### Abstract

The purpose of this paper is to extend the symmetric representation of the rigid body equations from the group SO∈(n) to other groups. These groups are matrix subgroups of the general linear group that are defined by a quadratic matrix identity. Their corresponding Lie algebras include several classical semisimple matrix Lie algebras. The approach is to start with an optimal control problem on these groups that generates geodesics for a left-invariant metric. Earlier work by Bloch, Crouch, Marsden, and Ratiu defines the symmetric representation of the rigid body equations, which is obtained by solving the same optimal control problem in the particular case of the Lie group SO∈(n). This paper generalizes this symmetric representation to a wider class of matrix groups satisfying a certain quadratic matrix identity. We consider the relationship between this symmetric representation of the generalized rigid body equations and the generalized rigid body equations themselves. A discretization of this symmetric representation is constructed making use of the symmetry, which in turn give rise to numerical algorithms to integrate the generalized rigid body equations for the given class of matrix Lie groups.

Original language | English (US) |
---|---|

Pages (from-to) | 469-500 |

Number of pages | 32 |

Journal | Foundations of Computational Mathematics |

Volume | 8 |

Issue number | 4 |

DOIs | |

State | Published - Aug 1 2008 |

Externally published | Yes |

### Fingerprint

### Keywords

- Generalized rigid body equations
- Geodesics
- Optimal control

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics
- Mathematics(all)
- Applied Mathematics
- Computational Mathematics

### Cite this

*Foundations of Computational Mathematics*,

*8*(4), 469-500. https://doi.org/10.1007/s10208-008-9025-1

**Optimal control and geodesics on quadratic matrix Lie groups.** / Bloch, Anthony M.; Crouch, Peter E.; Marsden, Jerrold E.; Sanyal, Amit.

Research output: Contribution to journal › Article

*Foundations of Computational Mathematics*, vol. 8, no. 4, pp. 469-500. https://doi.org/10.1007/s10208-008-9025-1

}

TY - JOUR

T1 - Optimal control and geodesics on quadratic matrix Lie groups

AU - Bloch, Anthony M.

AU - Crouch, Peter E.

AU - Marsden, Jerrold E.

AU - Sanyal, Amit

PY - 2008/8/1

Y1 - 2008/8/1

N2 - The purpose of this paper is to extend the symmetric representation of the rigid body equations from the group SO∈(n) to other groups. These groups are matrix subgroups of the general linear group that are defined by a quadratic matrix identity. Their corresponding Lie algebras include several classical semisimple matrix Lie algebras. The approach is to start with an optimal control problem on these groups that generates geodesics for a left-invariant metric. Earlier work by Bloch, Crouch, Marsden, and Ratiu defines the symmetric representation of the rigid body equations, which is obtained by solving the same optimal control problem in the particular case of the Lie group SO∈(n). This paper generalizes this symmetric representation to a wider class of matrix groups satisfying a certain quadratic matrix identity. We consider the relationship between this symmetric representation of the generalized rigid body equations and the generalized rigid body equations themselves. A discretization of this symmetric representation is constructed making use of the symmetry, which in turn give rise to numerical algorithms to integrate the generalized rigid body equations for the given class of matrix Lie groups.

AB - The purpose of this paper is to extend the symmetric representation of the rigid body equations from the group SO∈(n) to other groups. These groups are matrix subgroups of the general linear group that are defined by a quadratic matrix identity. Their corresponding Lie algebras include several classical semisimple matrix Lie algebras. The approach is to start with an optimal control problem on these groups that generates geodesics for a left-invariant metric. Earlier work by Bloch, Crouch, Marsden, and Ratiu defines the symmetric representation of the rigid body equations, which is obtained by solving the same optimal control problem in the particular case of the Lie group SO∈(n). This paper generalizes this symmetric representation to a wider class of matrix groups satisfying a certain quadratic matrix identity. We consider the relationship between this symmetric representation of the generalized rigid body equations and the generalized rigid body equations themselves. A discretization of this symmetric representation is constructed making use of the symmetry, which in turn give rise to numerical algorithms to integrate the generalized rigid body equations for the given class of matrix Lie groups.

KW - Generalized rigid body equations

KW - Geodesics

KW - Optimal control

UR - http://www.scopus.com/inward/record.url?scp=47249123754&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=47249123754&partnerID=8YFLogxK

U2 - 10.1007/s10208-008-9025-1

DO - 10.1007/s10208-008-9025-1

M3 - Article

AN - SCOPUS:47249123754

VL - 8

SP - 469

EP - 500

JO - Foundations of Computational Mathematics

JF - Foundations of Computational Mathematics

SN - 1615-3375

IS - 4

ER -