Hamiltonian Boundary Value Problems, Conformal Symplectic Symmetries, and Conjugate Loci

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New Zealand Journal of Mathematics

Vol. 48, (2018), Pages 83-99


Robert I McLachlan

Institute of Fundamental Sciences

Massey University

Palmerston North

New Zealand

+64 6 951 7652

mailto:r.mclachlan@massey.ac.nz

Christian Offen

Institute of Fundamental Sciences

Massey University

Palmerston North

New Zealand

+64 6 951 8707

mailto:c.offen@massey.ac.nz




Abstract In this paper we continue our study of bifurcations of solutions of boundary-value problems for symplectic maps arising as Hamiltonian diffeomorphisms. These have been shown to be connected to catastrophe theory via generating functions and ordinary and reversal phase space symmetries have been considered. Here we present a convenient, coordinate free framework to analyse separated Lagrangian boundary value problems which include classical Dirichlet, Neumann and Robin boundary value problems. The framework is then used to prove the existence of obstructions arising from conformal symplectic symmetries on the bifurcation behaviour of solutions to Hamiltonian boundary value problems. Under non-degeneracy conditions, a group action by conformal symplectic symmetries has the effect that the flow map cannot degenerate in a direction which is tangential to the action. This imposes restrictions on which singularities can occur in boundary value problems. Our results generalise classical results about conjugate loci on Riemannian manifolds to a large class of Hamiltonian boundary value problems with, for example, scaling symmetries.

Keywords Hamiltonian boundary value problems; singularities; conformal symplectic geometry; catastrophe theory; conjugate loci.

Classification (MSC2000) 37J20.

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