Lagrangian mechanics without ordinary differential equations

A variational proof is provided of the existence and uniqueness of evolutions of regular Lagrangian systems. Introduction Let Q be a smooth, finite dimensional manifold and L : TQ → R be a smooth Lagrangian. Evolutions of the Lagrangian system defined by L are by definition the C curves q : [0, h] → R which are critical points of the action Sh = ∫ h 0 L ◦ q(t) dt, subject to the constraint that q(0) and q(h) are constant. A typical route to existence and uniqueness (given that L is regular) of the Lagrangian evolutions, is to to show that derivatives q(t) of evolution curves q(t) are integral curves of the Lagrangian vector field XE , constructed either using the Euler-Lagrange equations in charts, or using the Lagrange two-form ωL, the energy E, and the equation iXEωL = dE. In any case, standard ODE theory provides existence and uniqueness of the initial value problem q(0) = q0, q (0) = v0. For a self-contained exposition, see [1]. Given two nearby q1, q2 ∈ Q, does there exist a unique evolution curve q(t) such that q(0) = q1 and q(h) = q2? This is the local boundary value problem of Lagrangian mechanics. The problem crops up in a variety of situations. For example: 1. If Q is a Riemannian manifold and L(v) = 12g(v, v), then the Lagrangian evolution curves are constant speed reparameterizations of the geodesics, and the local boundary value problem becomes that of locating the unique local geodesic connecting two sufficiently nearby points. Partially supported by the Natural Sciences and Engineering Research Council, Canada. 1 2. A solution to the local boundary value problem is required to construct type 1 generating functions St(q2, q1) for the Hamiltonian flow, which are defined by St(q2, q1) = ∫ t 0 L ◦ q(t) dt where q(t) is the evolution curve with q(0) = q1, q(t) = q2. After constructing the Lagrangian flow FE t , the solution to the local boundary value problem is obtained by solving the equations τQF XE t (vq1) = q2 for vq1 ∈ Tq1Q as a function of q1, q2, t, where τQ : TQ → Q is the canonical projection. This may appear to be a straightforward application of the implicit function theorem near t = 0, q1 = q2, but that is not quite so, because the equation fails to be appropriately regular there. With some care, however, the local boundary value problem can be solved by this route [6]. But, first solving the initial value problem seems like a rather circuitous route to the solution of the local boundary value problem, especially considering that the boundary values q1 and q2 actually occur as the constraints in the original variational formulation for the evolution. Regular constrained optimization problems have critical points which persist as smooth functions of the constraint values. Why not solve the boundary value problem directly, avoiding an excursion into the initial value problem via ODE theory? The obstruction to simply getting on with the job is a fundamental one: the problem of finding the critical points of Sh subject to the constraint q(0) = q0, q(t) = q1, is nonregular at h = 0, which precisely where one wants to perturb from. Indeed, the objective function Sh is actually zero when h = 0. More seriously, the constraint q(t) 7→ (