Some General Principles of Integration of Hamiltonian Systems of Differential Equations

In the following pages it is proposed to develop ab initio a theory of periodic solutions of Hamiltonian systems of differential equations. Such solutions are of theoretical importance for the following reason: that whereas the attempt to obtain, for a real Hamiltonian system, solutions valid for all real values of the independent variable leads in general to divergent series, for certain solutions which are formally periodic the series can be proved convergent. In the words of Poincare, “ce qui nous rend ces solutions périodiques si précieuses, c’est qu’elles sont, pour ainsi dire, la seule breche paroù nous puissions essayer de pénétrer dans une place jusqu'ici reputee inabordable. The existing theory of periodic solutions of differential equations was developed by Poincare mainly with reference to the equations of Celestial Mechanics. With a suitable choice of co-ordinates these are of the Hamiltonian form.

Download Full-text

On symmetry in periodic solutions of hamiltonian systems

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700028494 ◽

1965 ◽

Vol 5 (4) ◽

pp. 463-486

Author(s):

Arthur R. Jones

Keyword(s):

Differential Equations ◽

Periodic Solutions ◽

Hamiltonian Systems ◽

Symmetry Property ◽

Systems Of Differential Equations

In this paper the theory of periodic solutions of analytic Hamiltonian systems of differential equations, which is due to Cherry [5], is specialized to systems which have one symmetry property.

Download Full-text

A NEW TOPOLOGICAL INVARIANT OF TOPOLOGICAL HAMILTONIAN SYSTEMS OF DIFFERENTIAL EQUATIONS AND APPLICATIONS TO PROBLEMS IN PHYSICS AND MECHANICS

Mechanics, Analysis and Geometry: 200 Years After Lagrange ◽

10.1016/b978-0-444-88958-4.50009-4 ◽

1991 ◽

pp. 127-155

Author(s):

A.T. FOMENKO

Keyword(s):

Differential Equations ◽

Hamiltonian Systems ◽

Topological Invariant ◽

Systems Of Differential Equations

Download Full-text

Homoclinic solutions for coupled systems of differential equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500014335 ◽

1985 ◽

Vol 99 (3-4) ◽

pp. 319-328 ◽

Cited By ~ 4

Author(s):

P. Grindrod ◽

B. D. Sleeman

Keyword(s):

Differential Equations ◽

Hamiltonian Systems ◽

Reaction Diffusion ◽

Homoclinic Orbits ◽

Homoclinic Solutions ◽

Coupled Systems ◽

Reaction Diffusion Equations ◽

Diffusion Equations ◽

Systems Of Differential Equations ◽

Unmyelinated Nerve

SynopsisTopological ideas based on the notion of flows and Wazewski sets are used to establish the existence of homoclinic orbits to a class of Hamiltonian systems. The results, as indicated, are applicable to a variety of reaction diffusion equations including models of bundles of unmyelinated nerve axons.

Download Full-text

Numerical Integration of Nearly-Hamiltonian Systems

International Astronomical Union Colloquium ◽

10.1017/s0252921100062242 ◽

1978 ◽

Vol 41 ◽

pp. 159-173

Author(s):

Victor R. Bond

Keyword(s):

Phase Space ◽

Differential Equations ◽

Numerical Integration ◽

Hamiltonian Systems ◽

Hamiltonian Function ◽

Extended Phase Space ◽

Extended Phase ◽

Systems Of Differential Equations

AbstractConsideration is given to the solution by numerical integration of systems of differential equations that are derived from a Hamiltonian function in the extended phase space plus additional forces not included in the Hamiltonian (that is, nearly-Hamiltonian systems). An extended phase space Hamiltonian which vanishes initially will vanish on any solution of the system differential equations. Furthermore, it vanishes in spite of the additional forces, and defines a surface in the extended phase space upon which the solution is constrained.

Download Full-text