The Morse index theorem for mechanical systems with reflections (Q6566569)

The classical Morse index theorem on a smooth Riemannian manifold \((M, g)\) says that at a geodesic \( \alpha (t), t \in [0, T]\), the index of the second variation of the energy functional with fixed endpoints equals the total number (with multiplicity) of points on \(\alpha\) conjugate to \(\alpha (0)\) (see [\textit{M. Morse}, The calculus of variations in the large. Providence, RI: American Mathematical Society (AMS) (1934; JFM 60.0450.01)]). The authors consider a Lagrangian \(L\in C^\infty (TM)\) given by \N\[\NL(x,v)=\frac{1}{2}g_{ij}v^iv^j -V(x),\N\]\Nand the associated action functional \N\[\N J[\alpha]=\int_0^T L(\alpha (t),\dot\alpha (t))dt. \N\]\NThe authors prove a generalization of the above classical result, where reflections are allowed. Two cases are treated: either \(M\) is a manifold with boudary \(Y = \partial M\), or \(Y\) is an embedded interior hypersurface of \(M\). The paths under consideration are required to undergo reflection at \(Y\) in the case \(Y = \partial M\), or permitted to undergo either reflection or transmission in the case of an interior hypersurface. \N\NThe main result of this paper is a Morse index theorem for the functional \(J\), both for trajectories with fixed boundary points and for the problem of periodic trajectories. Then an addition formula is proved. This allows the authors to compute the difference of the Morse index (with fixed endpoint conditions) of a concatenation of two paths and the sum of the Morse indices of the individual paths. The result is expressed as the index of the Hessian of the sum of the action functions along the two paths.