Nonlinear Mechanics - Physics at Oregon State University

2.4. HAMILTON’S CHARACTERISTIC FUNCTION 25
Evidentally, β has something to do with initial conditions: ωβ = ϕ0, the
initial phase angle.
p = ∂S
∂q = √ 2mα − m 2 ω 2 q 2
= √ 2mα cos(ωt + ϕ0)
The maximum value of p is √ 2mE, so that makes sense too.
2.4 Hamilton’s Characteristic Function
There is another way to use the F2 generating function to turn a difficult
problem into an easy one. In the previous section we chose F2 = S = W −αt,
so that K = 0. It is also possible to to take F2 = W (q) so that
(
K = H qk, ∂W
)
= E = α1
(2.29)
∂qk
The W obtained in this way is called Hamilton’s characteristic function.
W = ∑
Wk(qk, α1, . . . , αn)
k
= W (q1, . . . , qn, E, α2, . . . , αn) = W (q1, . . . , qn, α1, . . . , αn) (2.30)
It generates a contact transformation with properties quite different from
that generated by S. The equations of motion are
Pk
˙ = − ∂K
∂Qk
˙Qk = ∂K
∂Pk
= ∂K
∂αk
= 0 (2.31)
= δk1
The new feature is that ˙ Q1 = 1 so Q1 = t − t0. In general
but now β1 = t − t0.
Qk = ∂W
∂αk
pk = ∂W
∂qk
as before.
The algorithm now works like this:
= βk
(2.32)
(2.33)
(2.34)