---
_id: '19943'
abstract:
- lang: eng
  text: '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. '
article_type: original
author:
- first_name: Robert I
  full_name: McLachlan, Robert I
  last_name: McLachlan
- first_name: Christian
  full_name: Offen, Christian
  id: '85279'
  last_name: Offen
  orcid: https://orcid.org/0000-0002-5940-8057
citation:
  ama: 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.
    doi:<a href="https://doi.org/10.53733/34 ">10.53733/34 </a>
  apa: McLachlan, R. I., &#38; 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. <a href="https://doi.org/10.53733/34 ">https://doi.org/10.53733/34
    </a>
  bibtex: '@article{McLachlan_Offen_2018, title={Hamiltonian boundary value problems,
    conformal symplectic symmetries, and conjugate loci}, volume={48}, DOI={<a href="https://doi.org/10.53733/34
    ">10.53733/34 </a>}, journal={New Zealand Journal of Mathematics}, author={McLachlan,
    Robert I and Offen, Christian}, year={2018}, pages={83–99} }'
  chicago: '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. <a href="https://doi.org/10.53733/34
    ">https://doi.org/10.53733/34 </a>.'
  ieee: '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, doi: <a href="https://doi.org/10.53733/34 ">10.53733/34
    </a>.'
  mla: 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, doi:<a href="https://doi.org/10.53733/34
    ">10.53733/34 </a>.
  short: R.I. McLachlan, C. Offen, New Zealand Journal of Mathematics 48 (2018) 83–99.
date_created: 2020-10-06T16:39:08Z
date_updated: 2023-09-21T07:29:24Z
ddc:
- '510'
department:
- _id: '636'
doi: '10.53733/34 '
extern: '1'
external_id:
  arxiv:
  - '1804.07479'
file:
- access_level: open_access
  content_type: application/pdf
  creator: coffen
  date_created: 2020-10-06T16:49:29Z
  date_updated: 2020-10-07T14:04:01Z
  file_id: '19946'
  file_name: Hamiltonian_Boundary_Value_Problems,_Conformal_Symplectic_Symmetries,_and_Conjugate_Loci.pdf
  file_size: 3126111
  relation: main_file
  title: Hamiltonian Boundary Value Problems, Conformal Symplectic Symmetries, and
    Conjugate Loci
file_date_updated: 2020-10-07T14:04:01Z
has_accepted_license: '1'
intvolume: '        48'
keyword:
- Hamiltonian boundary value problems
- singularities
- conformal symplectic geometry
- catastrophe theory
- conjugate loci
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://nzjmath.org/index.php/NZJMATH/article/view/34
oa: '1'
page: 83-99
publication: New Zealand Journal of Mathematics
publication_status: published
quality_controlled: '1'
status: public
title: Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate
  loci
type: journal_article
user_id: '85279'
volume: 48
year: '2018'
...