The objective of this note is to prove an existence result for brake orbits in classical Hamiltonian systems (which was first proved by S.V. Bolotin) by using Floer theory. To this end, we compute an open string analogue of symplectic homology (so called wrapped Floer homology) of some domains in cotangent bundles, which appear naturally in the study of classical Hamiltonian systems. The main part of the computations is to show invariance of wrapped Floer homology under certain handle attaching to domains.
"Handle attaching in wrapped Floer homology and brake orbits in classical Hamiltonian systems." Osaka J. Math. 50 (2) 363 - 396, June 2013.