Hamilton-Jacobi equation

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Derivation from variational principle

Instead of varying the action over trajectories with fixed endpoints, consider evaluating the action over a classical path connecting two given endpoints:

S[q_0, q_1, t_1] = \int_{t_0}^{t_1}\!dt\, L(q, \dot{q}, t)\,,

where q(t_0) = q_0\, and q(t_1) = q_1\, are the initial and final configurations of the system. Varying around the classical path, we find

\delta S\, = \int_{t_0}^{t_1}\!dt\, \left( \frac{\partial L}{\partial q} \delta q + \frac{\partial L}{\partial \dot q} \delta \dot q \right)\,,
= \int_{t_0}^{t_1}\!dt\, \left[ \frac{\partial L}{\partial q} - \frac{d}{dt} \left( \frac{\partial L}{\partial \dot q} \right) \right] \delta q + \left. \frac{\partial L}{\partial \dot q} \delta q \right|_{t_0}^{t_1}\,,
= \left. \frac{\partial L}{\partial \dot q} \delta q \right|_{t_0}^{t_1}\,,

by the Euler-Lagrange equations. Therefore,

\frac{\partial S}{\partial q_1 } = \left.\frac{\partial L}{\partial \dot q}\right|_{t=t_1} = p_1\,.

On the other hand, if we vary t_1\,, then

\frac{d S}{d t_1}\, = L(q(t_1), \dot q(t_1), t_1)\,,

but also

\frac{d S}{d t_1}\, = \frac{\partial S}{\partial t_1} + \frac{\partial S}{\partial q_1 } \dot q_1\,,
= \frac{\partial S}{\partial t_1} + p_1 \dot q_1\,,

so that

\frac{\partial S}{\partial t_1} = L( q_1, \dot q_1, t_1) - p_1 \dot q_1 = -H(q_1, p_1, t_1)\,.

Since our choice of t_1\, was arbitrary, we may set it equal to the present time t\,, to obtain the Hamilton-Jacobi equations

\frac{\partial S}{\partial t} + H\left(q, \frac{\partial S}{\partial \dot q}, t\right)=0\,.

The generalization to N\, degrees of freedom is trivial, and we get

\frac{\partial S}{\partial t} + H\left(q_1, \dots, q_N, \frac{\partial S}{\partial \dot q_1}, \dots, \frac{\partial S}{\partial \dot q_N}, t\right)=0\,.

Derivation from canonical transformation

Relation to Schrödinger equation

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