The Hamiltonian Structures of 3D ODE with Time-Independent Invariants

Guo Zhong-heng; Chen Yu-ming

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Article Navigation > Applied Mathematics and Mechanics > 1995 > 16(4): 283-288

Guo Zhong-heng, Chen Yu-ming. The Hamiltonian Structures of 3D ODE with Time-Independent Invariants[J]. Applied Mathematics and Mechanics, 1995, 16(4): 283-288.

Citation:

Guo Zhong-heng, Chen Yu-ming. The Hamiltonian Structures of 3D ODE with Time-Independent Invariants[J]. Applied Mathematics and Mechanics, 1995, 16(4): 283-288.

Citation:

Guo Zhong-heng, Chen Yu-ming. The Hamiltonian Structures of 3D ODE with Time-Independent Invariants[J]. Applied Mathematics and Mechanics, 1995, 16(4): 283-288.

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The Hamiltonian Structures of 3D ODE with Time-Independent Invariants

Guo Zhong-heng¹,
Chen Yu-ming²

1.
Department of Mathematics, Peking University, Beijing, 100871;
2.
Department of Applied Mathematics, Hu'nan University, Changsha, 410012

Received Date: 1994-04-01
Publish Date: 1995-04-15

Abstract

Abstract

We have proved that any 3-dimensional dynamical system of ordinary differential equations(in short, 3D ODE)With time-independent invariants can be rewritten as Haniltonian systems with respect to generalized Poisson brackets and the Hamiltonians are these invariants. As an example,we discuss the Kermack-Mckendrick modelfor epidemics in detail. The results we obtained are generalization of those obtained by Y. Nutku.
- Poisson bracket,
- Hamiltonian structure,
- bi-Hamiltonianstructure,
- invariant,
- the Kermack-Mckendrick model for epidemics

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References(7)

References

[1]	Nutku, Y., Bi-Hamiltonian structure of the Kermack-Mckendrick model for epidemics, J. Phys. A: Math. Gen., 23(1990), L1145-L1146.
[2]	Krishnaprasad, P. S. and J. E. Marsden, Hamiltonian structures and stability for rigid bodies with flexible attachments,Arch. Rational Mech. Anal., 98(1987), 71-93.
[3]	Andrey, L., The rate of entropy change in non-Hamiltonian systems, Phys. Lett. A., 111(1985), 45-46.
[4]	Gonzalez-Gascon, F., Note on a paper of Andrey concerning non-Hamiltonian systems, Phys. Lett. A., 114(1986), 61-62.
[5]	Nutku, Y., Hamiltonian structure of the Lotka-Volterra equations, Phvs. Lett. A., 145(1990), 27-28.
[6]	Olver, P. J., Applications of Lie Groups to Differential Eguations, Springer-Verlag, New York Inc. (1986).
[7]	John, F., Partial Differential Eguations, 4th ed., Springer-Verlag, New York(1982).