Euler-Poincaré equations with constraints

Question

It is well known that the stationary curves (say $\Xi(t)$) of a regular Lagrangian $\mathcal{L}$ on a compact, semi-simple Lie group $G$ have the property that $\xi(t) = \frac{d \Xi(t)}{dt} \Xi(t)^{-1}$ solves the Euler Poincaré equations:

$\frac{d}{dt}\frac{\partial \ell}{\partial \xi} = -ad^*_{\xi} \left(\frac{\partial \ell}{\partial \xi}\right)$

or in coordinates:

$\frac{d}{dt}\frac{\partial \ell}{\partial \xi^d} = -C^{b}_{ad}\frac{\partial \ell}{\partial \xi^b} \xi^a$

wherein $\ell$ is the restriction of $\mathcal{L}$ to the lie algebra of $G$, say $\mathfrak{g}$, and $C$ are the structure constants of $\mathfrak{g}$. I also further suppose that $\mathcal{L}$ is homogeneous degree, this entails that action of any curve does not depend on the parametrisation of that curve, i.e. only the image of the curve is taken into account. If I wanted to find constrained extremal curves of $\mathcal{L}$ (for example, determining the geodesics of a sub-Riemanian metric) I would typically impose a constraint using a Lagrange multiplier, $\lambda$ say.

I could solve the standard Euler Lagrange equations for $\mathcal{L}(\Xi, \frac{d \Xi}{dt}) + \lambda(f(\Xi, \frac{d \Xi}{dt}) - c)^2$ including an equation for the variation by $\lambda$. However, I want to know if it's possible to write this problem as an Euler Poincaré equation in the case that $f$ depends only on $\frac{d \Xi(t)}{dt} \Xi(t)^{-1}$. That is to say, the constraint is also right invariant.

I'm specifically interested in the geodesics of a Finsler metric on $SU(n)$ which are parallel to a linear affine distribution.

Answer 1

Generally no. The Euler-Poincaré equation derives all of its structure from the Lie bracket, and an arbitrary constraint does not respect this structure. The most simple counter-example is probably a subspace constraint. Let $V \subset \mathfrak{g}$ be a subspace which is not a sub-algebra. This subspace generates a constraint distribution on $TG$ in the obvious way, which we'll denote by $VG$. The constrained Euler-Lagrange equations on $TG$ are given by

$$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{g}} \right) - \frac{\partial L}{\partial g} \in (VG)^\perp \\ \dot{g} \in VG$$

Here $(VG)^\perp \subset T^*G$ is the annihilator of $VG$. These equations can be reduced to $\mathfrak{g}$. Here they take the form

$$\frac{d}{dt} \left( \frac{\partial \ell}{\partial \xi} \right) \pm {\rm ad}^*_\xi \left( \frac{\partial \ell}{\partial \xi} \right) \in V^\perp \\ \xi \in V$$

Here $V^\perp \subset\mathfrak{g}^*$ is the annihilator of $V$. In any case, the above equation is not an Euler-Poincaré equation, and It can not be transformed into one (although, one may call it a "non-holonomic Euler-Poincar\'e equation).