Routhian and Cyclic Coordinates

Routhian and Cyclic Coordinates in advanced mechanics. Theory, formulas, worked examples and applications.

Conceptual context

Why solve a complete mechanical problem with the Lagrangian when some of its coordinates do not even appear explicitly in it? That is the question answered by the Routhian: when a coordinate is cyclic (absent from the Lagrangian, even though its velocity does appear), its conjugate momentum is exactly conserved, and it can be eliminated from the problem by substituting it with a constant instead of a dynamical variable.

This leaf uses the hybrid construction with the convention $R$=Σ$p_\alpha$$\dot q_\alpha$−$L$: the complete Lagrangian is subtracted from the cyclic contribution. The opposite convention, also common in the literature, changes the overall sign. Once one convention is fixed, the Routhian treats the non-cyclic coordinates $q_i$ through Lagrange-type equations and the cyclic coordinates $q_\alpha$ through their conserved momenta $p_\alpha$.

🟢 Essential level

The essential idea is simple: if a coordinate does not appear in the Lagrangian, its conjugate momentum is conserved, and it no longer needs to be treated as one more dynamical unknown. Instead of solving a second-order differential equation for that coordinate, it suffices to substitute its constant momentum and keep a problem with fewer independent variables.

The Routhian $R$ formalizes that substitution through a partial Legendre transform. Under this leaf's convention it is the cyclic contribution minus the Lagrangian; it must not be mixed with the opposite-sign definition in the same derivation.

A celebrated example illustrating the idea: in the central-force problem (a planet orbiting a star), the orbital angle is a cyclic coordinate. Its conjugate momentum, the angular momentum, is conserved, and using the Routhian reduces the two-dimensional problem to an equivalent problem in a single radial coordinate, without losing any physical information.

A practical way to consolidate Routhian and Cyclic Coordinates is to separate the physical system, the independent variables, and the mathematical representation before calculating. Routhian and Cyclic Coordinates in advanced mechanics. The student should be able to explain what information the analytical description preserves, what it replaces from the Newtonian treatment, and which observable consequence would reveal an inconsistent choice.

🔵 Formal level

Basic formulas

This first formal layer presents the reference relation used to recognize the model and read its variables before entering more abstract derivations. Identify what each term represents, which assumptions make it valid, and how the chosen coordinates connect to the physical motion that the model is intended to describe.

The formal level rests on four relations:

Conserved cyclic momentum

Definition of the Routhian

Real student questions

"Why not use the complete Hamiltonian directly, if a Legendre transform must be done anyway?" Because the Routhian only transforms the cyclic coordinates, leaving the non-cyclic ones in their original Lagrangian form, which is usually simpler to solve when those coordinates still have complicated dynamics of their own.

"How do I know which coordinates are cyclic?" One checks directly whether the coordinate appears explicitly in the Lagrangian; if only its velocity appears, it is cyclic, regardless of whether its motion looks periodic or not.

Final synthesis

By the end of this leaf, the student should be able to identify cyclic coordinates in a Lagrangian, compute their conserved momentum, build the corresponding Routhian, and apply the reduced Euler-Lagrange equations to the remaining non-cyclic coordinates.

The relevant final check is one of consistency: verifying that solving the reduced problem with $R$ and then reconstructing the cyclic coordinates through their conserved momentum reproduces exactly the same trajectory as solving the complete problem with $L$ from the start.