Mathematical Methods for Physicists: A concise introduction - Site Map

THE CALCULUS OF VARIATIONS
must vanish at the endpoints of the integral, the variations of the parameter must
be so chosen that this condition is satis®ed.
To summarize, suppose a given system can be described by the action integral
I ˆ
Z t2
t 1
Lq … i …t†; _q i …t†; t†dt; _q ˆ dq=dt:
The Rayleigh±Ritz method requires the selection of a trial solution, ideally in the
form
q ˆ Xn
iˆ1
a i f i …t†;
…8:20†
which satis®es the appropriate conditions at both the initial and ®nal times, and
where as are undetermined constant coecients and the fs are arbitrarily chosen
functions. This trial solution is substituted into the action integral I and integration
is performed so that we obtain an expression for the integral I in terms of the
coecients. The integral I is then made `stationary' with respect to the assumed
solution by requiring that
@I
ˆ 0
…8:21†
@a i
after which the resulting set of n simultaneous equations is solved for the values of
the coecients a i . To illustrate this method, we apply it to two simple examples.
Example 8.5
A simple harmonic oscillator consists of a mass M attached to a spring of force
constant k. As a trial function we take the displacement x as a function t in the
form
x…t† ˆX1
nˆ1
A n sin n!t:
For the boundary conditions we have x ˆ 0; t ˆ 0, and x ˆ 0; t ˆ 2=!. Then the
potential energy and the kinetic energy are given by, respectively,
V ˆ 1
2 kx2 ˆ 1
2 k P1
T ˆ 1
2 M _x2 ˆ 1 P1
2
M!2
The action I has the form
I ˆ
Z 2=!
0
Ldt ˆ
Z 2=!
0
P 1
nˆ1 mˆ1
P 1
nˆ1 mˆ1
A n A m sin n!t sin m!t;
A n A m nm cos n!t cos m!t:
…TV†dt ˆ X 1
…kA 2 n Mn 2 A 2
2!
n! 2 †:
nˆ1
360