The preceding theorems are easily adapted to the case of cyclic systems. We have only to write
S = ∫t′t (R − V) dt = ∫t′t (T − κχ˙ − κ′χ˙′ − ... − V) dt
(12)
in place of (1), and
A = ∫ (2T − κχ˙ − κ′χ˙′ − ...) dt,
(13)
in place of (8); cf. § 7 ad fin. It is understood, of course, that in (12) S is regarded as a function of the initial and final values of the palpable co-ordinates q1, q2, ... qm, and of the time of transit τ, the cyclic momenta being invariable. Similarly in (13), A is regarded as a function of the initial and final values of q1, q2, ... qm, and of the total energy H, with the cyclic momenta invariable. It will be found that the forms of (4) and (9) will be conserved, provided the variations δq1, δq2, ... be understood to refer to the palpable co-ordinates alone. It follows that the equations (5), (6) and (10), (11) will still hold under the new meanings of the symbols.
9. Reciprocal Properties of Direct and Reversed Motions.
We may employ Hamilton’s principal function to prove a very remarkable formula connecting any two slightly disturbed Lagrange’s formula. natural motions of the system. If we use the symbols δ and Δ to denote the corresponding variations, the theorem is
| d | Σ (δpr·Δqr − Δpr·δqr) = 0; |
| dt |