## Abstract

This paper seeks to generalise one aspect of classical Krein theory for linear Hamiltonian systems by examining how the existence of a non-trivial, homogeneous, polynomial W of degree m ≥ 2 with (Ax, ∇W(x)) = 0,x ∈ ℝ^{N}, affects the spectrum of a real linear transformation A on ℝ^{N}. Amongst other things it is shown that (i) such a W exists if, and only if, the spectrum of A is linearly dependent over the natural numbers, and (ii) there exists such a W which is non-degenerate if, and only if, all the eigenvalues of A are imaginary and semi-simple. In classical Krein theory W is quadratic. Our enquiry is motivated by a theory of topological invariants for dynamical systems which have a first integral. Degenerate Hamiltonian systems are a special class where the present considerations are relevant.

Original language | English |
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Pages (from-to) | 99-122 |

Number of pages | 24 |

Journal | Journal of Geometry and Physics |

Volume | 19 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 1996 |

## Keywords

- Degenerate hamiltonian systems
- Flows with a first integral
- Homotopy invariant
- Krein theory
- Polynomial invariants

## ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Geometry and Topology