and therefore in the case of a conservative system free from extraneous force,
| d | (Τ + V) = 0 or Τ + V = const., |
| dt |
(16)
which is the equation of energy. For examples of the application of the formula (13) see [Mechanics], § 22.
3. Constrained Systems.
It has so far been assumed that the geometrical relations, if any, which exist between the various parts of the system Case of varying relations. are of the type § 1 (1), and so do not contain t explicitly. The extension of Lagrange’s equations to the case of “varying relations” of the type
x = ƒ(t, q1, q2, ... qn), y = &c., z = &c.,
(1)
was made by J.M.L. Vieille. We now have
| ẋ = | ∂x | + | ∂x | q˙1 + | ∂x | q˙2 + ..., &c., &c., |
| ∂t | ∂q1 | ∂q2 |