Question about the Hamilton Jaccobi Equation

[Zeno Marx]

Hi, I was wondering about the interpretation of the Hamilton Jaccobi equation.

Naively we have H + $\partial$S/$\partial$t = 0 where H is the Hamiltonian and S is the action. But the action is the time integral of the Lagrangian so you would expect $\partial$S/$\partial$t = L

Thus H + L = KE + PE + KE - PE = 2KE = 0

(PE = potential energy, KE = Kinetic energy)

Which simply says that then system is not moving which seems odd for something billed as an equation of motion. Something is clearly very wrong here but what?

 

[vanhees71]

The idea is to find new canonical coordinates $(Q,P)$ of phase space such that all trajectories are described by by $(Q,P)=\text{const}$. Thus you look for the corresponding canonical transformation from the original canonical coordinates $(q,p)$. You can easily show that up to "gauge invariance" this is achieved by the demand that
$$H'(t,Q,P)=0.$$
To that end we introducd the generating function $g(t,q,P)$ for this canonical transformation. The relation between the old and the new quantities is
$$p=\partial_q g, \quad Q=\partial_P g, \quad H'(t,Q,P)=H(t,q,p)+\partial_t g =H \left(t,q,\partial_q g \right)+\partial_t g\stackrel{!}{=}0.$$
This is the Hamilton-Jacobi equation.

For the case of a Hamilton function that is not explicitly dependent on time, the Hamiltonian is conserved along the possible trajectories of the system (energy conservation). Then from the Hamilton-Jacobi equation you get
$$H(t,q,p)=E=-\partial_t g=\text{const} \; g=-E t + S(q,E,P_2,\ldots,P_f),$$
where we use $E$ as one of the new canonical momenta, and $S$ is not explicitly time dependent.

Now we evaluate the action functional in terms of $g$:
$$A=\int_{t_1}^{t_2} \mathrm{d} t (\dot{q} \dot{p}-H)=\int_{t_1}^{t_2} \mathrm{d} t (\dot{q} \partial_q g+\partial_t g)=g_2-g_1,$$
where
$$g_j=g(t=t_j,q(t_j),P).$$
Note that $P=\text{const}$ according to the construction of the new canonical coordinates. As you see, indeed $g$ is closely related to the action.