The Reduced Euler-Lagrange Equations
Marsden, J. E. and J. Scheurle
Fields Inst. Commun., 1, 139-164
Abstract:
Marsden and Scheurle [1993] studied Lagrangian reduction in the
context of momentum map constraints--here meaning the reduction of the standard
Euler-Lagrange system restricted to a level set of a momentum map. This provides a
Lagrangian parallel to the reduction of symplectic manifolds. The present paper
studies the Lagrangian parallel of Poisson reduction for Hamiltonian systems. For
the reduction of a Lagrangian system on a level set of a conserved quantity, a key object
is the Routhian, which is the Lagrangian minus the mechanical connection paired with the
fixed value of the momentum map. For unconstrained systems, we use a velocity shifted
Lagrangian, which plays the role of the Routhian in the constrained theory. Hamilton's
variational principle for the Euler-Lagrange equations breaks up into two sets of
equations that represent a set of Euler-Lagrange equations with gyroscopic forcing that
can be written in terms of the curvature of the connection for horizontal variations, and
into the Euler-Poincaré equations for the vertical variations. This new set of
equations is what we call the reduced Euler-Lagrange equations, and it includes the
Euler-Poincaré and the Hamel equations as special cases. We illustrate this
methodology for a rigid body with internal rotors and for a particle moving in a magnetic
field.