A time-dependent generating function produces canonical time slices and shifts the Hamiltonian by $S_t$

Let $n\ge 1$, let $J\subseteq \mathbb{R}$ be an open interval, let $O_q,O_Q\subseteq {\mathbb{R}}^n$ be nonempty open sets (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement), and let $S:J\times O_q\times O_Q\to \mathbb{R}$ be of class $C^3$ with $\det S_{qQ}(t,q,Q)\mathrm{\ne }0$ for every $(t,q,Q)$ ($C^k$ Euclidean maps and diffeomorphisms). For each $t\in J$ let $S_t:=S(t,\cdot ,\cdot )$ and let $C_t$ be the first-kind map generated by $S_t$ (Generating functions of the first and second kind for a local canonical map). Let $H:J\times V\to \mathbb{R}$ be $C^2$ on a phase domain $V\subseteq {\mathbb{R}}^{2n}$ such that $(q,S_q(t,q,Q))\in V$ for every $(t,q,Q)\in J\times O_q\times O_Q$; thus every Hamiltonian argument in the identity below is in its declared domain. Then:

1. for every $t\in J$ the map $C_t$ is a canonical $C^2$ local diffeomorphism (The local map generated by a first- or second-kind generating function is a canonical $C^2$ local diffeomorphism);
2. on $J\times O_q\times O_Q$, with $p:=S_q(t,q,Q)$, $P:=-S_Q(t,q,Q)$ and $K(t,q,Q):=H(t,q,S_q(t,q,Q))+S_t(t,q,Q)$, the one-form identity $pdq-H(t,q,p)dt=PdQ-Kdt+dS$ holds identically: on the region where both descriptions are defined, the Poincare-Cartan one-forms of $H$ and of $K=H+S_t$ differ by the exact form $dS$, with all quantities read as functions of $(t,q,Q)$.

The words canonical and the finite-regularity form operations here use Finite-regularity differential forms, Hamiltonian fields and canonical maps in phase charts, with the displayed $C^2$ maps; smoothness is not inferred.

• The sign in $K=H+S_t$ is fixed by the convention $P=-S_Q$ of the first-kind generating function and by the identity $pdq-Hdt=PdQ-Kdt+dS$; the opposite sign convention for the generating function flips the sign of $S_t$ together with that of $S$.
• The statement is local in $(q,Q)$ and in $t$; it does not assert propagation of the generating function along the flow, which would be the Hamilton-Jacobi equation, treated in The Hamilton-Jacobi equation and a complete integral of it.