where ⅋ involves the velocities q˙1, q˙2, ... q˙m alone, and K the momenta κ, κ′, κ″, ... alone. For in virtue of (29) we have, from (25),
| Τ = R − (κ | ∂R | + κ′ | ∂R | + κ″ | ∂R | + ... ), |
| ∂κ | ∂κ′ | ∂κ″ |
(31)
and it is evident that the terms in R which are bilinear in respect of the two sets of variables q˙1, q˙2, ... q˙m and κ, κ′, κ″, ... will disappear from the right-hand side.
It may be noted that the formula (30) gives immediate proof of two important theorems due to Bertrand and to Lord Kelvin respectively. Let us suppose, in the first place, that the system is started by given impulses of certain types, Maximum and minimum energy. but is otherwise free. J.L.F. Bertrand’s theorem is to the effect that the kinetic energy is greater than if by impulses of the remaining types the system were constrained to take any other course. We may suppose the co-ordinates to be so chosen that the constraint is expressed by the vanishing of the velocities q˙1, q˙2, ... q˙m, whilst the given impulses are κ, κ′, κ″,... Hence the energy in the actual motion is greater than in the constrained motion by the amount ⅋.
Again, suppose that the system is started with prescribed velocity components q˙1, q˙2, ... q˙m, by means of proper impulses of the corresponding types, but is otherwise free, so that in the motion actually generated we have κ = 0, κ′ = 0, κ″ = 0, ... and therefore K = 0. The kinetic energy is therefore less than in any other motion consistent with the prescribed velocity-conditions by the value which K assumes when κ, κ′, κ″, ... represent the impulses due to the constraints.
Simple illustrations of these theorems are afforded by the chain of straight links already employed. Thus if a point of the chain be held fixed, or if one or more of the joints be made rigid, the energy generated by any given impulses is less than if the chain had possessed its former freedom.
2. Continuous Motion of a System.
We may proceed to the continuous motion of a system. The Lagrange’s equations. equations of motion of any particle of the system are of the form
mẍ = X, mÿ = Y, mz¨ = Z