L^2$ result. The uncuttable-refinement companion satellite (Remark 3.3: “Trotter product formula as a refinement theorem”) develops this connection systematically.

5. Safe vs Unsafe Uses of the Dirac Delta in Variational Mechanics

The preceding sections involve two related but logically distinct mathematical objects. Conflating them is a common source of error.

5.1 Dirac-supported variations (safe under regularity)

Using mollifier sequences $\rho_\varepsilon\to\delta$ as test functions against a continuous integrand is always safe — it is standard distribution theory. This is Theorem 2.1. No renormalization or regularization ambiguity arises; the $\varepsilon\to0$ limit is unique and controlled by continuity.

Remark 5.1a (Universality of mollifier convergence — the functional-analytic root of "safe"). The limit $\int F[q](t)\,\rho_\varepsilon(t-t_0)\,dt\to F[q](t_0)$ holds for every mollifier kernel $\rho\in C_c^\infty$ with $\int\rho=1$, because the integrand is continuous at $t_0$. This universality is the mathematical content of “safe”: the result is independent of the regularization kernel. By contrast, the diagonal Green function $G_0(0,0;E)$ of Section 5.2 evaluates a singular kernel at coincidence — the result depends on the UV cutoff $\Lambda$, and physical predictions require a renormalization condition to fix the scheme dependence. The safe/unsafe classification of this section thus reduces to a single functional-analytic test: is the functional being evaluated continuous at the relevant point?

5.2 Delta potentials (require renormalization)

A point interaction $V(q)=g\,\delta(q)$ in the Hamiltonian is a different object [AlbeverioGesztesyHoeghKrohnHolden2005]. In dimensions $d\ge2$, the naive coupling constant $g$ requires renormalization (the resolvent acquires a logarithmic or power-law divergence depending on $d$) [Jackiw1991DeltaPotentials]. In $d=1$ the delta potential is well-defined without renormalization, but this is an accident of low dimension, not a general principle. The companion note on delta objects treats the half-density kernel structure of point interactions in detail.

Example 5.2a (Dimensional hierarchy of point interactions). The classification is controlled by the diagonal Green function $G_0(0,0;E)$, which inherits its singularity from the heat kernel exponent $d/2$ (Remark 5.2b below). Concretely: in $d=1$, $G_0(0,0;E)$ is finite, so the rank-one perturbation formula gives a unique bound state at $E=-mg^2/(2\hbar^2)$ without renormalization. In $d=2$, $G_0(0,0;E)\sim\ln(\Lambda/\kappa)$ diverges logarithmically; the bare coupling must run as $g(\Lambda)\sim 2\pi\hbar^2/(m\ln(\Lambda/\alpha))$ to produce a finite amplitude, generating a transmutation scale $\alpha$ (dimensional transmutation) [Jackiw1991DeltaPotentials]. In $d=3$, $G_0(0,0;E)\sim\Lambda$ diverges linearly and the physical parameter is the scattering length $a$. For $d\ge4$, no non-trivial self-adjoint extension exists: $H^2(\mathbb{R}^d)$ functions need not be continuous (Morrey embedding requires $d<4$), so the point is invisible to the Laplacian [AlbeverioGesztesyHoeghKrohnHolden2005].

Remark 5.2a (The self-adjoint extension as a quantum impulse condition). In $d=1$, the delta potential $V=g\,\delta(x)$ defines a self-adjoint extension of $-\hbar^2/(2m)\,d^2/dx^2$ via the matching condition $\psi'(0^+)-\psi'(0^-)=(2mg/\hbar^2)\,\psi(0)$. This is the quantum counterpart of Theorem 3.2’s classical impulse jump $p(t_0^+)-p(t_0^-)=J$: the same integration-across-a-delta argument produces a derivative discontinuity in both cases. The structural difference is that the classical impulse $J$ is externally prescribed, whereas the quantum “impulse” $(2mg/\hbar^2)\,\psi(0)$ depends on the solution itself at the interaction point — the jump condition is a self-consistent boundary condition, not external forcing. This self-consistency is what makes the $d\ge2$ case nontrivial: the wavefunction’s diagonal singularity couples back into the matching condition, requiring renormalization to define the extension.

Remark 5.2b (The exponent $$d/2$$ links path-integral normalization to renormalization thresholds). The connection is made explicit by the Laplace representation: $G_0(0,0;E)\propto\int_0^\infty t^{-d/2}\,e^{-\kappa^2 t}\,dt$, where $\kappa^2=-2mE/\hbar^2$. The short-time singularity $t^{-d/2}$ is the same exponent that normalizes each segment of Section 4.5’s path integral. The integral converges iff $d<2$; at the marginal dimension $d=2$ the coupling acquires scale dependence and dimensional transmutation first appears. Thus the half-density exponent that makes the composition law intrinsic (Section 4.5) simultaneously controls whether a point interaction can be defined without renormalization.

5.3 Summary table

Object	Math status	Renormalization?
Mollifier probe of $F[q]$ (Thm 2.1)	Rigorous	No
Corner/impulse matching (Thms 3.1–3.2)	Rigorous	No
$\delta$ potential, $d=1$	Well-defined	No
$\delta$ potential, $d\ge 2$	Requires care	Yes
Products $\delta(t)^2$	Undefined	Always

Remark 5.3a (Why $$\delta^2$$ is undefined — the Schwartz impossibility). A classical theorem of Schwartz (1954) shows that no associative algebra containing the distributions $\mathcal{D}'(\mathbb{R})$ can extend the pointwise multiplication of continuous functions — in particular, $\delta(t)^2$ has no distributional meaning. This is the mathematical obstruction behind perturbative UV divergences in QFT: loop integrals involve products of propagators $G(x,y)^2$ at coincident points, and the diagonal singularity $G(x,x)\sim t^{-d/2}$ (Remark 5.2b) makes the product undefined in exactly the Schwartz sense. The table’s hierarchy is thus: rows 1–2 involve no products (safe); rows 3–4 involve a single diagonal evaluation (manageable via self-adjoint extensions, Section 5.2); row 5 involves genuine products of singular distributions, requiring the full machinery of renormalization theory.

6. Outlook

The stochastic-forcing interpretation of Section 4.5’s $N$-impulse model — random impulses with prescribed statistics — remains open as a bridge to stochastic mechanics.
Treat the piecewise-smooth trajectory as a weak solution and examine whether the Hamilton–Jacobi equation acquires viscosity-solution structure at the kink.
Connect the corner-condition analysis to broken geodesics in Riemannian geometry (Synge’s world function approach).
The impulse-work formula $J\cdot\bar{v}$ of Remark 3.2a uses the midpoint velocity $\bar{v}=(v_++v_-)/2$. In stochastic calculus, the midpoint (Stratonovich) convention arises from the symmetric limit of discrete approximations, while the pre-point (Itô) convention would give $\Delta H=J\cdot v_-$. The variational principle selects the Stratonovich result; the distinction becomes physical for the random-impulse model of item 1, where the Itô correction is non-vanishing.

References

[Jackiw1991DeltaPotentials] R. Jackiw, “Delta-function potentials in two- and three-dimensional quantum mechanics,” MIT-CTP-1937 (Jan 1991). Reprinted in M.A.B. Bég Memorial Volume (World Scientific, 1991), pp. 25–42. OA mirror: https://www.physics.smu.edu/scalise/P6335fa21/notes/Jackiw.pdf.
[Nauenberg2003KeplerArea] Michael Nauenberg, “Kepler’s Area Law in the Principia: Filling in some details in Newton’s proof of Prop. 1,” Historia Mathematica 30 (2003), 441–456. arXiv:math/0112048. DOI 10.1016/S0315-0860(02)00027-7. (Defends Newton’s continuum limit via Lemma 3; the polygonal construction has a well-defined limit parameterizing a continuous planar curve.)
[BatesWeinstein1997] Sean Bates and Alan Weinstein, “Lectures on the Geometry of Quantization,” Berkeley Mathematics Lecture Notes, vol. 8, AMS, 1997. ISBN 978-0-8218-0798-9. OA: https://math.berkeley.edu/~alanw/GofQ.pdf. (Canonical reference for half-density formalism in geometric quantization; half-density kernels and composition.)
[Hormander2003] Lars Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, 2nd ed., Springer, 2003. DOI 10.1007/978-3-642-61497-2. (Schwartz kernel theorem; distributional calculus for PDE Green functions.)
[AlbeverioGesztesyHoeghKrohnHolden2005] S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, 2nd ed., AMS Chelsea Publishing, 2005. ISBN 978-0-8218-3624-4. (Canonical reference for point interactions in quantum mechanics; self-adjoint extensions, delta potentials.)
[FeynmanHibbs1965] Richard P. Feynman and Albert R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, 1965. (Path integral as refinement limit of time-sliced amplitudes; foundational treatment.)

Object	Math status	Renormalization?
Mollifier probe of \(F[q]\) (Thm 2.1)	Rigorous	No
Corner/impulse matching (Thms 3.1–3.2)	Rigorous	No
\(\delta\) potential, \(d=1\)	Well-defined	No
\(\delta\) potential, \(d\ge 2\)	Requires care	Yes
Products \(\delta(t)^2\)	Undefined	Always