The Euler-Poincare equation

Question

Can anyone tell me very basically how the Euler-Poincare equation generalises the Euler-Lagrange equation? Further does anyone know if there is an "easy" relationship between the two, i.e. can anyone derive either equation from the other? Finally I would be jolly grateful if anyone had a good BASIC source of information at an introductory level to the subject.

Answer 1

1. The stationary action principle and the Euler-Lagrange (EL) equations are very broad and general constructions. The field variables in the variational principle could in principle map into some generic manifold $M$.

2. On the other hand, Euler-Poincare (EP) equations appear in the special situation where the manifold is a Lie group $M=G$, and the action is left-$G$- invariant. One next uses the exponential map to make the variables Lie algebra-valued (rather than Lie group-valued). The EP equations reads $$ \left( \frac{d}{dt}+ {\rm ad}^{\ast}_{\xi}\right) \frac{\delta \ell}{\delta\xi}~=~0, $$ where the variables $\xi$ are Lie algebra-valued. The Lie algebra-valued EP equations are equivalent to the Lie group-valued EL equations for the same problem. See Ref. 1 for further details.

3. The Euler (E) equations for a rigid body are a special case of the EP equations.

References:

1. J.E. Marsden and T.S. Ratiu, Intro to Mechanics and Symmetry, 2nd Eds, 1998; Section 13.5.