# An Index Theorem for Non‐Periodic Solutions of Hamiltonian Systems

@article{Piccione1999AnIT, title={An Index Theorem for Non‐Periodic Solutions of Hamiltonian Systems}, author={Paolo Piccione and Daniel V. Tausk}, journal={Proceedings of the London Mathematical Society}, year={1999}, volume={83} }

We consider a Hamiltonian setup M, ω, H, L, Γ, P, where M, ω is a symplectic manifold, L is a distribution of Lagrangian subspaces in M, P is a Lagrangian submanifold of M, H is a smooth time‐dependent Hamiltonian function on M, and Γ:[a,b] → M is an integral curve of the Hamiltonian flow H→ starting at P. We do not require any convexity property of the Hamiltonian function H. Under the assumption that Γ(b) is not P‐focal, we introduce the Maslov index imaslovΓ of Γ given in terms of the first…

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## References

SHOWING 1-10 OF 43 REFERENCES

A Generalized Index Theorem for Morse-Sturm Systems and Applications to semi-Riemannian Geometry

- Mathematics
- 1999

We prove an extension of the Index Theorem for Morse-Sturm systems of the form $-V''+RV=0$, where R is symmetric with respect to a (non positive) symmetric bilinear form, and thus the corresponding…

Morse theory for periodic solutions of hamiltonian systems and the maslov index

- Mathematics
- 1992

In this paper we prove Morse type inequalities for the contractible 1-periodic solutions of time dependent Hamiltonian differential equations on those compact symplectic manifolds M for which the…

A Maslov-type index theory for symplectic paths

- Mathematics
- 1997

In this paper, we extend the Maslov-type index theory defined in [7], [15], [10], and [18] to all continuous degenerate symplectic paths, give a topological characterization of this index theory for…

Stability of the Focal and Geometric Index in semi-Riemannian Geometry via the Maslov Index

- Mathematics
- 1999

We investigate the problem of the stability of the number of conjugate or focal points (counted with multiplicity) along a semi-Riemannian geodesic $\gamma$. For a Riemannian or a non spacelike…

The Morse index theorem

- Mathematics
- 1967

The use of a matrix Riccati equation to establish sufficiency theorems in the calculus of variations is well known (see [3], e.g.). In this note we extend the method to give an elementary proof of…

A note on the Morse index theorem for geodesics between submanifolds in semi-Riemannian geometry

- Mathematics
- 1999

The computation of the index of the Hessian of the action functional in semi-Riemannian geometry at geodesics with two variable endpoints is reduced to the case of a fixed final endpoint. Using this…

Critical Point Theory and Hamiltonian Systems

- Mathematics
- 1989

1 The Direct Method of the Calculus of Variations.- 2 The Fenchel Transform and Duality.- 3 Minimization of the Dual Action.- 4 Minimax Theorems for Indefinite Functional.- 5 A Borsuk-Ulam Theorem…

Morse theory

- Mathematics
- 1999

Topology on L a, b : Fix a broken λ λ , ... , λ . Its neighborhood consists of its deformations and smoothings. Key: every smooth trajectory has R symmetry. Can reduce to a level set in a Morse chart…

Morse‐type index theory for flows and periodic solutions for Hamiltonian Equations

- Mathematics
- 1984

An index theory for flows is presented which extends the classical Morse theory for gradient flows on compact manifolds. The theory is used to prove a Morse-type existence statement for periodic…

The Morse index theorem where the ends are submanifolds

- Mathematics
- 1988

In this paper the Morse Index Theorem is proven in the case where submanifolds P and Q are at the endpoints of a geodesic, -Y. At -Y, the index of the Hessian of the energy function defined on paths…