(8)
where X is the force corresponding to the co-ordinate χ. We may suppose X to be adjusted so as to make χ¨ = 0, and in the remaining equations nothing is altered if we write t for χ before, instead of after, the differentiations. The reason why the equation § 2 (15) no longer holds is that we should require to add a term Xχ˙ on the right-hand side; this represents the rate at which work is being done by the constraining forces required to keep χ˙ constant.
As an example, let x, y, z be the co-ordinates of a particle relative to axes fixed in a solid which is free to rotate about the axis of z. If φ be the angular co-ordinate of the solid, we find without difficulty
2Τ = m (ẋ² + ẏ² +z˙²) + 2φ˙m (xẏ − yẋ) + {I + m (x² + y²)} φ˙²,
(9)
where I is the moment of inertia of the solid. The equations of motion, viz.
| d | ∂Τ | − | ∂Τ | = X, | d | ∂Τ | − | ∂Τ | = Y, | d | ∂Τ | − | ∂Τ | = Z, | |||
| dt | ∂ẋ | ∂x | dt | ∂ẏ | ∂y | dt | ∂z˙ | ∂z |
(10)
and
| d | ∂Τ | − | ∂Τ | = Φ, | |
| dt | ∂φ˙ | ∂φ |