Hamilton-Jacobi equations
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In physics and mathematics, the Hamilton-Jacobi equation (HJE) is a particular canonical transformation of the classical Hamiltonian which results in a first order, non-linear differential equation whose solution describes the behavior of the system. This contrasts with Hamilton's equations of motion in that the HJE is a single differential equation of one variable for each coordinate, where Hamilton's equations are a system of first order equations, two for each coordinate. The HJE can be used to solve several problems elegantly, such as the Kepler problem. If we have a Hamiltonian of the form <math>H(q_1,\dots,q_n;p_1,\dots,p_n;t)</math> then the HJE for that system is
- <math>H\left(q_1,\dots,q_n;\frac{\partial S}{\partial q_1},\dots,\frac{\partial S}{\partial q_n};t\right) + \frac{\partial S}{\partial t}=0.</math>
In the HJE, S has the interesting property of being the classical action.
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Canonical transformations
The HJE follows directly from the observation that for any generating function <math>S(q,p',t)</math> (neglecting the index), the equations of motion are the same for H(q,p,t) and H'(q',p',t) provided that
- <math>
(1) \qquad {\partial S \over \partial q} = p, \qquad {\partial S \over \partial p'} = q', \qquad H' = H + {\partial S \over \partial t} </math>
and the new equations of motion become
- <math>
(2) \qquad {\partial H' \over \partial q'} = - {dp' \over dt}, \qquad {\partial H' \over \partial p'} = {dq' \over dt}. </math>
The HJE comes from the specific generating function S which makes H' identically zero. In this case, all its derivatives are also zero, and so
- <math> (3) \qquad {dp' \over dt} = {dq' \over dt} = 0. </math>
Thus, in the primed coordinate system, the system is perfectly stationary in phase space. However, we have not yet determined what generating function S accomplishes the transformation into the primed coordinate system, so we use the fact that
- <math>
H'(q',p',t) = H(q,p,t) + {\partial S \over \partial t} = 0. </math>
Since the Eq. (1) gives <math>p=\partial S/\partial q</math> this can be written
- <math>
H\left(q,{\partial S \over \partial q},t\right) + {\partial S \over \partial t} = 0, </math>
which is the HJE.
Solving
The HJE is frequently solved by separation of variables, so
- <math>S=S_1(q_1;\alpha_1,\dots,\alpha_n;t)+S_2(q_2;\alpha_1,\dots,\alpha_n;t)+\cdots+S_n(q_n;\alpha_1,\dots,\alpha_n;t)+at,</math>
where <math>\alpha_i</math> and <math>a</math> are the integration constants that arise from solving an (n + 1)-variable first order differential equation, and are also the canonical momenta p' in the primed coordinate frame. We use the variable name <math>\alpha</math> to emphasize the fact that in the primed coordinate frame, all the momenta are constants, as shown in Eq. (3). Therefore, from Eq. (1),
- <math>(4) \qquad q'=\beta={\partial S(q,\alpha,t) \over \partial \alpha}.</math>
At last, if we invert Eq. (4), we can write q in terms of the constants <math>\alpha</math> and <math>\beta</math> and also the time t. This completely solves the system - <math>\alpha</math> and <math>\beta</math> specify the initial conditions of the system, and the solution given by inverting Eq. (4) tells you the position at any future time. The reason there are two initial conditions for each coordinate is that each coordinate has an initial value but also an initial momentum, which must be worked into the solution.