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Central force fields and Kepler's laws

Published online by Cambridge University Press:  18 June 2018

Luis Blanco,
María García-García,
Joaquín Gutíerrez and
Andrea Rios
Luis Blanco
Affiliation:
Undergraduate, Escuela Técnica Superior de Ingenieros Industriales, Universidad Politécnica de Madrid, Madrid (Spain) e-mail: luis.blanco.diaz@alumnos.upm.es
María García-García
Affiliation:
Undergraduate, Escuela Técnica Superior de Ingenieros Industriales, Universidad Politécnica de Madrid, Madrid (Spain) e-mail: mariademadriguera@hotmail.com
Joaquín Gutíerrez
Affiliation:
Departamento de Matemáticas, Escuela Técnica Superior de Ingenieros Industriales, Universidad Politécnica de Madrid, C. José Gutiérrez Abascal 2, 28006 Madrid (Spain) e-mail: joaquin.gutierrez.alamo@gmail.com
Andrea Rios
Affiliation:
Undergraduate, Escuela Técnica Superior de Ingenieros Industriales, Universidad Politécnica de Madrid, Madrid (Spain) e-mail: andrea_michelle.laz@hotmail.com

Extract

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In this survey we give very simple proofs of Kepler's Laws and other facts about central force fields using only Newton's second law, Newton's law of universal gravitation, basic notions of vector calculus, and an elementary double integral.

Hopefully, this article will help undergraduate students of mathematics and engineering who wish to understand these fundamental scientific discoveries.

In many textbooks (see, for instance, [1, 2, 3, 4, 5]), Kepler's Laws are obtained using conservation of energy and angular momentum, differential equations, mobile reference systems, or notions not so well-defined such as differentials or ‘infinitesimal elements’. Some of the arguments appear to be rather involved if one is not accustomed to them, whereas the proof of Kepler's Laws may actually be obtained from quite simple facts.

Information

Type
Articles
Information
The Mathematical Gazette , Volume 102 , Issue 554 , July 2018, pp. 270 - 279
Copyright
Copyright © Mathematical Association 2018 

References

1. French, A. P., Newtonian mechanics (M.I.T. Introductory Physics Series), W. W. Norton & Co., New York (1971).Google Scholar
2. Kleppner, D. and Kolenkow, R., An introduction to mechanics (2nd edn.), Cambridge University Press (2014).Google Scholar
3. McGill, D. J. and King, W. W., Engineering mechanics – dynamics, PWS-KENT Publ. Co. (1989).Google Scholar
4. Morin, D., Introduction to classical mechanics with problems and solutions, Cambridge University Press (2008).Google Scholar
5. Riley, W. F. and Sturges, L. D., Engineering mechanics, dynamics, John Wiley & Sons, Inc., New York (1993).Google Scholar
6. Metzler, D., Derivation of Kepler's first law, video accessed 2018 at https://www.youtube.com/watch?v=SvOc3LUJcwAGoogle Scholar
7. Stewart, J., Calculus (2nd edn.), Brooks/Cole (1991).Google Scholar
8. Bertrand, J., Théorème relatif au mouvement d'un point attiré vers un centre fixe, C. R. Acad. Sci. LXXVII (1873) pp. 849–853.Google Scholar
9. Lederman, L. and Teresi, D., The god particle: if the universe is the answer, what is the question?, Houghton Mifflin Harcourt, New York (1993).Google Scholar
10. Braden, B., The surveyor's area formula, The College Math. J. 17 (1986) pp. 326–337.Google Scholar