If in Lagrange’s equations § 2 (10) we reverse the sign of the time-element dt, the equations are unaltered. The motion is therefore reversible; that is to say, if as the system is passing through any configuration its velocities q˙1, q˙2, ..., q˙m be all reversed, it will (if the forces be the same in the same configuration) retrace its former path. But it is important to observe that the statement does not in general hold of a gyrostatic system; the terms of (16), which are linear in q˙1, q˙2, ..., q˙m, change sign with dt, whilst the others do not. Hence the motion of a gyrostatic system is not reversible, unless indeed we reverse the cyclic motions as well as the velocities q˙1, q˙2, ..., q˙m. For instance, the precessional motion of a top cannot be reversed unless we reverse the spin.
The conditions of equilibrium of a system with latent cyclic motions Kineto-statics. are obtained by putting q˙1 = 0, q˙2 = 0, ... q˙m = 0 in (16); viz. they are
| Q1 = | ∂K | , Q2 = | ∂K | , ... |
| ∂q1 | ∂q2 |
(23)
These may of course be obtained independently. Thus if the system be guided from (apparent) rest in the configuration (q1, q2, ... qm) to rest in the configuration q1 + δq1, q2 + δq2, ..., qm + δqm, the work done by the forces must be equal to the increment of the kinetic energy. Hence
Q1δq1 + Q2δq2 + ... = δK,
(24)
which is equivalent to (23). The conditions are the same as for the equilibrium of a system without latent motion, but endowed with potential energy K. This is important from a physical point of view, as showing how energy which is apparently potential may in its ultimate essence be kinetic.
By means of the formulae (18), which now reduce to
| χ˙ = | ∂K | , χ˙′ = | ∂K | , χ˙″ = | ∂K | , ..., |
| ∂κ | ∂κ′ | ∂κ″ |