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Symmetries and conservation laws: Consequences of Noether’s theorem
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);
2.English translation by M. A. Tavel, “Invariant variation problem,” Transport Theory Stat. Mech.1 (3), 183–207 (1971). Both papers are available at 〈http://www.physics.ucla.edu/∼cwp/Phase2/Noether,_Amalie_Emmy@861234567.html〉.
3.N. Byers, “E. Noether’s discovery of the deep connection between symmetries and conservation laws,” Isr. Math. Conf. Proc.12, 67–82 (1999).
4.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. Hanc, S. Tuleja, and M. Hancova, “Simple derivation of Newtonian mechanics from the principle of least action,” Am. J. Phys. 71, 386–391 (2003).
9.The idea of using computers comes from E. F. Taylor. See E. F. Taylor, S. Vokos, J. M. O’Meara, and N. S. Thornber, “Teaching Feynman’s sum over paths quantum theory,” Comput. Phys. 12 (2), 190–199 (1998)
9.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.
11.Reference 4, Vol. II, Chap. 19, p. 19-8 or the more detailed discussion in Ref. 8.
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.
13.In that case it is necessary to consider the invariance of the action up to an additive constant (the difference in any arbitrary function between final position and initial position of a particle), which will give conservation of motion of the center of mass. See also Refs. 17 or 18.
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. Havas and J. Stachel, “Invariances of approximately relativistic Lagrangians and the center of mass theorem. I,” Phys. Rev. 185 (5), 1636–1647 (1969).
18.N. C. Bobillo-Ares, “Noether’s theorem in discrete classical mechanics,” Am. J. Phys. 56 (2), 174–177 (1988).
19.C. M. Giordano and A. R. Plastino, “Noether’s theorem, rotating potentials, Jacobi’s integral of motion,” Am. J. Phys. 66 (11), 989–995 (1998).
20.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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