American Institute of Mathematical Sciences

Advanced
March  2010, 2(1): 51-68. doi: 10.3934/jgm.2010.2.51

Geodesic boundary value problems with symmetry

Colin J. Cotter 1, and  Darryl D. Holm 2,

1. 

Department of Aeronautics, Imperial College London, London SW7 2AZ, United Kingdom
2. 

Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

Received  December 2009 Revised  February 2010 Published  April 2010

This paper shows how commuting left and right actions of Lie groups on a manifold may be used to complement one another in a variational reformulation of optimal control problems as geodesic boundary value problems with symmetry. In such problems, the endpoint boundary condition is only specified up to the right action of a symmetry group. In this paper we show how to reformulate the problem by introducing extra degrees of freedom so that the endpoint condition specifies a single point on the manifold. We prove an equivalence theorem to this effect and illustrate it with several examples. In finite-dimensions, we discuss geodesic flows on the Lie groups $SO(3)$ and $SE(3)$ under the left and right actions of their respective Lie algebras. In an infinite-dimensional example, we discuss optimal large-deformation matching of one closed oriented curve to another embedded in the same plane. In the curve-matching example, the manifold Emb$(S^1, \mathbb{R}^2)$ comprises the space $S^1$ embedded in the plane $\mathbb{R}^2$. The diffeomorphic left action Diff$(\mathbb{R}^2)$ deforms the curve by a smooth invertible time-dependent transformation of the coordinate system in which it is embedded, while leaving the parameterisation of the curve invariant. The diffeomorphic right action Diff$(S^1)$ corresponds to a smooth invertible reparameterisation of the $S^1$ domain coordinates of the curve. As we show, this right action unlocks an important degree of freedom for geodesically matching the curve shapes using an equivalent fixed boundary value problem, without being constrained to match corresponding points along the template and target curves at the endpoint in time.
Keywords: Shape space, symmetry reduction., geodesics on manifolds.
Mathematics Subject Classification: Primary: 70H33, 58E50; Secondary: 92C5.
Citation: Colin J. Cotter, Darryl D. Holm. Geodesic boundary value problems with symmetry. Journal of Geometric Mechanics, 2010, 2 (1) : 51-68. doi: 10.3934/jgm.2010.2.51
[1]

Alex Eskin, Maryam Mirzakhani. Counting closed geodesics in moduli space. Journal of Modern Dynamics, 2011, 5 (1) : 71-105. doi: 10.3934/jmd.2011.5.71
[2]

R. Bartolo, Anna Maria Candela, J.L. Flores. Timelike Geodesics in stationary Lorentzian manifolds with unbounded coefficients. Conference Publications, 2005, 2005 (Special) : 70-76. doi: 10.3934/proc.2005.2005.70
[3]

Samir Chowdhury, Facundo Mémoli. Explicit geodesics in Gromov-Hausdorff space. Electronic Research Announcements, 2018, 25: 48-59. doi: 10.3934/era.2018.25.006
[4]

Martin Bauer, Philipp Harms, Peter W. Michor. Sobolev metrics on shape space of surfaces. Journal of Geometric Mechanics, 2011, 3 (4) : 389-438. doi: 10.3934/jgm.2011.3.389
[5]

E. García-Toraño Andrés, Bavo Langerock, Frans Cantrijn. Aspects of reduction and transformation of Lagrangian systems with symmetry. Journal of Geometric Mechanics, 2014, 6 (1) : 1-23. doi: 10.3934/jgm.2014.6.1
[6]

Keith Burns, Eugene Gutkin. Growth of the number of geodesics between points and insecurity for Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 403-413. doi: 10.3934/dcds.2008.21.403
[7]

Giovanni De Matteis, Gianni Manno. Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 453-481. doi: 10.3934/cpaa.2014.13.453
[8]

Jan J. Dijkstra and Jan van Mill. Homeomorphism groups of manifolds and Erdos space. Electronic Research Announcements, 2004, 10: 29-38.

[9]

L. Búa, T. Mestdag, M. Salgado. Symmetry reduction, integrability and reconstruction in $k$-symplectic field theory. Journal of Geometric Mechanics, 2015, 7 (4) : 395-429. doi: 10.3934/jgm.2015.7.395
[10]

Claudio Meneses. Linear phase space deformations with angular momentum symmetry. Journal of Geometric Mechanics, 2019, 11 (1) : 45-58. doi: 10.3934/jgm.2019003
[11]

Hui Liu, Yiming Long, Yuming Xiao. The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3803-3829. doi: 10.3934/dcds.2018165
[12]

Jeffrey K. Lawson, Tanya Schmah, Cristina Stoica. Euler-Poincaré reduction for systems with configuration space isotropy. Journal of Geometric Mechanics, 2011, 3 (2) : 261-275. doi: 10.3934/jgm.2011.3.261
[13]

Sebastián Ferrer, Francisco Crespo. Alternative angle-based approach to the $\mathcal{KS}$-Map. An interpretation through symmetry and reduction. Journal of Geometric Mechanics, 2018, 10 (3) : 359-372. doi: 10.3934/jgm.2018013
[14]

Martin Bauer, Philipp Harms, Peter W. Michor. Sobolev metrics on shape space, II: Weighted Sobolev metrics and almost local metrics. Journal of Geometric Mechanics, 2012, 4 (4) : 365-383. doi: 10.3934/jgm.2012.4.365
[15]

Matteo Novaga, Diego Pallara, Yannick Sire. A symmetry result for degenerate elliptic equations on the Wiener space with nonlinear boundary conditions and applications. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 815-831. doi: 10.3934/dcdss.2016030
[16]

Alexander Nabutovsky and Regina Rotman. Lengths of geodesics between two points on a Riemannian manifold. Electronic Research Announcements, 2007, 13: 13-20.

[17]

Eva Glasmachers, Gerhard Knieper, Carlos Ogouyandjou, Jan Philipp Schröder. Topological entropy of minimal geodesics and volume growth on surfaces. Journal of Modern Dynamics, 2014, 8 (1) : 75-91. doi: 10.3934/jmd.2014.8.75
[18]

Abbas Bahri. Attaching maps in the standard geodesics problem on $S^2$. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 379-426. doi: 10.3934/dcds.2011.30.379
[19]

Amin Boumenir. Determining the shape of a solid of revolution. Mathematical Control & Related Fields, 2019, 9 (3) : 509-515. doi: 10.3934/mcrf.2019023
[20]

Hiroaki Yoshimura, Jerrold E. Marsden. Dirac cotangent bundle reduction. Journal of Geometric Mechanics, 2009, 1 (1) : 87-158. doi: 10.3934/jgm.2009.1.87

2018 Impact Factor: 0.525

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences