Taylor Series and Getting the General Solutions for the Equations of Motion Using Poisson Bracket Relations

Ibtisam Frhan Al-Maaitah

doi:doi:10.11648/j.wjap.20210603.12

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Taylor Series and Getting the General Solutions for the Equations of Motion Using Poisson Bracket Relations

Ibtisam Frhan Al-Maaitah

Published in World Journal of Applied Physics (Volume 6, Issue 3)

Received: 26 June 2021 Accepted: 28 July 2021 Published: 4 August 2021

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Abstract

In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called canonical transformations, which map canonical coordinate systems into canonical coordinate systems. In this work we study some examples from the classical mechanics of particles and apply mathematical method for building the equation of motion. In the present paper Poisson Brackets and their properties are presented, by using Poisson brackets and their properties we calculate some brackets. We use the Poisson bracket with Hamiltonians to express the time dependence of a function u (t), the main idea Taylor series is taken as the required solution for equation of motion using the properties of the Poisson Brackets, We have examined examples from the classical mechanics to illustrate the idea such as motion with a constant acceleration, simple harmonic oscillator, freely falling particle. The solutions are compatible with what is known in classical mechanics. The work is fundamental and sheds new light onto classical mechanics. Poisson brackets are a powerful and sophisticated tool in the Hamiltonian formalism of Classical Mechanics. They also happen to provide a direct link between classical and quantum mechanics.

Published in	World Journal of Applied Physics (Volume 6, Issue 3)
DOI	10.11648/j.wjap.20210603.12
Page(s)	47-51
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), 2021. Published by Science Publishing Group

Keywords

Poisson Brackets, JACOBI Identity, Taylor Series, Leibniz rule, Harmonic Oscillator, Hamilton Function, Generalized Coordinates, Generalized Momenta

References

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[2]	L. D. Landau; E. M. Lifshitz, Course of T heoretical Physics, Vol. 1–Mechanics. Franklin Book Company. (1972). ISBN978-0-08-016739-8.
[3]	Edward A. Desloge, Classical Mechanics, Volume1, John Wiley and sons, (1982).
[4]	Atamp. Arya, Introduction to classical Mechanics, Allyn and Bacon, (1990).
[5]	Dare A. Wells, Lagrangian Dynamics, New York McGraw–Hill, (1967).
[6]	Casetta, L. Poisson brackets formulation for the dynamics of aposition-dependent mass particle. Acta Mech 228, 4491–4496 (2017).
[7]	N. M. Bodunov and V. I. Khaliulin, Journal of Physics 1679 (2020) 032086.
[8]	S. G. Dani (2012)." Ancient Indian Mathematics–A Conspectus". Resonance. 17 (3): 236–246.
[9]	M. Abramowitz and I. A. Stegun, (Eds.). Hand book of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 9 th printing. New York: Dover, p. 880, 1972.
[10]	G. Arfken, "Taylor's Expansion." 5.6 in Mathematical Methods for Physicists, 3^rded. Orlando, FL: Academic Press, pp. 303-313, 1985.
[11]	E. N. Moore, Theoretical Mechanics, New York, John Wiley and sons, (1983).
[12]	R. a. Becker Introduction to Theoretical Mechanics, New York: McGraw-Hill, (1945).
[13]	Jerry B. Marion, Classical Dynamics of Particles and Systems, 2^nded., Academic press. (1970).
[14]	Ket hR. Symon, Mechanice 3^rded. Addison–Wesley Publishing cob, (1980).
[15]	Grant R. Fowles and GeorgeL. Cassiday, Analytical Mechanics 5^thed. Saunders Golden Sunburst series, (1993).
[16]	E. F. Taylor, Introductory Mechanics, New York, John Wiley and sons, (1963).
[17]	Kittel, Knight, Ruderman, Helmholz and Moyer, Mechanics (Berkeley Physics Course Vo. l1).

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APA Style

Ibtisam Frhan Al-Maaitah. (2021). Taylor Series and Getting the General Solutions for the Equations of Motion Using Poisson Bracket Relations. World Journal of Applied Physics, 6(3), 47-51. https://doi.org/10.11648/j.wjap.20210603.12

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ACS Style

Ibtisam Frhan Al-Maaitah. Taylor Series and Getting the General Solutions for the Equations of Motion Using Poisson Bracket Relations. World J. Appl. Phys. 2021, 6(3), 47-51. doi: 10.11648/j.wjap.20210603.12

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AMA Style

Ibtisam Frhan Al-Maaitah. Taylor Series and Getting the General Solutions for the Equations of Motion Using Poisson Bracket Relations. World J Appl Phys. 2021;6(3):47-51. doi: 10.11648/j.wjap.20210603.12

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@article{10.11648/j.wjap.20210603.12,
  author = {Ibtisam Frhan Al-Maaitah},
  title = {Taylor Series and Getting the General Solutions for the Equations of Motion Using Poisson Bracket Relations},
  journal = {World Journal of Applied Physics},
  volume = {6},
  number = {3},
  pages = {47-51},
  doi = {10.11648/j.wjap.20210603.12},
  url = {https://doi.org/10.11648/j.wjap.20210603.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.wjap.20210603.12},
  abstract = {In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called canonical transformations, which map canonical coordinate systems into canonical coordinate systems. In this work we study some examples from the classical mechanics of particles and apply mathematical method for building the equation of motion. In the present paper Poisson Brackets and their properties are presented, by using Poisson brackets and their properties we calculate some brackets. We use the Poisson bracket with Hamiltonians to express the time dependence of a function u (t), the main idea Taylor series is taken as the required solution for equation of motion using the properties of the Poisson Brackets, We have examined examples from the classical mechanics to illustrate the idea such as motion with a constant acceleration, simple harmonic oscillator, freely falling particle. The solutions are compatible with what is known in classical mechanics. The work is fundamental and sheds new light onto classical mechanics. Poisson brackets are a powerful and sophisticated tool in the Hamiltonian formalism of Classical Mechanics. They also happen to provide a direct link between classical and quantum mechanics.},
 year = {2021}
}

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TY  - JOUR
T1  - Taylor Series and Getting the General Solutions for the Equations of Motion Using Poisson Bracket Relations
AU  - Ibtisam Frhan Al-Maaitah
Y1  - 2021/08/04
PY  - 2021
N1  - https://doi.org/10.11648/j.wjap.20210603.12
DO  - 10.11648/j.wjap.20210603.12
T2  - World Journal of Applied Physics
JF  - World Journal of Applied Physics
JO  - World Journal of Applied Physics
SP  - 47
EP  - 51
PB  - Science Publishing Group
SN  - 2637-6008
UR  - https://doi.org/10.11648/j.wjap.20210603.12
AB  - In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called canonical transformations, which map canonical coordinate systems into canonical coordinate systems. In this work we study some examples from the classical mechanics of particles and apply mathematical method for building the equation of motion. In the present paper Poisson Brackets and their properties are presented, by using Poisson brackets and their properties we calculate some brackets. We use the Poisson bracket with Hamiltonians to express the time dependence of a function u (t), the main idea Taylor series is taken as the required solution for equation of motion using the properties of the Poisson Brackets, We have examined examples from the classical mechanics to illustrate the idea such as motion with a constant acceleration, simple harmonic oscillator, freely falling particle. The solutions are compatible with what is known in classical mechanics. The work is fundamental and sheds new light onto classical mechanics. Poisson brackets are a powerful and sophisticated tool in the Hamiltonian formalism of Classical Mechanics. They also happen to provide a direct link between classical and quantum mechanics.
VL  - 6
IS  - 3
ER  -

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Author Information

Ibtisam Frhan Al-Maaitah

Applied Physics Department, Faculty of Science, Tafila Technical University, Tafila, Jordan

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Ibtisam Frhan Al-Maaitah. (2021). Taylor Series and Getting the General Solutions for the Equations of Motion Using Poisson Bracket Relations. World Journal of Applied Physics, 6(3), 47-51. https://doi.org/10.11648/j.wjap.20210603.12

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ACS Style

Ibtisam Frhan Al-Maaitah. Taylor Series and Getting the General Solutions for the Equations of Motion Using Poisson Bracket Relations. World J. Appl. Phys. 2021, 6(3), 47-51. doi: 10.11648/j.wjap.20210603.12

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AMA Style

Ibtisam Frhan Al-Maaitah. Taylor Series and Getting the General Solutions for the Equations of Motion Using Poisson Bracket Relations. World J Appl Phys. 2021;6(3):47-51. doi: 10.11648/j.wjap.20210603.12

Copy | Download

@article{10.11648/j.wjap.20210603.12,
  author = {Ibtisam Frhan Al-Maaitah},
  title = {Taylor Series and Getting the General Solutions for the Equations of Motion Using Poisson Bracket Relations},
  journal = {World Journal of Applied Physics},
  volume = {6},
  number = {3},
  pages = {47-51},
  doi = {10.11648/j.wjap.20210603.12},
  url = {https://doi.org/10.11648/j.wjap.20210603.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.wjap.20210603.12},
  abstract = {In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called canonical transformations, which map canonical coordinate systems into canonical coordinate systems. In this work we study some examples from the classical mechanics of particles and apply mathematical method for building the equation of motion. In the present paper Poisson Brackets and their properties are presented, by using Poisson brackets and their properties we calculate some brackets. We use the Poisson bracket with Hamiltonians to express the time dependence of a function u (t), the main idea Taylor series is taken as the required solution for equation of motion using the properties of the Poisson Brackets, We have examined examples from the classical mechanics to illustrate the idea such as motion with a constant acceleration, simple harmonic oscillator, freely falling particle. The solutions are compatible with what is known in classical mechanics. The work is fundamental and sheds new light onto classical mechanics. Poisson brackets are a powerful and sophisticated tool in the Hamiltonian formalism of Classical Mechanics. They also happen to provide a direct link between classical and quantum mechanics.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Taylor Series and Getting the General Solutions for the Equations of Motion Using Poisson Bracket Relations
AU  - Ibtisam Frhan Al-Maaitah
Y1  - 2021/08/04
PY  - 2021
N1  - https://doi.org/10.11648/j.wjap.20210603.12
DO  - 10.11648/j.wjap.20210603.12
T2  - World Journal of Applied Physics
JF  - World Journal of Applied Physics
JO  - World Journal of Applied Physics
SP  - 47
EP  - 51
PB  - Science Publishing Group
SN  - 2637-6008
UR  - https://doi.org/10.11648/j.wjap.20210603.12
AB  - In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called canonical transformations, which map canonical coordinate systems into canonical coordinate systems. In this work we study some examples from the classical mechanics of particles and apply mathematical method for building the equation of motion. In the present paper Poisson Brackets and their properties are presented, by using Poisson brackets and their properties we calculate some brackets. We use the Poisson bracket with Hamiltonians to express the time dependence of a function u (t), the main idea Taylor series is taken as the required solution for equation of motion using the properties of the Poisson Brackets, We have examined examples from the classical mechanics to illustrate the idea such as motion with a constant acceleration, simple harmonic oscillator, freely falling particle. The solutions are compatible with what is known in classical mechanics. The work is fundamental and sheds new light onto classical mechanics. Poisson brackets are a powerful and sophisticated tool in the Hamiltonian formalism of Classical Mechanics. They also happen to provide a direct link between classical and quantum mechanics.
VL  - 6
IS  - 3
ER  -

Copy | Download