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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 Hamilton 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. Then we say that on the curve chosen, are satisfied the Hamilton 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 Hamilton 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.]

Then we can prove that S is function of $S(q^{\alpha}(t), t ,q^{\alpha}(t_0), t_0 )$, so that: $$ dS = \frac{\partial S}{\partial q^{\alpha}}d q^{\alpha} +\frac{\partial S}{\partial t} dt $$

Equating the two results, we obtain:

$$ \frac{\partial S}{\partial t} + H \left( q^{\alpha} , \frac{\partial S}{\partial q^{\alpha}}, t \right) = 0 $$

Which is the Hamilton-Jacobi equation.

Third question Is this reasoning formally correct ? It doesn't feel quite right to me. And also, more importantly, do you know any book that treats the argument in this way, or similar, that is more rigorous?

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  1. On one hand, Hamilton's principal function $S(q,\alpha,t)$ and Hamilton-Jacobi (HJ) equation$$H(q,p,t)~=~-\frac{\partial S}{\partial t}, \qquad p_j~=~\frac{\partial S}{\partial q^j}, \tag{1}$$ is usually defined via a canonical transformation of type 2. Here $S=F_2$ is a generating function. The new momenta $P_i=\alpha_i$ are the integration constants, and constants of motion. The Kamiltonian $K\equiv 0$ vanishes identically. The total time derivative $$ \frac{dS}{dt}~=~\dot{q}^j\frac{\partial S}{\partial q^j}+\frac{\partial S}{\partial t} ~\stackrel{(1)}{=}~ \dot{q}^jp_j-H~=~L \tag{2} $$ is equal to the Lagrangian $L$ on-shell. As a consequence, the Hamilton's principal function $S(q,\alpha, t)$ can be interpreted as an action on-shell. See also this related Phys.SE post.

  2. On the other hand, the (Dirichlet) on-shell action $S(q_f,t_f;q_i,t_i)$ satisfies $$h_f~=~-\frac{\partial S}{\partial t_f}, \qquad p_f~=~\frac{\partial S}{\partial q_f}.\tag{3} $$ For a proof of eq. (3), see e.g. my Phys.SE answer here.

  3. Eq. (3) looks deceptively like eq. (1). However, the devil is in the details. To lift eq. (3) to eq. (1), there still remains an identification problem of finding the new momenta $P_i=\alpha_i$ in terms the final and initial data $(q_f,t_f;q_i,t_i)$.

  4. Finally let us mention that Caratheodory’s variational method of equivalent Lagrangians can be used to derive the HJ equation in a quite different approach, see Ref. 1. (There is a similar identification issue with this method.)

References:

  1. H.A. Kastrup, Canonical theories of Lagrangian dynamical systems in physics, Phys. Rep. 101 (1983) 1; Section 2.4.
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