I have a Lagrangian on $SU(n)$, which is not invariant.
Given the Lagrangian $\mathcal{L}[U_t, \dot{U}_t] = \langle \dot{U}_t, \nabla J \big|_{U_t} \rangle$
I need to find the curves of stationary action, or at least a differential equation they satisfy.
Herein:
1) $\langle, \rangle$ is the trace inner product
2) $J(U) = |\langle U, G \rangle|^2$ where $G \in SU(n)$
3) I already know that $\nabla J = Tr(G^{\dagger}U)G - Tr(GU^{\dagger})UG^{\dagger}U$
4) $\dot{U}$ must be constrained to take the form $\dot{U} = (A + w(t)B)U_t$, where $A,B$ are two algebra elements (which generate the whole algebra) and $w$ is a function. For the more physical individuals, this is Schrödinger’s equation for the dipole approximation. For those of mathematical proclivities, it is an affine control system.
Finding the EL equation in coordinates seems a lost cause because of the lack of useful coodinates on $SU(n)$. However, the variations of $\delta \dot{U}_t = \delta w(t) B U_t + (A+w(t)B)\delta U_t$ is simple, and $\delta \nabla J \big|_{U_t}$ can be found without too much difficulty using:
$\delta U_t = U_t \int_0^{t} U_s^{\dagger} B U_s \delta w(s) ds$.
and is only slightly unwieldy.
Is it possible, using some coordinate free EL equation, or by any other method, to find the curves I'm seeking, or better still, the $w$ corresponding to these curves.
