Abstract
In this chapter we introduce the Morse index and nullity for periodic orbits of Euler-Lagrange systems of Tonelli type, an orbit being regarded as an extremal point of the action functional. These indices were first introduced by Morse [Mor96] in the study of closed geodesics. As we will see in the forthcoming chapters, they play a crucial role in the proof of existence and multiplicity results of periodic orbits. In Section 2.1 we give the definition of Morse index and nullity of a periodic orbit, and we prove that they are always finite. In Section 2.2 we outline the beautiful iteration theory of Bott, which studies the behavior of the Morse indices as a periodic orbit is iterated. Finally, in Section 2.3 we describe the relation between the Morse index and the Maslov index from symplectic geometry, which is an index for periodic orbits of general Hamiltonian systems.
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