This must be equal to the rate at which the forces acting on the system do work, viz. to
where the first term represents the work done in virtue of the rotation.
We have still to notice the modifications which Lagrange’s equations undergo when the co-ordinates Constrained systems. are not all independently variable. In the first place, we may suppose them connected by a number of relations of the type
These may be interpreted as introducing partial constraints into a previously free system. The variations in the expressions (6) and (7) of § 2 which are to be equated are no longer independent, but are subject to the relations
Introducing indeterminate multipliers , one for each of these equations, we obtain in the usual manner equations of the type
in place of § 2 (10). These equations, together with (28), serve to determine the co-ordinates and the multipliers .
When does not occur explicitly in the relations (28) the system is said to be holonomic. The term connotes the existence of integral (as opposed to differential) relations between the co-ordinates, independent of the time.
Again, it may happen that although there are no prescribed relations between the co-ordinates , yet from the circumstances of the problem certain geometrical conditions are imposed on their variations, thus
where the coefficients are functions of and (possibly) of . It is assumed that these equations are not integrable as regards the variables ; otherwise, we fall back on the previous conditions. Cases of the present type arise, for instance, in ordinary dynamics when we have a solid rolling on a (fixed or moving) surface. The six co-ordinates which serve to specify the position of the solid at any instant are not subject to any necessary relation, but the conditions to be satisfied at the point of contact impose three conditions of the form (31). The general equations of motion are obtained, as before, by the method of indeterminate multipliers, thus
The co-ordinates , and the indeterminate multipliers , are determined by these equations and by the velocity-conditions corresponding to (31). When does not appear explicitly in the coefficients, these velocity-conditions take the forms
Systems of this kind, where the relations (31) are not integrable, are called non-holonomic.
4. Hamiltonian Equations of Motion.
In the Hamiltonian form of the equations of motion of a conservative system with unvarying relations, the kinetic energy is supposed expressed in terms of the momenta and the co-ordinates , as in § 1 (19). Since the symbol now denotes a variation extending to the co-ordinates as well as to the momenta, we must add to the last member of § 1 (21) terms of the types
Since the variations may be taken to be independent, we infer the equations § 1 (23) as before, together with
Hence the Lagrangian equations § 2 (14) transform into
If we write
so that denotes the total energy of the system, supposed expressed in terms of the new variables, we get
If to these we join the equations
which follow at once from § 1 (23), since does not involve , we obtain a complete system of differential equations of the first order for the determination of the motion.
The equation of energy is verified immediately by (5) and (6), since these make
The Hamiltonian transformation is extended to the case of varying relations as follows. Instead of (4) we write
and imagine to be expressed in terms of the momenta , the co-ordinates , and the time. The internal forces of the system are assumed to be conservative, with the potential energy . Performing the variation on both sides, we find
terms which cancel in virtue of the definition of being omitted. Since may be taken to be independent, we infer
It follows from (11) that
The equations (10) and (12) have the same form as above, but is no longer equal to the energy of the system.
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 , provided the remaining co-ordinates and the velocities, including of course the velocities , are unaltered. Secondly, there are no forces acting on the system of the types . This case arises, for example, when the system includes gyrostats which are free to rotate about their axes, the co-ordinates 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 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 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 reduce, in virtue of the above hypotheses, to the forms
where are the constant momenta corresponding to the cyclic co-ordinates . These equations are linear in ; solving them with respect to these quantities and substituting in the remaining Lagrangian equations, we obtain differential equations to determine the remaining co-ordinates . The object of the present investigation is to ascertain the general form of the resulting equations. The retained co-ordinates may be called (for distinction) the palpable co-ordinates of the system; in many practical questions they are the only co-ordinates directly in evidence.
If, as in § 1 (25), we write
and imagine to be expressed by means of (2) as a quadratic function of with coefficients which are in general functions of the co-ordinates , then, performing the operation on both sides, we find