4

0. Some Aspects of Theoretical Mechanics

0.3. The Hamilton-Jacobi equation

Yet another formulation of the problem passes from the solution of a system

of ordinary differential equations to the solution of a partial differential

equation. The resulting partial differential equation is the Hamilton-Jacobi

equation

for the action function S. Here, giving a solution which is dependent on £,

the n variables g, and the n initial parameters a,

s

*(«-£0+£-°dt+)jtdq

S = S(q, t, a),

is equivalent to giving a solution q = q(t), p = p(t) of (2). Here we present

only the following consideration:

Let S = S(q, i, a) be a solution of (3) with

\dqidakJ

Then the n equations

d S K fi 1

-—- = be, £ = l,...,n,

in the & are solvable in the qi = ipi(t,a,b), i = 1,..., n. This allows one to

write

dqe

as a function of t, a, and 6:

Vl = ile(t, a, b).

These qi,pi satisfy Hamilton's equations (2), since differentiating

„ / dS , . \ dS

n

(+) H(q,-(q,t,a),t)+- = 0

with respect to ai gives

dH

dpk daedqk ' daen dt

y , dH d2S d2S

t—-^ rim. Fin n r)rii. r)n f)+

k

OS

And differentiating — — = bt with respect to t gives

oat

y

d2S

.

d2S

_

Q

*-£ dqkdat

k

dtdai

,n.