| ∂ | (T − V) = − | ∂H | , | ∂ | (T − V) = − | ∂H | , .... |
| ∂q1 | ∂q1 | ∂q2 | ∂q2 |
(11)
It follows from (11) that
| ṗ1 = − | ∂H | , ṗ2 = − | ∂H | , .... |
| ∂q1 | ∂q2 |
(12)
The equations (10) and (12) have the same form as above, but H is no longer equal to the energy of the system.
5. Cyclic Systems.
A cyclic or gyrostatic system is characterized by the following properties. In the first place, the kinetic energy is not affected if we alter the absolute values of certain of the co-ordinates, which we will denote by χ, χ′, χ″, ..., provided the remaining co-ordinates q1, q2, ... qm and the velocities, including of course the velocities χ˙, χ˙′, χ˙″, ..., are unaltered. Secondly, there are no forces acting on the system of the types χ, χ′, χ″, .... This case arises, for example, when the system includes gyrostats which are free to rotate about their axes, the co-ordinates χ, χ′, χ″, ... then being the angular co-ordinates of the gyrostats relatively to their frames. Again, in theoretical hydrodynamics we have the problem of moving solids in a frictionless liquid; the ignored co-ordinates χ, χ′, χ″, ... then refer to the fluid, and are infinite in number. The same question presents itself in various physical speculations where certain phenomena are ascribed to the existence of latent motions in the ultimate constituents of matter. The general theory of such systems has been treated by E.J. Routh, Lord Kelvin, and H.L.F. Helmholtz.
If we suppose the kinetic energy Τ to be expressed, as in Lagrange’s method, in terms of the co-ordinates and Routh’s equations. the velocities, the equations of motion corresponding to χ, χ′, χ″, ... reduce, in virtue of the above hypotheses, to the forms
| d | ∂Τ | = 0, | d | ∂Τ | = 0, | d | ∂Τ | = 0, ..., | |||
| dt | ∂χ˙ | dt | ∂χ˙′ | dt | ∂χ˙″ |