Variational Principles in Classical Mechanics - Revised Second Edition, 2019

15.4. HAMILTON-JACOBI THEORY 423
where
µ
s
2 µ 2 Γ
Γ
= 0 = 0
s1 − = 2 0
2 − ()
0 2
Transforming back to the original variable gives
() = − Γ
2 sin ( + ) ()
where and are given by the initial conditions. Equation is identical to the solution for the underdamped
linearly-damped linear oscillator given previously in equation 335.
Case 2:
2 =1, that is, Γ
2 0
=1
r h
In this case = 1 − ¡ ¢
2
i
2
=0and thus equation simplifies to
= − − 2
4 + √
and
=
= − + √
0
Therefore the solution is
() = − Γ
2 ( + ) ()
where F and G are constants given by the initial conditions. This is the solution for the critically-damped
linearly-damped, linear oscillator given previously in equation 338.
Case 3:
2 1, that is, Γ
2 0
1
r h¡
Define a real constant where =
¢ 2
i
2 − 1 = , then
= − − 2
4 + Z p(
+ 2 2 )
Then
This last integral gives
=
= − + 1 0
Z
p
( + 2 2 )
sinh −1 µ
√
= 0 ( + ) ≡ +
where
Then the original variable gives
s µ 2
= 0 = 0 − 1
2 0
() = − Γ
2 sinh ( + ) ()
This is the classic solution of the overdamped linearly-damped, linear harmonic oscillator given previously in
equation 337 The canonical transformation from a non-autonomous to an autonomous system allowed use
of Hamiltonian mechanics to solve the damped oscillator problem.
Note that this example used Bateman’s non-standard Lagrangian, and corresponding Hamiltonian, for
handling a dissipative linear oscillator system where the dissipation depends linearly on velocity. This nonstandard
Lagrangian led to the correct equations of motion and solutions when applied using either the
time-dependent Lagrangian, or time-dependent Hamiltonian, and these solutions agree with those given in
chapter 35 which were derived using Newtonian mechanics.