Invariant connections in geometric mechanics: reduction, nonlocality, and curvature effects

We present a systematic study on the role of invariant connections in geometric mechanics. We first develop a comprehensive theory of reduction under left- and right-invariant connections on Lie groups, showing that the Euler–Poincaré and Lie–Poisson equations are independent of the connection. We then introduce a novel connection-dependent variational principle, where the Lagrangian depends on the velocity parallel-transported back to the initial point of the curve. For Cartan–Schouten connections, this leads to an integro-differential Euler–Poincaré equation that exhibits two distinct sources of nonlocality in time: a path-dependent term encoded in the parallel transport and a future-dependent term arising from a curvature integral. We reformulate this equation as a two-point boundary value problem and present a two-level numerical scheme. The general theory is illustrated on the Heisenberg group, where the equations simplify due to nilpotency, and on the rotation group, where the full integro-differential structure is retained.