Symmetries and Conservation Laws for Hamiltonian Systems
American Journal of Applied Mathematics
Volume 4, Issue 3, June 2016, Pages: 132-136
Received: Apr. 19, 2016; Accepted: May 3, 2016; Published: May 14, 2016
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Estomih Shedrack Massawe, Department of Mathematics, University of Dar es Salaam, Dar es Salaam, Tanzania
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In this paper, it is shown that symmetry of a physical system is a transformation which may be applied to the state space without altering the system or its dynamical interaction in any way. The theory is applied to generalize the concept of symmetries and conservation laws with external to Hamiltonian systems with external forces. By this we obtain a generalized Noether’s Theorem which states that for Hamiltonian systems with external forces, a symmetry law generates a conservation law and vice versa.
Symmetries, Conservation Laws, Hamiltonian Systems
To cite this article
Estomih Shedrack Massawe, Symmetries and Conservation Laws for Hamiltonian Systems, American Journal of Applied Mathematics. Vol. 4, No. 3, 2016, pp. 132-136. doi: 10.11648/j.ajam.20160403.13
Copyright © 2016 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
V. I. Arnold, Mathematical Methods of Classical Mechanics (Translation of the 1974 Russian Edition), Springer, New York, 1978.
C. M. Marle, On Symmetries and Constants of Motion in Hamiltonian Systems with Non-Holonomic Constraints.
J. E. Marsden and A. Weinstern, Reduction of symplectic manifolds with symmetry, Reports on Mathematical Physics, 5, pp. 121-130, 1974.
A. J. Van der Schaft, Symmetries and Conservation Laws for Hamiltonian Systems with Inputs and Outputs: A Generalization of Noethers Theorem, Systems & Control Letters, Vol 1, pp. 108-115, 1981.
L. P. Landau and E. M. Lifschitz, Mechanics, Pergamon Press, Oxford, 1976.
A. J. Van der Schaft, System Theoretic Description of Physical System, Doctoral Thesis, Mathematical Centrum, Amsterdam, 1984.
V. I. Arnold and S. P. Novikov, Dynamical Systems IV, Springer-Verlag, Berlin, Heidelberg, 1990.
A. J. Van der Schaft, Symmetries, Conservation Laws and Time Reveribility for Hamiltonian Systems with External Forces, Journal of Mathematical Physics, Vol 24, pp. 2095-2101.
J. C. Willems and A. J. Van der Schaft, Modelling of Dynamical Systems Using External and Internal Variables with Application to Hamiltonian Systems, Dynamical Systems and Microphysics, pp. 233-263, Academic Press, New York, 1982.
S. P. Banks, Mathematical Theories of Nonlinear Systems, Prentice Hall, New York, 1988.
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