The endpoint action function of a smooth family of extremals has differential $dW=pdq-Hdt$

Let $n\ge 1$, let $J\subseteq \mathbb{R}$ and $U\subseteq {\mathbb{R}}^n$ be open and nonempty, and let $\mathcal{L}:J\times U\times {\mathbb{R}}^n\to \mathbb{R}$ be a fibre-full $C^2$ time-dependent Lagrangian chart that is hyperregular, with inverse Legendre branch $v$ and Hamiltonian $H(t,q,p)=p\cdot v(t,q,p)-\mathcal{L}(t,q,v(t,q,p))$ (The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity, A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations). Fix $t_0\in J$ and $q_0\in U$, let $T\subseteq J\setminus \{t_0\}$ be open and nonempty, and let $Q\subseteq U$ be open and nonempty, let $D\subseteq T\times Q\times J$ be open, and let $X:D\to U$ be a $C^2$ map such that for every $(t,q)\in T\times Q$ the points $(t,q,t)$ and $(t,q,t_0)$ lie in $D$, the closed interval with endpoints $t_0$ and $t$ is contained in the $s$-section $D_{t,q}:=\{s\in J:(t,q,s)\in D\}$, which is required to be an open interval, and:

1. $X(t,q,t)=q$ and $X(t,q,t_0)=q_0$;
2. $s\mapsto X(t,q,s)$ is a solution of the Euler–Lagrange equation of $\mathcal{L}$ on the open interval $D_{t,q}$, i.e. $(s,X(t,q,s),{\mathrm{\partial }}_{s}X(t,q,s))\in J\times U\times {\mathbb{R}}^n$ and $E_{\mathcal{L}}(X(t,q,\cdot ))=0$ on $D_{t,q}$.

Define

$p(t,q):={\mathcal{L}}_v(t,q,{\mathrm{\partial }}_{s}X(t,q,t)),W(t,q):={\int }_{t_0}^t\mathcal{L}(s,X(t,q,s),{\mathrm{\partial }}_{s}X(t,q,s))ds((t,q)\in T\times Q)\mathrm{.}$

Then $W$ is of class $C^1$ on $T\times Q$ and

$W_q(t,q)=p(t,q),W_t(t,q)=-H(t,q,p(t,q))((t,q)\in T\times Q)\mathrm{.}$

• The two identity clauses determine the first variation of $W$ completely: $dW=pdq-Hdt$ at every point of $T\times Q$, where $p$ is the momentum of the selected extremal. This is a conditional statement: it presumes the $C^2$ family $X$ of extremals with the endpoint property, and its proof never asserts that such a family exists.

• The existence of families with the required endpoint property is the subject of A smooth local family of extremals with prescribed endpoint exists when the endpoint momentum map has invertible derivative, where the hypothesis used is invertibility of the derivative of the endpoint map with respect to the initial momentum. Several extremals may share the same endpoint data while a selected branch still forms a single-valued $C^2$ family. A singular derivative of the endpoint map prevents applying that inverse-function argument, but does not by itself prove that no such family exists. The present theorem applies to every given family satisfying its hypotheses.

• The identities $W_q=p$ and $W_t=-H(t,q,W_q)$ are the Hamilton–Jacobi relations for the endpoint action on the chart. The second follows from the first by substitution, since $p=W_q$; the theorem proves the pair directly.