### Abstract

This paper gives a comprehensive analysis of the inverse problem of Lagrangian dynamics for the geodesic equations of the canonical linear connection on Lie groups of dimension four. Starting from the Lie algebra, in every case a faithful four-dimensional representation of the algebra is given as well as one in terms of vector fields and a representation of the linear group of which the given algebra is its Lie algebra. In each case the geodesic equations are calculated as a starting point for the inverse problem. Some results about first integrals of the geodesics are obtained. It is found that in three classes of algebra, there are algebraic obstructions to the existence of a Lagrangian, which can be determined directly from the Lie algebra without the need for any representation. In all other cases there are Lagrangians and indeed whole families of them. In many cases a formula for the most general Hessian of a Lagrangian is obtained.

Original language | English |
---|---|

Pages (from-to) | 395-425 |

Number of pages | 31 |

Journal | Journal of Lie Theory |

Volume | 14 |

Issue number | 2 |

Publication status | Published - 2004 |

Externally published | Yes |

### Fingerprint

### Keywords

- Canonical symmetric connection
- Euler-Lagrange equations
- First integral of geodesics
- Lagrangian
- Lie algebra
- Lie group

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Lie Theory*,

*14*(2), 395-425.

**Variationality of four-dimensional lie group connections.** / Ghanam, Ryad; Thompson, G.; Miller, E. J.

Research output: Contribution to journal › Article

*Journal of Lie Theory*, vol. 14, no. 2, pp. 395-425.

}

TY - JOUR

T1 - Variationality of four-dimensional lie group connections

AU - Ghanam, Ryad

AU - Thompson, G.

AU - Miller, E. J.

PY - 2004

Y1 - 2004

N2 - This paper gives a comprehensive analysis of the inverse problem of Lagrangian dynamics for the geodesic equations of the canonical linear connection on Lie groups of dimension four. Starting from the Lie algebra, in every case a faithful four-dimensional representation of the algebra is given as well as one in terms of vector fields and a representation of the linear group of which the given algebra is its Lie algebra. In each case the geodesic equations are calculated as a starting point for the inverse problem. Some results about first integrals of the geodesics are obtained. It is found that in three classes of algebra, there are algebraic obstructions to the existence of a Lagrangian, which can be determined directly from the Lie algebra without the need for any representation. In all other cases there are Lagrangians and indeed whole families of them. In many cases a formula for the most general Hessian of a Lagrangian is obtained.

AB - This paper gives a comprehensive analysis of the inverse problem of Lagrangian dynamics for the geodesic equations of the canonical linear connection on Lie groups of dimension four. Starting from the Lie algebra, in every case a faithful four-dimensional representation of the algebra is given as well as one in terms of vector fields and a representation of the linear group of which the given algebra is its Lie algebra. In each case the geodesic equations are calculated as a starting point for the inverse problem. Some results about first integrals of the geodesics are obtained. It is found that in three classes of algebra, there are algebraic obstructions to the existence of a Lagrangian, which can be determined directly from the Lie algebra without the need for any representation. In all other cases there are Lagrangians and indeed whole families of them. In many cases a formula for the most general Hessian of a Lagrangian is obtained.

KW - Canonical symmetric connection

KW - Euler-Lagrange equations

KW - First integral of geodesics

KW - Lagrangian

KW - Lie algebra

KW - Lie group

UR - http://www.scopus.com/inward/record.url?scp=4544333637&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4544333637&partnerID=8YFLogxK

M3 - Article

VL - 14

SP - 395

EP - 425

JO - Journal of Lie Theory

JF - Journal of Lie Theory

SN - 0949-5932

IS - 2

ER -