Nonlinear Mechanics - Physics at Oregon State University

2.2. THE SECOND GENERATING FUNCTION 21
K = ωP P = E/ω
We have achieved our goal. Q is cyclic, and the equations of motion are
trivial.
˙Q = ∂K
q =
∂P = ω Q = ωt + Q0 (2.13)
√ 2E
mω 2 sin(ωt + Q0) p = √ 2mE cos(ωt + Q0) (2.14)
2.2 The Second Generating Function
There’s an old recipe for tiger stew that begins, “First catch the tiger.” In
our quest for the tiger, we now turn our attention to the second generating
function, F2 = F2(q, P, t). F2 is obtained from F1 by means of a Legendre
transformation. 1
F2(q, P ) = F1(q, Q) + ∑
(2.15)
k
QkPk
We are looking for transformation equations analogous to (refe2.8) and (2.9).
Since L = ¯ L + ˙ F1,
∑
pk ˙qk − H = ∑
Pk ˙ Qk − K + d
dt (F2 − ∑ QkPk)
k
k
k
= − ∑ Qk ˙
Pk − K + ˙
F2
Substitute
F2
˙ = ∑
[
∂F2
˙qk +
∂qk k
∂F2
]
Pk
˙ +
∂Pk
∂F2
∂t
−H = −K + ∑
[( ) ( ) ]
∂F2
∂F2
− pk ˙qk + − Qk Pk
˙ +
∂qk
∂Pk
∂F2
∂t
We are working on the assumption that ˙q and ˙
P are not independent variables.
We enforce this by requiring that
∂F2
∂qk
∂F2
∂Pk
= pk
= Qk
(2.16)
(2.17)
K(q, P ) = H(q(Q, P ), P ) + ∂
∂t F2(q(Q, P ), P ) (2.18)
1 When in doubt, do a Legendre transformation.