Routhian mechanics

Definition

Let $n, k \ge 1$, let $q = (q_1, \ldots, q_n)$ and $s = (s_1, \ldots, s_k)$ be generalized coordinates, and let $L(q, s, \dot q, \dot s, t)$ be a Lagrangian of class $C^2$. Assume that each $s_j$ is a cyclic coordinate, that is, $\partial L / \partial s_j = 0$, so that $L = L(q, \dot q, \dot s, t)$. Let $p_j = \partial L / \partial \dot s_j$ for $j = 1, \ldots, k$. Assume that for fixed $(q, \dot q, t)$ the map $\dot s \mapsto p$ is invertible, with inverse $\dot s = u(q, \dot q, p, t)$ of class $C^1$. The Routhian is the function $$R(q, \dot q, p, t) = \sum_{j=1}^k p_j \, u_j(q, \dot q, p, t) - L\big(q, \dot q, u(q, \dot q, p, t), t\big).$$ Routhian mechanics describes the motion by the Lagrange equations of $R$ in the variables $q$, with $p$ held constant: $$\frac{d}{dt} \frac{\partial R}{\partial \dot q_i} - \frac{\partial R}{\partial q_i} = 0, \qquad i = 1, \ldots, n,$$ together with $\dot p_j = 0$ and $\dot s_j = \partial R / \partial p_j$.