1
$\begingroup$

Let the action functional $S[q]$ given by \begin{equation}\label{eq16} S[q]=\int\limits^{t_2}_{t_1}L\left(q^i(t),\dot{q}^i(t)\right)dt.\tag{1} \end{equation} Also, we know that using Legendre Transform the hamiltonian $H(q^i,p_i)$ is related with $L(q^i,\dot{q}^i)$ by \begin{equation} L(q^i,\dot{q}^i)=p_i q^i - H(q^i,p_i) \quad \text{with} \quad p_i=\frac{\partial L}{\partial \dot{q}^i}\tag{2} \end{equation} Thus, replacing this last equation inside the action function we have \begin{align}\label{eq105} S(q,t)&=\int\limits^t_{t_0}p_i(t)\dot{q}^i(t)dt-\int\limits^t_{t_0}H(q^i(t),p_i(t))dt\tag{3}\\ S(q,t)&=\int\limits^{q(t)}_{q(t_0)}p_i(t)dq^i-\int\limits^t_{t_0}H(q^i(t),p_i(t))dt.\tag{4} \end{align} Finally, we also know the diferential form of $S(q,t)$ is by definition \begin{equation}\label{eq110} dS(q^i,t)=\frac{\partial S(q^i,t)}{\partial q^i}dq^i + \frac{\partial S(q^i,t)}{\partial t}dt\tag{5} \end{equation} which give us these following relations \begin{equation} p_i=\frac{\partial S(q^i,t)}{\partial q^i} \quad \text{and} \quad -H(q^i,p_i)=\frac{\partial S(q^i,t)}{\partial q^i}.\tag{6} \end{equation} Replacing $p_i$ relation inside $H$ give us the Hamilton-Jacobi equation \begin{equation}\label{eq116} \frac{\partial S(q^i,t)}{\partial t}+H\left(q^i,\frac{\partial S(q^i,t)}{\partial q^i}\right)=0.\tag{7} \end{equation}

Question When working with action functional we already know that physical motions are those curves that are the an extremum of $S[q]$. But here we actually wrote $S[q]$ as a function $S(q,t)$ of $(q,t)$ and this give us Hamilton-Jacobi equation. My question is:

  • Every solution of Hamilton-Jacobi equation is an extremum of $S[q]$?

  • Also, there's an relation (even conceptually) between extremum of $S[q]$ and general solutions of Hamilton-Jacobi equation?

$\endgroup$
0
$\begingroup$
  1. Eq. (1) is the off-shell action functional $S[q]$.

  2. Eqs. (3)-(4) are presumably the (Dirichlet) on-shell action function $S(q_f,t_f;q_i,t_i)$. It satisfies eqs. (5)-(6), which are proven in a Lemma of my Phys.SE answer here.

  3. Hamilton's principal function $S(q,\alpha,t)$ is the solution to Hamilton-Jacobi equation (7).

OP's main questions (v2) seems a bit like asking to compare apples and oranges. Presumably they want to ask about relationships between the 3 above objects denoted with the same letter $S$. This is explained in my Phys.SE answer here.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.