Principle of Least Action
The least-action principle is an assertion about the
nature of motion that provides an alternative approach to mechanics
completely independent of Newton's laws. Not only does the least-action
principle offer a means of formulating classical mechanics that is
more flexible and powerful than Newtonian mechanics, [but also] variations
on the least-action principle have proved useful in general relativity
theory, quantum field theory, and particle physics. As a result,
this principle lies at the core of much of contemporary theoretical
physics. The principle of least action (more correctly, the principle of stationary action) has wide applicability in undergraduate physics education, from mechanics in introductory classes through electricity and magnetism, quantum mechanics, special and general relativity—and it provides a deep foundation for advanced subjects and current research. Interactive Software The software ActionClockTicks by Slavo Tuleja uses the action principle to construct trajectories between fixed initial and final points for projectile motion, satellite motion, simple harmonic oscillator, and a moon shot. The total time for the trajectory is set by the user. Play this program online or download it to your computer. The open source JAVA code of the program is also available. |
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Action Summary by Jozef Hanc, Jon Ogborn, Edwin Taylor,
with software by Slavomir Tuleja. A conference proceeding that
summarizes action, starting with dramatized appearances of the
originators from Fermat and Maupertuis to Hamilton and Feynman.
Subsequent papers introduce action formalism, apply it to teaching
introductory quantum mechanics, and trace the scope of applications
of action from quantum field theory to cosmology. "A Call to Action," Edwin F. Taylor. Guest Editorial, American Journal of Physics,Vol. 71, No. 5, May 2003, pages 423-425. Outlines the case for using the principle of least action in undergraduate physics classes. [ Read on-line as HTML ] [ Download PDF file (~39K) ] [ Back to overview ] "Deriving the nonrelativistic principle of least action from the Schwarzschild metric and the Principle of Maximal Aging" Edwin F. Taylor. Unpublished appendix to "A Call to Action." In general relativity a particle moves along the worldline of maximal proper time (maximal aging). In the limit of small spacetime curvature and low velocity this reduces to the principle of least action, as shown in this paper for a free particle external to a spherically symmetric, nonrotating center of gravitational attraction. [ Download PDF file (~138K) ] [ Back to overview ] "Action: Forcing Energy to Predict Motion," Dwight E. Neuenschwander, Edwin F. Taylor, and Slavomir Tuleja, The Physics Teacher, Vol. 44, March 2006, pages 146-152. Scalar energy is employed to predict motion instead of the vector Newton's law of motion. [ Download PDF file (~260K) ] [ Back to overview ] "Variational mechanics in one and two dimensions" Jozef Hanc, Edwin F. Taylor, Slavomir Tuleja. American Journal of Physics, Vol. 73, No. 7, July 2005, pages 603-610. Heuristic derivations of the Euler-Maupertuis abbreviated action and the Hamilton action. [ Read on-line as HTML ] [ Download PDF file (~110K) ] [ Back to overview ] "Symmetries and conservation laws: Consequences of Noether's theorem," Jozef Hanc, Slavomir Tuleja, and Martina Hancova, American Journal of Physics, Vol. 72, No. 4, April 2004, pages 428-435. Derives conservation laws from symmetry operations using the principle of least action. These derivations are examples of Noether's Theorem. [ Read on-line as HTML ] [ Download PDF file (~337K) ] [ Back to overview ] "Deriving Lagrange's equations using elementary calculus," Jozef Hanc, Edwin F. Taylor, and Slavomir Tuleja. American Journal of Physics, Vol. 72, No. 4, April 2004, pages 510-513. Lagrange's equations, alternatives to F=ma, are usually derived from the principle of least action using the calculus of variations. This paper derives them using elementary calculus. [ Read on-line as HTML ] [ Download PDF file (~83K) ] [ Back to overview ] "Simple derivation of Newtonian mechanics from the principle of least action," Jozef Hanc, Slavomir Tuleja, Martina Hancova. American Journal of Physics, Vol. 71. No. 4, April 2003, pages 386 - 391. Derives Newton's laws of motion from the principle of least action. [ Read on-line as HTML ] [ Download PDF file (~250K) ] [ Back to overview ] "Quantum physics explains Newton's laws of motion," Jon Ogborn and Edwin F. Taylor, Physics Education, Vol. 40, No. 1, 2005, pages 26-34. Richard Feynman expressed the quantum mechanics of particle motion in the command, "Explore all paths." In the limit of large mass, this command goes over into Newton's law of motion and the principle of least action. [ Download PDF file (~140K) ] [ Back to overview ] "From conservation of energy to the principle of least action: A story line," Jozef Hanc and Edwin F. Taylor, American Journal of Physics, Vol. 72, No. 4, Aril 2004, pages 514-521. Conservation of energy is sufficient to predict motion in one dimension and for systems whose motion can be expressed as one independent coordinate. This prediction can also be used to introduce the principle of least action and Lagrange's equations. [ Read on-line as HTML ] [ Download PDF file (~200K) ] [ Back to overview ] "Getting the Most Action from the Least Action: A proposal," Thomas A. Moore, American Journal of Physics, Vol. 72, No. 4, April 2004, pages 522-527. The principle of least action is a powerful addition to upper undergraduate courses for physics majors, modifying the selection of topics and presenting advanced topics in a more contemporary way. [ Read on-line as HTML ] [ Download PDF file (~87K) ] [ Back to overview ] "When action is not least," C. G. Gray and Edwin F. Taylor, American Journal of Physics, Vol. 75, No. 5, May 2007, pages 434-458. Action is a minimum along a sufficiently short worldline in all potentials and along worldlines of any length in some potentials. For long enough worldlines in a majority potentials, however, the action is a saddle point, that is, a minimum with respect to some nearby alternative curves and a maximum with respect to others. The action is never a true maximum. [ Download PDF file (~1.0MB) ] [ Back to overview ]
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