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Symmetries and Conservation Laws for Hamiltonian Systems

Received: 19 April 2016     Accepted: 3 May 2016     Published: 14 May 2016
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Abstract

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.

Published in American Journal of Applied Mathematics (Volume 4, Issue 3)
DOI 10.11648/j.ajam.20160403.13
Page(s) 132-136
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Symmetries, Conservation Laws, Hamiltonian Systems

References
[1] V. I. Arnold, Mathematical Methods of Classical Mechanics (Translation of the 1974 Russian Edition), Springer, New York, 1978.
[2] C. M. Marle, On Symmetries and Constants of Motion in Hamiltonian Systems with Non-Holonomic Constraints.
[3] J. E. Marsden and A. Weinstern, Reduction of symplectic manifolds with symmetry, Reports on Mathematical Physics, 5, pp. 121-130, 1974.
[4] 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.
[5] L. P. Landau and E. M. Lifschitz, Mechanics, Pergamon Press, Oxford, 1976.
[6] A. J. Van der Schaft, System Theoretic Description of Physical System, Doctoral Thesis, Mathematical Centrum, Amsterdam, 1984.
[7] V. I. Arnold and S. P. Novikov, Dynamical Systems IV, Springer-Verlag, Berlin, Heidelberg, 1990.
[8] 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.
[9] 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.
[10] S. P. Banks, Mathematical Theories of Nonlinear Systems, Prentice Hall, New York, 1988.
Cite This Article
  • APA Style

    Estomih Shedrack Massawe. (2016). Symmetries and Conservation Laws for Hamiltonian Systems. American Journal of Applied Mathematics, 4(3), 132-136. https://doi.org/10.11648/j.ajam.20160403.13

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    ACS Style

    Estomih Shedrack Massawe. Symmetries and Conservation Laws for Hamiltonian Systems. Am. J. Appl. Math. 2016, 4(3), 132-136. doi: 10.11648/j.ajam.20160403.13

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    AMA Style

    Estomih Shedrack Massawe. Symmetries and Conservation Laws for Hamiltonian Systems. Am J Appl Math. 2016;4(3):132-136. doi: 10.11648/j.ajam.20160403.13

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  • @article{10.11648/j.ajam.20160403.13,
      author = {Estomih Shedrack Massawe},
      title = {Symmetries and Conservation Laws for Hamiltonian Systems},
      journal = {American Journal of Applied Mathematics},
      volume = {4},
      number = {3},
      pages = {132-136},
      doi = {10.11648/j.ajam.20160403.13},
      url = {https://doi.org/10.11648/j.ajam.20160403.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20160403.13},
      abstract = {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.},
     year = {2016}
    }
    

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    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
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    AB  - 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.
    VL  - 4
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Author Information
  • Department of Mathematics, University of Dar es Salaam, Dar es Salaam, Tanzania

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