Real transformations with polynomial invariants

E. N. Dancer, J. F. Toland

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1 Citation (SciVal)


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 ∈ ℝN, affects the spectrum of a real linear transformation A on ℝN. 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 languageEnglish
Pages (from-to)99-122
Number of pages24
JournalJournal of Geometry and Physics
Issue number2
Publication statusPublished - Jun 1996


  • 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


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