# A Smooth Pseudo-Gradient for the Lagrangian Action Functional

@article{Abbondandolo2008ASP, title={A Smooth Pseudo-Gradient for the Lagrangian Action Functional}, author={Alberto Abbondandolo and Matthias Schwarzy}, journal={Advanced Nonlinear Studies}, year={2008}, volume={9}, pages={597 - 623} }

Abstract We study the action functional associated to a smooth Lagrangian function on the tangent bundle of a manifold, having quadratic growth in the velocities. We show that, although the action functional is in general not twice differentiable on the Hilbert manifold consisting of H1 curves, it is a Lyapunov function for some smooth Morse-Smale vector field, under the generic assumption that all the critical points are non-degenerate. This fact is suffcient to associate a Morse complex to…

## 54 Citations

Geodesics and Jacobi fields of pseudo-Finsler manifolds

- Mathematics
- 2014

In this paper, we derive the first and the second variation of the energy functional for a pseudo-Finsler metric using the family of affine connections associated to the Chern connection. This opens…

Linear instability for periodic orbits of non-autonomous Lagrangian systems

- Physics, Mathematics
- 2019

Inspired by the classical Poincaré criterion about the instability of orientation preserving minimizing closed geodesics on surfaces, we investigate the relation intertwining the instability and the…

Lectures on the free period Lagrangian action functional

- Mathematics
- 2013

In this expository article we study the question of the existence of periodic orbits of prescribed energy for classical Hamiltonian systems on compact configuration spaces.We use a variational…

A multiplicity result for Euler–Lagrange orbits satisfying the conormal boundary conditions

- MathematicsJournal of Fixed Point Theory and Applications
- 2020

In this paper, we study the multiplicity problem for Euler–Lagrange orbits that satisfy the conormal boundary conditions for a suitable class of reversible Lagrangian functions on compact manifolds.…

Minimax periodic orbits of convex Lagrangian systems on complete Riemannian manifolds

- Mathematics
- 2020

In this paper we study the existence of periodic orbits with prescribed energy levels of convex Lagrangian systems on complete Riemannian manifolds. We extend the existence results of Contreras by…

An Alternative Variational Principle for Geodesics of a Randers Metric

- Mathematics
- 2009

Abstract We present an alternative variational principle for the geodesics of a Randers metric. We define a functional I on the manifold of H1,2 curves joining two points on a Randers manifold (M, F)…

Symplectic homology of convex domains and Clarke's duality

- Mathematics
- 2019

We prove that the Floer complex that is associated with a convex Hamiltonian function on $\mathbb{R}^{2n}$ is isomorphic to the Morse complex of Clarke's dual action functional that is associated…

Linear instability of periodic orbits of free period Lagrangian systems

- Mathematics
- 2021

In this paper we provide a sufficient condition for the linear instability of a periodic orbit for a free period Lagrangian system on a Riemannian manifold. The main result establish a general…

Calculus of Variations, Conjugate Points and Morse Index

- Mathematics
- 2015

Let us re-examine the classical conditions of Calculus of Variations geared to obtain a strong minimum (in the topology of the uniform convergence) for an arbitrary Lagrangian function…

The Role of the Legendre Transform in the Study of the Floer Complex of Cotangent Bundles

- Mathematics
- 2013

Consider a classical Hamiltonian H on the cotangent bundle T*M of a closed orientable manifold M, and let L:TM → R be its Legendre-dual Lagrangian. In a previous paper we constructed an isomorphism Φ…

## References

SHOWING 1-10 OF 20 REFERENCES

Periodic solutions of Lagrangian systems on a compact manifold

- Mathematics
- 1986

Abstract Let M be a smooth n -dimensional manifold and let TM be its tangent bundle. We consider a time periodic Lagrangian of period T, l t : TM → R , and we seek T -periodic solutions of the…

LECTURES ON THE MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

- Mathematics
- 2006

After reviewing some classical results about hyperbolic dynamics in a Banach setting, we describe the Morse complex for gradient-like flows on an infinite-dimensional Banach manifold M, under the…

High action orbits for Tonelli Lagrangians and superlinear Hamiltonians on compact configuration spaces

- Mathematics
- 2006

Abstract Multiplicity results for solutions of various boundary value problems are known for dynamical systems on compact configuration manifolds, given by Lagrangians or Hamiltonians which have…

On the global stable manifold

- Mathematics
- 2005

We give an alternative proof of the stable manifold theorem as an application of the (right and left) inverse mapping theorem on a space of sequences. We investigate the diffeomorphism class of the…

The homology of path spaces and Floer homology with conormal boundary conditions

- Mathematics
- 2008

Abstract.We define the Floer complex for Hamiltonian orbits on the cotangent bundle of a compact manifold which satisfy non-local conormal boundary conditions. We prove that the homology of this…

Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric

- Mathematics, Physics
- 2009

We show that the index of a lightlike geodesic in a conformally standard stationary spacetime is equal to the index of its spatial projection as a geodesic of a Finsler metric associated to the…

On the Floer homology of cotangent bundles

- Mathematics
- 2004

This paper concerns Floer homology for periodic orbits and for a Lagrangian intersection problem on the cotangent bundle of a compact orientable manifold M. The first result is a new uniform estimate…

Riemannian Geometry

- Nature
- 1927

THE recent physical interpretation of intrinsic differential geometry of spaces has stimulated the study of this subject. Riemann proposed the generalisation, to spaces of any order, of Gauss's…

A remark on regularization in Hilbert spaces

- Mathematics
- 1986

We present here a simple method to approximate uniformly in Hilbert spaces uniformly continuous functions byC1,1 functions. This method relies on explicit inf-sup-convolution formulas or equivalently…

Morse-palais lemma for nonsmooth functionals on normed spaces

- Mathematics
- 2007

Using elementary differential calculus we get a version of the Morse-Palais lemma. Since we do not use powerful tools in functional analysis such as the implicit theorem or flows and deformations in…