Nonlinear Mechanics - Physics at Oregon State University

2.1. CONTACT TRANSFORMATIONS 19
to find F , and given F , (2) how to find the transformation (q, p) → (Q, P ).
We deal with (2) now and postpone (1) to later sections.
Consider the variables q, Q, p, and P . Any two of these constitute a
complete set, so there are four kinds of generating functions usually called
F1(q, Q, t), F2(q, P, t), F3(p, Q, t), and F4(p, P, t). All four are discussed in
Goldstein. F1 provides a good introduction. Most of our work will make
use of F2.
Starting with F1(q, Q) (2.3) becomes
Since
we get with the help of (4)
¯L(Q, ˙ Q, t) = L(q, ˙q, t) − d
dt F1(q, Q, t) (2.7)
∂ ¯ L
∂ ˙qk
∂ ¯ L
∂ ˙qk
= ∂L
−
∂ ˙qk
= ∂L
∂ ˙ Qk
∂ ˙
F1
∂ ˙qk
= 0,
= ˙pk − ∂F1
∂qk
∂ ¯ L
∂ ˙ Qk
= Pk = ∂L
∂ ˙ Qk
∂F1 ˙
−
∂ ˙ Qk
This yields the two transformation equations
Pk = − ∂F1
∂Qk
pk = ∂F1
∂qk
= 0.
= − ∂F1
∂Qk
(2.8)
(2.9)
A straightforward set of substitutions gives our final formula for the Kamiltonian.
K = ∑
[
− ∂F
˙Qk − L +
∂Qk
∂F
˙qk +
∂qk
∂F
]
˙Qk +
Qk
∂F
∂t
To be more explicit
k
= −L + ∑
k
pk ˙qk + ∂F
∂t
K(Q, P ) = H(q(Q, P ), p(Q, P ), t) + ∂
∂t F1(q(Q, P ), Q, t) (2.10)
Summary: