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High action orbits for Tonelli Lagrangians and superlinear Hamiltonians on compact configuration spaces. (English) Zbl 1114.37034
Multiplicity results for solutions of various boundary value problems are known for dynamical systems on compact configuration manifolds, given by Lagrangian or Hamiltonian systems which have quadratic growth in the velocities or momenta. The paper under review extends some of these results to the classical setting of Tonelli Lagrangians, or to the Hamiltonians which are superlinear in the momenta and have a coercive action integrand.
More precisely, let $$M$$ be a compact manifold, the configuration space of a (possibly nonautonomous) – Lagrangian system, given by a Lagrangian $$L:\mathbb R\times TM\to \mathbb R$$. Here $$TM$$ is the tangent bundle of $$M$$, and its elements are denoted by $$(q,v)$$, where $$q\in M$$, and $$v$$ is a tangent vector at $$q$$. A classical assumption is that $$L$$ should be a Tonelli Lagrangian, meaning that $$L$$ is fiberwise $$C^2$$-strictly convex (that is $$\partial_{vv} L>0$$), and has superlinear growth on each fiber.
Tonelli assumptions imply existence results for minimal orbits, such as the existence of an orbit having the minimal action connecting two given points on $$M$$, or the existence of a periodic orbit of minimal action in every conjugacy class of the fundamental group of $$M$$ (if $$L$$ is assumed to be periodic in time).
However, existence and multiplicity results for orbits with large action and large Morse index have been known only for a smaller class of Lagrangians, namely Lagrangians that behave quadratically in $$v$$ for large $$| v|$$. In the present paper it is proved that such results hold for any Tonelli Lagrangian which induces a complete vector field. To formulate the results, fix a submanifold $$Q$$ of $$M\times M$$ and consider orbits $$\gamma :[0,1]\to M$$ satisfying the following (nonlocal) boundary condition: $(\gamma (0),\gamma (1))\in Q ,$ $D_vL(0,\gamma (0),\gamma '(0))[\xi_0]= D_vL(1,\gamma (1),\gamma '(1))[\xi_1],$ for every vector $$(\xi_0,\xi_1)\in T(M\times M)$$ which is tangent to $$Q$$ at $$(\gamma (0),\gamma (1))$$. Note that if $$Q$$ is the diagonal submanifold, then the above condition yields to periodic orbits; if $$Q$$ is a singleton $$(q_1,q_2)$$, then the above condition yields to the orbits joining $$q_1$$ to $$q_2$$.
The main result of the paper is that the number of orbits of a complete Tonelli Lagrangian system, satisfying the boundary conditions above, is greater than the cuplength of the space $$C_Q([0,1],M)$$ of continuous paths $$\gamma :[0,1]\to M$$ such that $$(\gamma (0),\gamma (1))\in Q$$ (the cuplength of a topological space $$X$$ is the maximum length of a non-vanishing cup product of elements of degree at least one in the cohomology ring of $$X$$). Moreover, if $$C_Q([0,1],M)$$ has infinitely many Betti numbers, there is a sequence of such orbits with diverging action.

##### MSC:
 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) 49J15 Existence theories for optimal control problems involving ordinary differential equations 58E99 Variational problems in infinite-dimensional spaces
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