I'm trying to understand the Euler-Poincare equations, which reduce the Euler-Lagrange equations for certain Lagrangians on a Lie group. I'm reading Darryl Holm's "Geometric mechanics and symmetry", where he suddenly uses what seems to be a variational (functional?) derivative, which I'm having a hard time understanding. The Euler-Poincare reduction theorem (and equation) goes as follows:
Where I'm guessing that $\frac{\delta l}{\delta \xi}$ is a variational derivative. It must be a covector, and in computations it seems to reduce to partial derivatives of $l$ wrt. $\xi$ (in suitable coordinates) - I have also seen other sources where the EP-equation is stated in terms of partial derivatives $\frac{\partial ...}{\partial ...}$ instead of $\frac{\delta ...}{\delta ...}$.
He defines it in two different ways. The first definition is found in one of the later chapters of the book (and it seems to be in a more general setting):
And in another book ("Geometric mechanics - part 2") he defines it as:
My questions are:
- which definition should I focus on?
- Is the pairing in the latter definition the usual covector-vector pairing?
Any explanations, hints or references would be greatly appreciated! Thanks.
