But this is a relatively rare situation, and uses specific geometric arguments. In negatively curved (hyperbolic) spaces it is well known that geodesics generated by solving the Euler–Lagrange equations are globally optimal and unique. When the points are antipodal, all great arcs connecting them are globally optimal, hence there is a severe lack of uniqueness. The Euler–Lagrange equations yield great arcs as the solution, but for most pairs of points on the sphere there are two great arcs connecting them, one of which is globally minimal, and one of which takes the long way around. A simple example of this is geodesics on the sphere. Otherwise, global optimality may not exist, may not be unique, or may not be provable. But in many cases testable sufficient conditions for global optimality cannot be established unless the integrand is convex. Stronger necessary conditions for local optimality such as those due to Jacobi and Legendre are also well known. In classical variational calculus, the Euler–Lagrange (E–L) Equations provide necessary conditions for local optimality for functionals that have integrands that are sufficiently regular, under the assumption that the extremal solutions are also sufficiently regular. This general theory is then applied to several topics such as optimal framing of curves in three-dimensional Euclidean space, optimal motion interpolation, and optimal reparametrization of video sequences to compare salient actions. Surprisingly, it is possible to prove global optimality in some nonconvex cases where even the regularity conditions required for classical necessary conditions do not hold. This article therefore reviews several nonstandard cases where unique globally optimal solutions can be guaranteed, and establishes a “bootstrapping” process to build new globally optimal variational solutions on larger spaces from existing ones on smaller spaces. The difficulties compound when integrands and/or the optimal paths are not sufficiently regular, since in this case the classical necessary conditions no longer apply. This is also true for variational problems on Lie groups, with the Euler–Poincaré equation establishing necessary conditions. Whereas in a coordinate-dependent setting the Euler–Lagrange equations establish necessary conditions for solving variational problems in which both the integrands of functionals and the resulting paths are assumed to be sufficiently smooth, uniqueness and global optimality are generally hard to prove in the absence of convexity conditions, and often times they may not even exist.
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