article
Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci
Robert I
McLachlan
author
Christian
Offen
author 85279https://orcid.org/0000-0002-5940-8057
636
department
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.
https://ris.uni-paderborn.de/download/19943/19946/Hamiltonian_Boundary_Value_Problems,_Conformal_Symplectic_Symmetries,_and_Conjugate_Loci.pdf
application/pdfno
2018
eng
Hamiltonian boundary value problemssingularitiesconformal symplectic geometrycatastrophe theoryconjugate loci
New Zealand Journal of Mathematics
4883-99
yes
McLachlan, Robert I, and Christian Offen. “Hamiltonian Boundary Value Problems, Conformal Symplectic Symmetries, and Conjugate Loci.” <i>New Zealand Journal of Mathematics</i> 48 (2018): 83–99.
@article{McLachlan_Offen_2018, title={Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci}, volume={48}, journal={New Zealand Journal of Mathematics}, author={McLachlan, Robert I and Offen, Christian}, year={2018}, pages={83–99} }
R. I. McLachlan and C. Offen, “Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci,” <i>New Zealand Journal of Mathematics</i>, vol. 48, pp. 83–99, 2018.
R.I. McLachlan, C. Offen, New Zealand Journal of Mathematics 48 (2018) 83–99.
McLachlan RI, Offen C. Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci. <i>New Zealand Journal of Mathematics</i>. 2018;48:83-99.
McLachlan, Robert I., and Christian Offen. “Hamiltonian Boundary Value Problems, Conformal Symplectic Symmetries, and Conjugate Loci.” <i>New Zealand Journal of Mathematics</i>, vol. 48, 2018, pp. 83–99.
McLachlan, R. I., & Offen, C. (2018). Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci. <i>New Zealand Journal of Mathematics</i>, <i>48</i>, 83–99.
199432020-10-06T16:39:08Z2020-10-07T14:04:01Z