Versal unfoldings of equivariant linear Hamiltonian vector fields

1993 ◽  
Vol 114 (3) ◽  
pp. 559-573 ◽  
Author(s):  
Ian Melbourne
Keyword(s):  
Vector Fields ◽  
Hamiltonian Vector ◽  
Hamiltonian Vector Fields ◽  
Versal Deformations ◽  
Equivariant Version ◽  
Symplectic Matrices

AbstractWe prove an equivariant version of Galin's theorem on versal deformations of infinitesimally symplectic matrices. Matrix families of codimension zero and one are classified, and the results are used to study the movement of eigenvalues in one parameter families.

Normal forms for linear Hamiltonian vector fields commuting with the action of a compact Lie group

1993 ◽  
Vol 114 (2) ◽  
pp. 235-268 ◽  
Author(s):  
Ian Melbourne ◽  
Michael Dellnitz
Keyword(s):  
Lie Group ◽  
Vector Fields ◽  
Normal Forms ◽  
Hamiltonian Vector ◽  
Compact Lie Group ◽  
The Real ◽  
Hamiltonian Vector Fields ◽  
Symplectic Action ◽  
Symplectic Matrices ◽  
Real Complex

AbstractWe obtain normal forms for infinitesimally symplectic matrices (or linear Hamiltonian vector fields) that commute with the symplectic action of a compact Lie group of symmetries. In doing so we extend Williamson's theorem on normal forms when there is no symmetry present.Using standard representation-theoretic results the symmetry can be factored out and we reduce to finding normal forms over a real division ring. There are three real division rings consisting of the real, complex and quaternionic numbers. Of these, only the real case is covered in Williamson's original work.

scholarly journals Hamiltonian Vector Fields on Weil Bundles

2015 ◽  
Vol 7 (3) ◽  
Author(s):  
Norbert Mahoungou Moukala ◽  
Basile Guy Richard Bossoto
Keyword(s):  
Vector Fields ◽  
Hamiltonian Vector ◽  
Hamiltonian Vector Fields

A weight-homogenous condition to the real Jacobian conjecture in

2021 ◽  
pp. 1-9
Author(s):  
Francisco Braun ◽  
Claudia Valls
Keyword(s):  
Differential Equations ◽  
Vector Fields ◽  
Qualitative Theory ◽  
Hamiltonian Vector ◽  
Jacobian Conjecture ◽  
Local Diffeomorphism ◽  
The Real ◽  
Hamiltonian Vector Fields ◽  
Real Linear

Abstract It is known that a polynomial local diffeomorphism $(f,\, g): {\mathbb {R}}^{2} \to {\mathbb {R}}^{2}$ is a global diffeomorphism provided the higher homogeneous terms of $f f_x+g g_x$ and $f f_y+g g_y$ do not have real linear factors in common. Here, we give a weight-homogeneous framework of this result. Our approach uses qualitative theory of differential equations. In our reasoning, we obtain a result on polynomial Hamiltonian vector fields in the plane, generalization of a known fact.

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