Variational mechanics in one and two dimensions

American Journal of Physics

Volumn 73, Issue 7, 2005, Pages 603-610

  Hanc, Jozef ^a   Taylor, Edwin F ^b   Tuleja, Slavomir ^c  

^a TECHNICAL UNIVERSITY OF KO ICE   (Slovakia)

^b MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^c Gymnazium Arm Gen L Svobodu   (Slovakia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 24144436091     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1848516     Document Type: Article

References (18)

1. We use the name least action instead of the technically correct stationary action for several reasons: (a) Many cases involve a local minimum of the action, (b) The value of the action is always a minimum for a sufficiently small segment of the curve, (c) The word least is self-descriptive, but stationary requires additional explanation. (d) The word least does not lead to the error that the value of either form of action, Eqs. (4) and (17), can be a maximum for an actual path, which it cannot. (e) Least action is the name most often used in the historical literature on the subject. We recommend that the term stationary action be introduced, with careful explanation, not long after the term least action itself.
2. For linear gravitational potential energy near the Earth's surface, we can integrate Newton's second law to derive an analytic expression for the basketball trajectory and hence the required direction of launch. However, in more complicated potentials we are reduced to trial and error to find a path that passes through the basket. Minimizing the Maupertuis-Euler abbreviated action finds the trajectory in one stroke. A similar comment applies to the Moon shot described in the following paragraph: Minimizing Hamilton's action gives us the worldline directly.
3. Jozef Hanc and Edwin F. Taylor, "From conservation of energy to the principle of least action: A story line," Am. J. Phys. 72, 514-521 (2004).
4. Derivations of the action outlined in this paper were stimulated by an interactive Java program developed by one of the authors (ST). This display numerically integrates Eq. (2), solving for the one-dimensional motion of a particle in a time-independent potential. In its extended form, the program shows all three panels in Fig. 1. The program is available at 〈http://vscience.euweb.cz/worldlines/Worldlines.html〉.
5. o are used by L. D. Landau and E. M. Lifschitz, Mechanics (Butterworth-Heinemann, London, 1999), Vol. 1, 3rd ed., p. 141
6. and Herbert Goldstein, Charles Poole, and John Safko, Classical Mechanics (Addison-Wesley, San Francisco, 2002), 3rd ed., pp. 359, 434. We have named the corresponding variational principle the abbreviated principle of least action, rather than the more technically correct principle of least abbreviated action, believing that "least abbreviated" might be incorrectly interpreted as "augmented."
7. Wolfgang Yourgrau and Stanley Mandelstam, Variational Principles in Dynamics and Quantum Theory (Dover, New York, 1979), pp. 24-29.
8. Richard P. Feynman, Robert B. Leighton, and Matthew Sands, The Feynman Lectures on Physics (Addison-Wesley, San Francisco, 1964), Vol. II, p. 19-8.
9. The minimization procedure for constructing a trajectory or worldline described in the caption to Fig. 5 is conceptually simple but not the most effective in practice. In both cases it is more efficient to start with a trial segmented curve with equal increments along the horizontal axis. Then we vary only the y-coordinates of intermediate points to minimize the action, obtaining the actual path. The minimization of action with respect to coordinates along the horizontal axis is not necessary because the result is just points uniformly distributed on the horizontal axis, which was our initial assumption. We do not discuss here the proof of this statement or the convergence of the algorithm, because it goes beyond the scope of the present paper.
10. According to the official rules of the National Basketball Association, a basket is scored after the final buzzer provided the ball is launched before the buzzer sounds.
11. For Hamilton's development of the principle of least action, see two of his papers at 〈http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics/〉
12. Landau and Lifschitz, Ref. 5, p. 141, Eq. (44.3); Goldstein et al., Ref. 5, p. 359.
13. Our derivation can be reversed to show the equivalence of Newton's second law and the principle of least action. See also Goldstein et al., Ref. 5, p. 35.
14. The example of a bead sliding along a uniformly rotating rod is in Goldstein et al., Ref. 5, pp. 28-29. Additional example is a pendulum whose string support is slowly pulled up through a small hole. See Cornelius Lanczos, The Variational Principle of Mechanics (Dover, New York, 1986), 4th ed., p. 124.
15. Further examples of name-encrusted terminology: d'Alembert's principle, Hamiltonian, Hamilton's principle, Hamilton's equations, Hamilton-Jacobi equation, Jacobi identity, Jacobi principle, Jacobi condition, Jacobi's theorem, Lagrangian, Poisson bracket, Poisson's equations, Hilbert integral, Legendre condition, Poincare invariants, Cartheodory's method, Bernoulli's method, Clebsch condition, Clebsch relation, Clebsch transformation, Descartes-Snell rule, Noether's theorem, Rayleigh's dissipation function, Routh's procedure, Staeckel conditions, Weierstrass condition, Weierstrass-Erdmann corner condition.
16. Kip S. Thorne, Black Holes and Time Warps: Einstein's Outrageous Legacy (Norton, New York, 1994), pp. 256-257;
17. John Archibald Wheeler with Kenneth Ford, Geons, Black Holes, and Quantum Foam: A Life in Physics (Norton, New York, 1998), pp. 296-297.
18. Landau and Lifschitz, Ref. 5, Chap. 1; Landau and Lifschitz rechristened Hamilton's principal function as the action in the first Russian edition in 1958, and in a 1940 textbook, a precursor of Ref. 5; See also Feynman, Ref. 7.