Many optimal control problems involving orientations and translations are naturally posed on Lie groups, including applications in robotics, rigid-body motion, vehicle dynamics, and multi-body systems. Their Hamilton–Jacobi equations have the usual difficulties of nonlinearity, non-smoothness, and curse-of-dimensionality, and also with geometric difficulties such as noncommutativity and the lack of global coordinates. We present two structure-preserving approaches. The first is a global Hopf-type formula for left-invariant problems, reducing the value function to a finite-dimensional minimization over the initial covector and yielding a grid-free solver with cost linear in the group dimension. The second is a Peter–Weyl dynamic programming scheme on compact groups, for more general problems, where a positive finite-rank Fejér-type filter restores monotonicity and gives a chart-free method with uniform convergence for Lipschitz data. We conclude with ongoing work on high-dimensional products of groups, including (classical and tropical) tensor methods.