A polar representation and the Morse index theorem (Q1573609)

The paper deals with a time-dependent linear Hamiltonian system \[ \dot Q = BQ + CP, \dot P = -AQ - B^* P \tag{1} \] and its polar representation, where \(A, B\) and \(C\) are time-dependent \(n \times n\) matrices, \(A\) and \(C\) are symmetric. The dot means derivative with respect to \(\tau \in [0, T].\) The author studies the behavior of the eigenvalues of symmetric operator \({\mathcal L}: H \to H\) defined by the action \[ \langle \gamma, {\mathcal L}\gamma \rangle = 2 \int_0^t L(\gamma(\tau), \dot \gamma(\tau), \tau) d\tau + (\gamma(0), N\gamma(0)), \] where \(L\) is the Lagrangian associated with system (1), \(N\) is a symmetric \(n \times n\) matrix, \(H\) is some Hilbert space of continuous functions \(\gamma: [0, t] \to \mathbb{R}^n\) with inner product \(\langle .,.\rangle.\) The Morse index theorem for \(\mathcal L\) is proved.