3 research outputs found
A Note on the First Integrals of Vector Fields with Integrating Factors and Normalizers
We prove a sufficient condition for the existence of explicit first integrals
for vector fields which admit an integrating factor. This theorem recovers and
extends previous results in the literature on the integrability of vector
fields which are volume preserving and possess nontrivial normalizers. Our
approach is geometric and coordinate-free and hence it works on any smooth
orientable manifold
Birational maps from polarization and the preservation of measure and integrals
The main result of this paper is the discretization of Hamiltonian systems of
the form , where is a constant symmetric matrix
and is a polynomial of degree in
any number of variables . The discretization uses the method of polarization
and preserves both the energy and the invariant measure of the differential
equation, as well as the dimension of the phase space. This generalises earlier
work for discretizations of first order systems with , and of second order
systems with and .Comment: Updated to final pre-publication versio