Is there any general theorem which specifies conditions where the critical solution of an action is unique (for given boundary conditions)?

Question

Consider a classical mechanical system with generalized coordinates $q_i$, $i \in \{1,\dots\,n\}$. And Lagrangian $L$. Given a path $\gamma$ (with coordinates $\gamma_i$) and two times $t_1$ and $t_2$ you can calculate the action of $\gamma$:

$$ S[\gamma] := \int_{t_1}^{t_2} L(\gamma_i(t),\dot \gamma_i(t),t) \,\mathrm{d}t $$

Is there any general theorem which specifies conditions where the critical solution of an action is unique (for given boundary conditions)? I am also looking for a (reference with a) proof.

Answer 1

Consider a smooth 2D sphere and a point of positive mass constrained to move on that without friction. The only force is the reactive force normal to the sphere. The trajectories of the motions of the point are geodesics of the spherical surface. Therefore if you fix the north and south poles as boundary conditions you find infinitely many solutions satisfying both Euler-Lagrange equations and the given boundary conditions, thus also making stationary the action.

Arnold in his celebrated book about mathematical methods of classical mechanics states -- without proof -- that if the endpoints are sufficiently close to each other the stationary curve is unique.

Actually, in case of a completely kinetic Lagrangian (also constrained), as the motions are geodesical, the uniqueness theorem you are looking for is just a version of the theorem stating the existence of geodesically convex neighborhoods in Riemannian differential geometry. It seems to me that Arnold says that the result generalizes to the case where also a potential energy term is added.

Answer 2

According to sec. 4 in Calculus of Variations, Gelfand, Dover 1991, a theorem due to Bernstein concerns existence and uniqueness of solutions to the equation $y'' = F(x,y,y')$:

If the functions $F$, $\partial_y F$ and $\partial_{y'} F$ are continuous at every finite point $(x,y)$ for any finite $y'$, and if a constant $k>0$ and functions $\alpha\equiv \alpha(x,y)\geq 0$, $\beta\equiv \beta(x,y) \geq 0$, which are bounded in every finite region of the plane, can be found such that $\partial_y F(x,y,y') > k$ and $|F(x,y,y')| \leq \alpha y'^2+\beta$, then one and only one integral curve of equation $y'' = F(x,y,y')$ passes through any two points $(a,A)$, $(b,B)$, $a\neq b$.

Mechanical systems are of the form $L = \frac{m}{2} \dot{x}^2 - V(x)$, therefore these conditions imply strictly convex $V(x)$ and an inequality relating forces and kinetic energy. Note that this theorem provides a sufficient, not necessary condition.