The Inverse Problem of the Calculus of Variation

Estomih Shedrack Massawe

doi:doi:10.11648/j.sjams.20160402.15

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The Inverse Problem of the Calculus of Variation

Estomih Shedrack Massawe

Published in Science Journal of Applied Mathematics and Statistics (Volume 4, Issue 2)

Received: 8 February 2016 Accepted: 9 March 2016 Published: 28 March 2016

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Abstract

In this paper, it is intended to determine the necessary and sufficient conditions for the existence and hence the construction of a Lagrangian of a dynamical system from its equations of motion. The existence of a Lagrangian is vital importance for the Hamiltonian description of a dynamical system since via the Legendre transformation we get the Hamiltonian of the system [1, 2]. It is also intended to show that the solution of the realization problem for the Hamiltonian system reduces to solving an inverse problem.

Published in	Science Journal of Applied Mathematics and Statistics (Volume 4, Issue 2)
DOI	10.11648/j.sjams.20160402.15
Page(s)	48-51
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), 2024. Published by Science Publishing Group

Keywords

Inverse Problems, Calculus of Variation, Realization Problem, Hamiltonian Systems

References

[1]	A. J. Van der Schaft, “Controllability and observability or affine nonlinear nonlinear Hamiltonian systems”, IEEE Trans, Automomatic Control, Vol AC-27, pp. 490-492, 1992.
[2]	W. M. Tulczyjew, “The Legendre transformation, Annales de I’Institut Henri Poincare–Section A – Vol XXII no. 1, pp 102-114, 1977.
[3]	S. Bernet and R. G. Cameroon, “Introduction to mathematical control systems”, Oxford University Press, New York, 1985.
[4]	J. C. Willems and J. C Van der Shaft, “Modelling of dynamical systems using external and internal variables with applications to Hamiltonian systems, Dynamical systems and Microphysics, pp 233-263, Academic Press, New York, 1982.
[5]	R. A. Abraham, J. E., Marsden, T. Ratiu, “Manifolds, Tensor analys and Applications, 2^nd edition, Springer-Verlag, New York, 1988.
[6]	S. P. Banks, “Mathematical theories of Nonlinear systems”, Prencice Hall, New York, 1988.
[7]	R. M. Santili, Foundations of Theoretical Mechanics I, Springer-Verlag, New York Inc., 1978.
[8]	A. J. Van der Schaft, System theoretic description of physical systems, Doctoral Thesis, Mathematical Centrum, Amsterdam, 1984.
[9]	W. M. Tulczyjew, Lagrangian submanifolds, statics and dynamics of mechanical systems, Dynamical systems and Microphysics, pp 3-25, Academic Press, New York, 1982.
[10]	A. J. Van der Schaft, Controllability and observability for affine nonlinear Hamiltonian systems, IEEE Trans, Automatic Control, Vol AC-27, pp 490-492, 1982.

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Estomih Shedrack Massawe. (2016). The Inverse Problem of the Calculus of Variation. Science Journal of Applied Mathematics and Statistics, 4(2), 48-51. https://doi.org/10.11648/j.sjams.20160402.15

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Estomih Shedrack Massawe. The Inverse Problem of the Calculus of Variation. Sci. J. Appl. Math. Stat. 2016, 4(2), 48-51. doi: 10.11648/j.sjams.20160402.15

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Estomih Shedrack Massawe. The Inverse Problem of the Calculus of Variation. Sci J Appl Math Stat. 2016;4(2):48-51. doi: 10.11648/j.sjams.20160402.15

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@article{10.11648/j.sjams.20160402.15,
  author = {Estomih Shedrack Massawe},
  title = {The Inverse Problem of the Calculus of Variation},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {4},
  number = {2},
  pages = {48-51},
  doi = {10.11648/j.sjams.20160402.15},
  url = {https://doi.org/10.11648/j.sjams.20160402.15},
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  abstract = {In this paper, it is intended to determine the necessary and sufficient conditions for the existence and hence the construction of a Lagrangian  of a dynamical system from its equations of motion. The existence of a Lagrangian is vital importance for the Hamiltonian description of a dynamical system since via the Legendre transformation  we get the Hamiltonian of the system [1, 2]. It is also intended to show that the solution of the realization problem for the Hamiltonian system reduces to solving an inverse problem.},
 year = {2016}
}

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TY  - JOUR
T1  - The Inverse Problem of the Calculus of Variation
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N1  - https://doi.org/10.11648/j.sjams.20160402.15
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AB  - In this paper, it is intended to determine the necessary and sufficient conditions for the existence and hence the construction of a Lagrangian  of a dynamical system from its equations of motion. The existence of a Lagrangian is vital importance for the Hamiltonian description of a dynamical system since via the Legendre transformation  we get the Hamiltonian of the system [1, 2]. It is also intended to show that the solution of the realization problem for the Hamiltonian system reduces to solving an inverse problem.
VL  - 4
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Author Information

Estomih Shedrack Massawe

Department of Mathematics, College of Natural and Applied Sciences, University of Dar es Salaam, Dar es Salaam, Tanzania

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Estomih Shedrack Massawe. (2016). The Inverse Problem of the Calculus of Variation. Science Journal of Applied Mathematics and Statistics, 4(2), 48-51. https://doi.org/10.11648/j.sjams.20160402.15

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ACS Style

Estomih Shedrack Massawe. The Inverse Problem of the Calculus of Variation. Sci. J. Appl. Math. Stat. 2016, 4(2), 48-51. doi: 10.11648/j.sjams.20160402.15

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AMA Style

Estomih Shedrack Massawe. The Inverse Problem of the Calculus of Variation. Sci J Appl Math Stat. 2016;4(2):48-51. doi: 10.11648/j.sjams.20160402.15

Copy | Download

@article{10.11648/j.sjams.20160402.15,
  author = {Estomih Shedrack Massawe},
  title = {The Inverse Problem of the Calculus of Variation},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {4},
  number = {2},
  pages = {48-51},
  doi = {10.11648/j.sjams.20160402.15},
  url = {https://doi.org/10.11648/j.sjams.20160402.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20160402.15},
  abstract = {In this paper, it is intended to determine the necessary and sufficient conditions for the existence and hence the construction of a Lagrangian  of a dynamical system from its equations of motion. The existence of a Lagrangian is vital importance for the Hamiltonian description of a dynamical system since via the Legendre transformation  we get the Hamiltonian of the system [1, 2]. It is also intended to show that the solution of the realization problem for the Hamiltonian system reduces to solving an inverse problem.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - The Inverse Problem of the Calculus of Variation
AU  - Estomih Shedrack Massawe
Y1  - 2016/03/28
PY  - 2016
N1  - https://doi.org/10.11648/j.sjams.20160402.15
DO  - 10.11648/j.sjams.20160402.15
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 48
EP  - 51
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20160402.15
AB  - In this paper, it is intended to determine the necessary and sufficient conditions for the existence and hence the construction of a Lagrangian  of a dynamical system from its equations of motion. The existence of a Lagrangian is vital importance for the Hamiltonian description of a dynamical system since via the Legendre transformation  we get the Hamiltonian of the system [1, 2]. It is also intended to show that the solution of the realization problem for the Hamiltonian system reduces to solving an inverse problem.
VL  - 4
IS  - 2
ER  -

Copy | Download