where the coefficients are functions of q1, q2, ... qn and (possibly) of t. It is assumed that these equations are not integrable as regards the variables q1, q2, ... qn; 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
| d | ∂T | − | ∂T | = Qr + λAr + μBr + ... | |
| dt | ∂q˙r | ∂qr |
(32)
The co-ordinates q1, q2, ... qn, and the indeterminate multipliers λ, μ, ..., are determined by these equations and by the velocity-conditions corresponding to (31). When t does not appear explicitly in the coefficients, these velocity-conditions take the forms
A1q˙1 + A2q˙2 + ... = 0, B1q˙1 + B2q˙2 + ... = 0, &c.
(33)
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 p1, p2, ... and the co-ordinates q1, q2, ..., as in § 1 (19). Since the symbol δ 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
| ∂T | δq1 + | ∂T` | δq2 + .... |
| ∂q1 | ∂q2 |