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.
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