Routhian mechanics

A reduction of Lagrangian mechanics that fixes conserved momenta of cyclic coordinates while keeping an Euler-Lagrange description for the remaining coordinates. It is named for Edward Routh. The Routhian applies a partial Legendre transform to the eliminated velocities.

Use the convention $R=L-{\sum }_i{p}_i{\dot{q}}_i$ for the eliminated cyclic coordinates, expressing their velocities in terms of their fixed conjugate momenta. Terms involving derivatives of those eliminated velocities cancel because $\partial L/\partial {\dot{q}}_i=p_i$. Thus the remaining coordinates obey the usual Euler-Lagrange equations for R. For a particle of fixed angular momentum $\ell$ in a central potential, this gives $R=m{\dot{r}}^2/2-V(r)-{\ell }^2/(2m{r}^2)$, rather than the expression obtained by direct substitution into L alone. An opposite overall sign convention is also used, with correspondingly signed mixed Hamiltonian equations.