Substituting in § 2 (10), we have
| d | ∂R | − | ∂R | = Q1, | d | ∂R | − | ∂R | = Q2, ... | ||
| dt | ∂q˙1 | ∂q1 | dt | ∂q˙2 | ∂q2 |
(8)
These are Routh’s forms of the modified Lagrangian equations. Equivalent forms were obtained independently by Helmholtz at a later date.
The function R is made up of three parts, thus
R = R2, 0 + R1, 1 + R0, 2, ...
(9)
where R2, 0 is a homogeneous quadratic function of q˙1, q˙2, ... q˙m, R0, 2 is Kelvin’s equations. a homogeneous quadratic function of κ, κ′, κ″, ..., whilst R1, 1 consists of products of the velocities q˙1, q˙2, ... q˙m into the momenta κ, κ′, κ″.... Hence from (3) and (7) we have
| T = R − (κ | ∂R | + κ′ | ∂R | + κ″ | ∂R | + ...) = R2, 0 − R0, 2. |
| ∂κ | ∂κ′ | ∂κ″ |
(10)