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Periodic orbit types of flows associated with optical homogeneous Lagrangians. (Type des orbites périodiques des flots associés à des lagrangiens optiques homogènes.) (English) Zbl 1130.37336

Summary: Let \(L : TM \rightarrow \mathbb R\) be a Lagrangian function which is optical and homogeneous in the fiber of the tangent bundle of an \(n\)-dimensional orientable manifold \(M\). Let \(\gamma\) be a 1-periodic loop which is a non degenerate critical point of the Lagrangian action with index \(p\) (there corresponds to \(\gamma\) a periodic point \((x,v)\) of the Euler-Lagrange flow). Then the Lefschetz number of the Poincaré first return map in the energy hypersurface near \((x,v)\) is \((-1)^{n-1-p}\), and thus if \(2n_h\) is the number of real hyperbolic Floquet multipliers of \((x,v)\) without reflection, then \(n_h = n-1+p\pmod2\).
Then we explain how to deduce from this result that every optical and superlinear Lagrangian function defined on the tangent bundle of an compact orientable evendimensional manifold whose \(\pi_1\) is non trivial has on every energy level above the so-called “critical one” at least one periodic orbit which is either degenerate or has one Floquet multiplier which is hyperbolic.

MSC:

37C27 Periodic orbits of vector fields and flows
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
58E99 Variational problems in infinite-dimensional spaces
70H03 Lagrange’s equations
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