The Morse Index Theorem of Saddle Point Reduction and Applications to the Boundary Value Problems of Hamiltonian Systems on the Half Line

In the framework of saddle point reduction, we established a connection between the Maslov index of solutions to boundary value problems for Hamiltonian systems on $$[0, \infty )$$ and the Morse index of the corresponding functional defined through saddle point reduction. As an application, we demonstrated that boundary value problems for Hamiltonian systems on $$[0, \infty )$$ admit at least two nontrivial connecting orbits, utilizing the Morse index formula, saddle point reduction, and the three critical points theorem.