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:H\left(q_1,\dots,q_n; rac{\partial S}{\partial q_1},\dots, rac{\partial S}{\partial q_n};t ight) + rac{\partial S}{\partial t}=0.

In the HJE, S has the interesting property of being the Classical Action .


CANONICAL TRANSFORMATIONS


The HJE follows directly from the observation that for any generating function S(q,p',t) (neglecting the index), the equations of motion are the same for H(q,p,t) and H'(q',p',t) provided that

:
(1) \qquad
{\partial S \over \partial q} = p, \qquad
{\partial S \over \partial p'} = q', \qquad
H' = H + {\partial S \over \partial t}


and the new equations of motion become

:
(2) \qquad {\partial H' \over \partial q'} = - {dp' \over dt}, \qquad
{\partial H' \over \partial p'} = {dq' \over dt}.


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

: (3) \qquad {dp' \over dt} = {dq' \over dt} = 0.

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

:
H'(q',p',t) = H(q,p,t) + {\partial S \over \partial t} = 0.


Since the Eq. (1) gives p=\partial S/\partial q this can be written

:
H\left(q,{\partial S \over \partial q},t ight) + {\partial S \over \partial t} = 0,


which is the HJE.


SOLVING


The HJE is frequently solved by Separation Of Variables , so

: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,

where \alpha_i and a 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 \alpha to emphasize the fact that in the primed coordinate frame, all the momenta are constants, as shown in Eq. (3). Therefore, from Eq. (1),

:(4) \qquad q'=\beta={\partial S(q,\alpha,t) \over \partial \alpha}.

At last, if we invert Eq. (4), we can write q in terms of the constants \alpha and \beta and also the time t. This completely solves the system - \alpha and \beta 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.


REFERENCES





SEE ALSO