where ω has been written for φ. These are the equations which we should have obtained by applying Lagrange’s rule at once to the formula
2Τ = m (ẋ² + ẏ² + z˙²) + 2mω (xẏ − yẋ) + mω² (x² + y²),
(15)
which gives the kinetic energy of the particle referred to axes rotating with the constant angular velocity ω. (See [Mechanics], § 13.)
More generally, let us suppose that we have a certain group of co-ordinates χ, χ′, χ″, ... whose absolute values do not affect the expression for the kinetic energy, and that by suitable forces of the corresponding types the velocity-components χ˙, χ˙′, χ˙″, ... are maintained constant. The remaining co-ordinates being denoted by q1, q2, ... qn, we may write
2T = ⅋ + T0 + 2(α1q˙1 + α2q˙2 + ...) χ˙ + 2(α′1q˙1 + α′2q˙2 + ...) χ˙′ + ...,
(16)
where ⅋ is a homogeneous quadratic function of the velocities q˙1, q˙2, ... q˙n of the type § 1 (8), whilst Τ0 is a homogeneous quadratic function of the velocities χ˙, χ˙′, χ˙″, ... alone. The remaining terms, which are bilinear in respect of the two sets of velocities, are indicated more fully. The formulae (10) of § 2 give n equations of the type
| d | ( | ∂⅋ | ) − | ∂⅋ | + (r, 1) q˙1 + (r, 2) q˙2 + ... − | ∂T0 | = Qr |
| dt | ∂qr | ∂qr | ∂qr |
(17)