Symmetries and Conservation Laws for Hamiltonian Systems

Estomih Shedrack Massawe

doi:doi:10.11648/j.ajam.20160403.13

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Symmetries and Conservation Laws for Hamiltonian Systems

Estomih Shedrack Massawe

Published in American Journal of Applied Mathematics (Volume 4, Issue 3)

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.

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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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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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TY  - JOUR
T1  - Symmetries and Conservation Laws for Hamiltonian Systems
AU  - Estomih Shedrack Massawe
Y1  - 2016/05/14
PY  - 2016
N1  - https://doi.org/10.11648/j.ajam.20160403.13
DO  - 10.11648/j.ajam.20160403.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 132
EP  - 136
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20160403.13
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
IS  - 3
ER  -

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Author Information

Estomih Shedrack Massawe

Department of Mathematics, University of Dar es Salaam, Dar es Salaam, Tanzania

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Plain Text BibTeX RIS

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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - Symmetries and Conservation Laws for Hamiltonian Systems
AU  - Estomih Shedrack Massawe
Y1  - 2016/05/14
PY  - 2016
N1  - https://doi.org/10.11648/j.ajam.20160403.13
DO  - 10.11648/j.ajam.20160403.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 132
EP  - 136
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20160403.13
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
IS  - 3
ER  -

Copy | Download