A problem in deriving the Hamilton Jacobi equation from a variational principle
as I've already said, I have a problem in understanding a reasoning from which we derive the Hamilton Jacobi equation from a variational principle. Let's take the <PERSON> functional:
$$ S = \int_{t_0}^{t_1} [ p_{\alpha}\dot{q}^{\alpha} - H(q^{\alpha},p_{\alpha},t) ] dt $$
The first variation on the phase space of this functional is, in the most general form:
$$ (\delta S){\bar{\gamma}} = [p\alpha \delta q^{\alpha} - H \delta t]{t_0}^{t_1} + \int{t_0}^{t_1} \left{ \left[ \dot{q}^{\alpha} - \frac{\partial H}{\partial p_\alpha} \right]{\bar{\gamma}}\pi\alpha - \left[ \dot{p}{\alpha} + \frac{\partial H}{\partial q^\alpha} \right]{\bar{\gamma}}\eta_\alpha \right} dt $$
Where the variation of the functional S is evaluated on the deformation of the curve ${\bar{\gamma}} \to \gamma$ in the phase space:
$$ \bar{\gamma} : \begin{cases} q^{\alpha} = \overline{q}^{\alpha}(t) \ p_{\alpha} = \overline{p}{\alpha}(t) \ A = { \overline{q}^{\alpha}(t_0);\overline{p}{\alpha}(t_0) } \ B = { \overline{q}^{\alpha}(t_1);\overline{p}{\alpha}(t_1) } \ \end{cases} \qquad t \in [t_0,t_1] $$ $$ \gamma : \begin{cases} q^{\alpha} = \overline{q}^{\alpha}(t) + \lambda\eta{\alpha}(t) \ p_{\alpha} = \overline{p}{\alpha}(t) + \lambda\pi{\alpha}(t)\ A' = { \overline{q}^{\alpha}(t_0 + \lambda \delta t_0) + \lambda \eta_{\alpha}(t_0 + \lambda \delta t_0) ; \overline{p}{\alpha}(t_0 + \lambda \delta t_0) + \lambda \pi{\alpha}(t_0 + \lambda \delta t_0) } \ B' = { \overline{q}^{\alpha}(t_1 + \lambda \delta t_1)+\lambda \eta_{\alpha}(t_1 + \lambda \delta t_1); \overline{p}{\alpha}(t_1 + \lambda \delta t_1) + \lambda \pi{\alpha}(t_1 + \lambda \delta t_1) } \ \end{cases} \qquad t \in [t_0 +\lambda \delta t_0,t_1 +\lambda \delta t_1] $$
Where $\eta_{\alpha}$ and $\pi_{\alpha}$ are regular function. Now, in my notes we choose $\bar{\gamma}$ e we let $A=A'$, so that we have an initial fixed point. 955
Then we say that on the curve chosen, are satisfied the <PERSON> equation, so that the variation of S becomes only:
$$ (\delta S){\bar{\gamma}} = [p\alpha \delta q^{\alpha} - H \delta t]_{t_0}^{t_1} $$
[First question Is this legit ? If the <PERSON> equation are derived from the same variational principle, can we say ''a priori'' that them are valid on a particular path on the phase space ? ]
Then we consider the point B movable, so that it depends from time. In this way, S isn't a functional anymore, but instead is a function of time. So the variation can be interpreted as a differential:
$$ dS= p_\alpha d q^{\alpha} - H d t $$
[Second question I wish to have a mathematical proof for that, because for me isn't trivial as it sounds. 346