An Index Theorem for Non Periodic Solutions of Hamiltonian Systems
Abstract
We consider a {\em Hamiltonian setup} $\sextuple$, where
(M,ω)
is a symplectic manifold,
L
is a distribution of Lagrangian subspaces in
M
,
P
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 $\Hf$ 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 it is introduced the Maslov index $\maslov(\Gamma)$ of
Γ
given in terms of the first relative homology group of the Lagrangian Grassmannian; under generic circumstances $\maslov(\Gamma)$ is computed as a sort of {\em algebraic count} of the
P
-focal points along
Γ
. We prove the following version of the Index Theorem: under suitable hypotheses, the Morse index of the Lagrangian action functional restricted to suitable variations of
Γ
is equal to the sum of $\maslov(\Gamma)$ and a {\em convexity term} of the Hamiltonian
H
relative to the submanifold
P
. When the result is applied to the case of the cotangent bundle
M=T
M
∗
of a semi-Riemannian manifold
(M,g)
and to the geodesic Hamiltonian
H(q,p)=
1
2
g
−1
(p,p)
, we obtain a semi-Riemannian version of the celebrated Morse Index Theorem for geodesics with variable endpoints in Riemannian geometry.