Skip to main content

What Is the Lagrangian Counting?

International Journal of Theoretical Physics volume 42, pages 363–381 (2003)Cite this article

Abstract

The calculus of variations provides an exhaustive descriptive account of the least-action principle and other icons of nature's parsimoniousness. However, when we seek explanations rather than mere formal descriptions of variational principles, almost invariably we discover a combinatorial origin; the best known examples are thermodynamics and Darwinian evolution. When it comes to the least-action principle, however, it is surprising that reductionistic attacks of this kind have been virtually absent. An eminent exception is, of course, Richard Feynman's explanation in terms of quantum path integrals, but even there, though the spirit of the approach is combinatorial, the nature of the objects that are “counted” is somewhat elusive; one is still explaining a mystery through another mystery. Feynman himself stresses that “whoever thinks they understand quantum mechanics, they don't,” and ultimately admits “I don't know what action is.” The challenge we proposes is to devise “classical” models (that is, models based on ordinary counting of large numbers of discrete objects rather than superposition of complex amplitudes) of classical analytical mechanics. Never mind what method nature actually uses; how come models of this kind—which, for example, were a dime a dozen for the laws of perfect gases—are so hard to come by for the least-action principle?

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

REFERENCES

  • Arnold, V. (1978). Mathematical Methods of Classical Mechanics, Springer, Berlin.

    Google Scholar 

  • Brush, G. (1976). The Kind of Motion We Call Heat: A History of the Kinetic Theory of Gases in the 19th Century, North-Holland, Amsterdam.

    Google Scholar 

  • Cox, R. T. (1946). Probability, frequency, and reasonable expectation. American Journal of Physics 14, 1-13.

    Google Scholar 

  • Denn, M. (1978). Optimization by Variational Methods, Rober Krieger, Huntington, New York.

    Google Scholar 

  • Feynman, R. (1948). Space-time approach to non-relativistic quantum mechanics, Reviews of Modern Physics 20, 367-387.

    Google Scholar 

  • Feynman, R. (1963). Lectures on Physics, Vol. I, Addison-Wesley, Reading, MA.

    Google Scholar 

  • Feynman, R., and Hibbs, A. (1965). Quantum Mechanics and Path Integrals, McGraw-Hill, New York.

    Google Scholar 

  • Forsyth, A. R. (1960). Calculus of Variations, Dover, New York.

    Google Scholar 

  • Goldstein, H. (1980). Classical Mechanics, 2nd edn., Addison-Wesley, Reading, MA.

    Google Scholar 

  • Hildebrandt, S. and Tromba, A. (1996). The Parsimonious Universe: Shape and form in the natural world. Springer, Berlin.

    Google Scholar 

  • Jaynes, E. (1957). Information theory and statistical mechanics. Physical Review 106, 620-630

    Google Scholar 

  • Jaynes, E. (1957). Information theory and Statical Mechanics. Physical Review 108, 171-190.

    Google Scholar 

  • Jaynes, E. (2003). Probability: The Logic of Science, Cambridge University Press, Cambridge, UK.

    Google Scholar 

  • Landauer, R. (1952). Path concepts in Hamilton-Jacobi theory, American Journal of Physics 20, 363-367.

    Google Scholar 

  • Lanczos, C. (1970). The Variational Principles of Mechanics, 4th edn., Dover, New York.

    Google Scholar 

  • Margolus, N. (March 1988). Physics and computation, PhD Thesis, MIT/LCS/TR-415, MIT.

  • Sagan, H. (1969). Introduction to the Calculus of Variations, Dover, New York.

    Google Scholar 

  • Smith, M. (May 1994). Cellular automata methods in mathematical physics, PhD Thesis, MIT/LCS/TR-615, MIT.

  • Sussman, G. Wisom, J., and Mayer, M. (2001). Structure and Interpretation of Classical Mechanics, MIT Press, Cambridge, MA.

    Google Scholar 

  • Toffoli, T. (1998). Action, or the fungibility of computation. In Feynman and Computation A. Hey ed. Perseus, Reading, Massachusetts, pp. 348-392.

    Google Scholar 

  • Weinstock, R. (1974). Calculus of Variations—With Applications to Physics and Engineering, Dover, New York.

    Google Scholar 

  • Yourgrau, W. and Mandelstam, S. (1979). Variational Principles in Dynamics and Quantum Theory, Dover, New York.

    Google Scholar 

  • Youssef, S. (October, 2001). Physics with exotic probability theory. arXiv.org, hep-th/0110253.

Download references

Author information

Authors and Affiliations

  1. Electrical and Computer Engineering, Boston University, Boston, Massachusetts

    Tommaso Toffoli

Authors
  1. Tommaso Toffoli
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Toffoli, T. What Is the Lagrangian Counting?. International Journal of Theoretical Physics 42, 363–381 (2003). https://doi.org/10.1023/A:1024411819910

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1024411819910

  • least action principle
  • action as amount of computation
  • Lagrangian mechanics