Mathematical Methods for Physicists: A concise introduction - Site Map

VARIATIONAL PROBLEMS
Figure 8.7.
polar coordinates r and in the plane of motion to specify the position of the
electron relative to the nucleus, the kinetic and potential energies are, respectively,
Then
and
The Hamiltonian H is
T ˆ 1
2 m…_r2 ‡ r 2 _ 2 †; V ˆ Ze2
r :
L ˆ T V ˆ 1
2 m…_r2 ‡ r 2 _ 2 †‡ Ze2
r
p r ˆ @L
@ _r ˆ m_rp
H ˆ 1
2m
p ˆ @L
@ _ ˆ mr2 _ :
p2 r ‡ p2
r 2 !
Ze2
r :
Replacing p r and p in the Hamiltonian by @I=@r and @I=@, respectively, we
obtain, by Eq. (8.36), the Hamilton±Jacobi equation
"
1 @I 2
‡ 1 #
@I 2
2m @r r 2 Ze2 ‡ @I
@ r @t ˆ 0:
Variational problems with several independent variables
The functional f in Eq. (8.1) contains only one independent variable, but very
often f may contain several independent variables. Let us now extend the theory
to this case of several independent variables:
ZZZ
I ˆ f fu; u x ; u y ; u z ; x; y; z† dxdydz;
…8:39†
V
where V is assumed to be a bounded volume in space with prescribed values of
u…x; y; z† at its boundary S; u x ˆ @u=@x, and so on. Now, the variational problem
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