We derive conservation laws from symmetry operations using the principle of least action. These derivations, which are examples of Noether’s theorem, require only elementary calculus and are suitable for introductory physics. We extend these arguments to the transformation of coordinates due to uniform motion to show that a symmetry argument applies more elegantly to the Lorentz transformation than to the Galilean transformation.
REFERENCES
1.
R. P. Feynman and S. Weinberg, Elementary Particles and the Laws of Physics (Cambridge U.P., Cambridge, 1999), p. 73.
2.
In reality, there are two Noether’s theorems and their converses. The first one refers to the invariance of the action with respect to a group of symmetries where the symmetry transformations depend analytically on many arbitrary finite parameters. The second theorem deals with the invariance of the action with respect to a group for which the transformations depend on arbitrary functions and their derivatives instead of on arbitrary parameters. Our paper considers one-parameter symmetry transformations. Therefore, it is connected with the first theorem. See E. Noether, “Invariante Variationsprobleme,” Nachr. v. d. Ges. d. Wiss. zu Göttingen, Math-phys. Klasse, 235–257 (1918);
English translation by
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R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison–Wesley, Reading, MA, 1963), Vol. I, Chap. 11, p. 11-1 or Chap. 52, p. 52-1.
5.
More accurately, the principle says that a particle moves along that path for which the action has a stationary value. So it is frequently and correctly called the principle of stationary action. See I. M. Gelfand and S. V. Fomin, Calculus of Variations (Prentice–Hall, Englewood Cliffs, NJ, 1963), Sec. 32.2 or D. J. Morin, 〈http://www.courses.fas.harvard.edu/∼phys16/handouts/textbook/ch5.pdf〉, Chap. 5.
6.
Generally such a quantity is called a cyclic or ignorable coordinate; H. Goldstein, Classical Mechanics (Addison–Wesley, New York, 1970), p. 48 or Ref. 5.
7.
Every quantity that depends on position coordinates and velocities and whose value does not change along actual trajectories is called a constant of the motion.
8.
We recommend a more detailed described procedure for introducing action in
J.
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, S.
Tuleja
, and M.
Hancova
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The idea of using computers comes from E. F. Taylor. See
E. F.
Taylor
, S.
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, J. M.
O’Meara
, and N. S.
Thornber
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,” Comput. Phys.
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), 190
–199
(1998
)or E. F. Taylor, Demystifying Quantum Mechanics, 〈http://www.eftaylor.com〉. Our software is based on Taylor’s.
10.
R. P. Feynman, The Character of Physical Law (Random House, New York, 1994), Chap. 4.
12.
Strictly speaking, in these and the following cases we should use the more traditional notation of partial instead of total derivatives. But in all cases it is clear which coordinates are variable and which are fixed.
14.
The relativistic formula for the action is given in Ref. 4, Vol. II, Chap. 19. We use the concept of invariance of mass that is used by E. F. Taylor and J. A. Wheeler, in Spacetime Physics: Introduction to Special Relativity (W. H. Freeman, New York, 1992), 2nd ed.
15.
The conservation law corresponding to the Lorentz transformation is derived in L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergamon, London, 1975), Vol. 2, pp. 41–42.
16.
E. F. Taylor and J. A. Wheeler, Exploring Black Holes: An Introduction to General Relativity (Addison–Wesley Longman, New York, 2000), Chaps. 1 and 4; also available at 〈http://www.eftaylor.com〉. The authors use a very similar, easy, and effective variational method.
17.
P.
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C. M.
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The substance of this article was used by the authors as subjects for student projects dealing with a special topic on the principle of least action in a semester quantum mechanics course for future teachers of physics at the Faculty of Science, P. J. Safarik University, Kosice, Slovakia. To obtain our materials and corresponding software, see 〈http://leastaction.topcities.com〉 (the mirror site 〈http://www.LeastAction.host.sk〉) or see Edwin Taylor’s website: 〈http://www.eftaylor.com/leastaction.html〉, which also includes our newest, continually updated and expanded materials.
21.
L. D. Landau and E. M. Lifshitz, Mechanics (Butterworth–Heinemann, Oxford, 1976), Sec. 1.2.
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