The modification of the Hamiltonian principle appropriate to Extension to cyclic systems. the case of cyclic systems has been given by J. Larmor. If we write, as in § 1 (25),
R = Τ − κχ˙ − κ′χ˙′ − κ″χ˙″ − ...,
(15)
we shall have
δ ∫t′t (R − V) dt = 0,
(16)
provided that the variation does not affect the cyclic momenta κ, κ′, κ″, ..., and that the configurations at times t and t′ are unaltered, so far as they depend on the palpable co-ordinates q1, q2, ... qm. The initial and final values of the ignored co-ordinates will in general be affected.
To prove (16) we have, on the above understandings,
δ ∫t′t (R − V) dt = ∫t′t (δT − κδχ˙ − ... − δV) dt
| = ∫t′t ( | ∂T | δq˙1 + ... + | ∂T | δq1 + ... − δV) dt, |
| ∂q˙1 | ∂q1 |