Finding extremals of Lagrangian actions

Given a smooth m-manifold M, a smooth Lagrangian L:TM→R and endpoints x0,xT∈M, we look for an extremal x:[0,T]→M of the action ∫0TL(x(t),ẋ(t))dt satisfying x(0)=x0 and x(T)=xT. When interpolating between endpoints, this amounts to a 2-point boundary value problem for the Euler–Lagrange equation. Single or multiple shooting is one of the most popular methods to solve boundary value problems, but the efficiency of shooting and the quality of solutions depends heavily on initial guesses. In the present paper, by dividing the interval [0,T] into several sub-intervals, on which extremals can be found efficiently by shooting when good initial guesses are available from the geometry of a variational problem, we then adjust all junctions by finding zeros of vector fields associated with the velocities at junctions with Newton’s method. We discuss the cases where L is the difference between kinetic energy and potential, M is a hypersurface in Euclidean space, or M is a Lie group. We make some comparisons in numerical experiments for a double pendulum, for obstacle avoidance by a moving particle on the 2-sphere, and for obstacle avoidance by a planar rigid body.