The Cluster's Routhian

A page about the Routhian — a hybrid between the Lagrangian and the Hamiltonian.

The Routhian

The Routhian is a function that is partially a Lagrangian (for some coordinates) and partially a Hamiltonian (for others). It is useful when some coordinates are cyclic (ignorable) and others are not. In the cluster, the Routhian describes the cluster's dynamics by treating some pages as coordinates and others as momenta.

The reduction

The Routhian reduces the number of equations of motion. By treating cyclic coordinates as momenta, you eliminate them from the Lagrangian equations. In the cluster, this means you can describe the dynamics with fewer equations by treating some pages as fixed.

The cyclic coordinate

A cyclic coordinate is one that does not appear in the Lagrangian. Its conjugate momentum is conserved. In the cluster, a cyclic page is one that does not change — its content is fixed. The conserved momentum is the page's influence on the rest of the cluster.

The partial Legendre transform

The Routhian is obtained by a partial Legendre transform of the Lagrangian. In the cluster, the Legendre transform converts some edit velocities (momenta) into edit potentials (coordinates). The Routhian is the result.

This Routhian

This page is a Routhian. It treats some of the cluster's pages as fixed and others as dynamic. The reduction simplifies the description without losing the essential dynamics.