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| Content Provider | Springer Nature Link |
|---|---|
| Author | Arnaud, Marie Claude |
| Copyright Year | 2006 |
| Abstract | Soit L : T M → ℝ un lagrangien optique et homogène dans la fibre défini sur le fibré tangent d’une variété orientable de dimension n et γ un lacet régulier 1-périodique qui est un point critique non dégénéré d’indice p de l’action lagrangienne associée à L (il lui correspond alors un point périodique (x, v) du flot d’Euler-Lagrange (φt )). Soit T une transversale en (x, v) au champ de vecteurs dans la surface d’énergie et P l’application de premier retour de Poincaré dans cette transversale; on montre alors que le nombre de Lefschetz pour P en (x, v) est (−1)n−1+p. On en déduit que si 2nh est le nombre de multiplicateurs de Floquet réels strictement positifs et non nuls, alors: nh = n − 1 + p (mod 2).On explique comment déduire qu’un lagrangien optique quelconque défini sur le fibré tangent d’une variété orientable compacte de dimension paire de π1 non trivial a une une orbite périodique qui est soit dégénérée, soit a un exposant de Floquet hyperbolique dans tout niveau d’énergie au dessus du niveau critique de Mañé.Let L : T M → 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 γ be a 1-periodic loop which is a non degenerate critical point of the Lagrangian action with index p (there corresponds to γ 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 2nh is the number of real hyperbolic Floquet multipliers of (x, v) without reflection, then nh = n − 1 + p (mod 2).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 π1 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. |
| Starting Page | 153 |
| Ending Page | 190 |
| Page Count | 38 |
| File Format | |
| ISSN | 16787544 |
| Journal | Bulletin of the Brazilian Mathematical Society |
| Volume Number | 37 |
| Issue Number | 2 |
| e-ISSN | 16787714 |
| Language | French |
| Publisher | Springer-Verlag |
| Publisher Date | 2006-01-01 |
| Publisher Place | Berlin, Heidelberg |
| Access Restriction | One Nation One Subscription (ONOS) |
| Subject Keyword | Lagrangian functions periodic orbits Floquet multipliers Lyapunov stability broken geodesics Jacobi metric Mathematics Mathematical and Computational Physics |
| Content Type | Text |
| Resource Type | Article |
| Subject | Mathematics |
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