Communications in Mathematical Physics

, Volume 128, Issue 3, pp 565–592 | Cite as

Isoholonomic problems and some applications

  • R. Montgomery


We study the problem of finding the shortest loops with a given holonomy. We show that the solutions are the trajectories of particles in Yang-Mills potentials (Theorem 4), or, equivalently, the projections of Kaluza-Klein geodesics (Theorem 2). Applications to quantum mechanics (Berry's phase, Sect. 3) and the optimal control of deformable bodies (Sect. 6) are touched upon.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
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  1. Aharonov, Y., Anandan, J.: Phase change during cyclic quantum evolution. Phys. Rev. Lett.58, 1593–1596 (1987)CrossRefPubMedGoogle Scholar
  2. Ambrose, W., Singer, I. M.: A theorem on holonomy. Trans. AMS75, 428–453 (1953)Google Scholar
  3. Arnol'd, V. I.: Some remarks on flows of frames. Sov. Math, translations of Doklady. USSR,2, 562–564 (1961)Google Scholar
  4. Arnol'd, V. I., Kozlov, V. V., Neishtadt, A. I. (1988): Dynamical systems III. vol. 3. In: The Encyclopaedia of Mathematical Sciences series. Berlin, Heidelberg, New York: Springer 1988Google Scholar
  5. Avron, J. E., Sadun, L., Segert, J., Simon, B.: Chern numbers and Berry's phases in fermi systems. Commun. Math. Phys.124, 595–627 (1989)CrossRefGoogle Scholar
  6. Baillieul, J. B.: Geometric methods for nonlinear optimal control problems. J. Optimization Th. Applications25, 519–548 (1975)CrossRefGoogle Scholar
  7. Balachandran, A. P., Borchardt, S., Stern, A.: Lagrangian and Hamiltonian descriptions of Yang-Mills particles. Phys. Rev.D17, 3247–3256 (1978)CrossRefGoogle Scholar
  8. Bär, C.: Carnot-Caratheodory-Metriken. Diplomarbeit, Bonn 1988Google Scholar
  9. Bär, C.: Geodesics for Carnot-Caratheodory Metrics. Preprint 1989Google Scholar
  10. Berry, M. V.: Quantal phase factors accompanying adiabatic changes. J. Phys. A.18, 15–27 (1984)Google Scholar
  11. Bliss, G. A.: Lectures on calculus of variations. Chicago, IL: Univ. of Chicago Press 1946Google Scholar
  12. Bliss, G. A.: The problem of Lagrange in the calculus of variations. Am. J. Math.52, 674–713 (1930)Google Scholar
  13. Brockett, R. W.: Control theory and singular Riemannian geometry. In: New directions in applied mathematics. Hilton, P. J., Young, G. S. (eds). Berlin, Heidelberg, New York: Springer 1981Google Scholar
  14. Carathéodory, C.: Calculus of variations and partial differential equations of the first, order, vol. 2. Holden-Day, S.F., CA 1967Google Scholar
  15. Cesari, L.: Optimization—Theory and applications. Berlin, Heidelberg, New York: Springer 1983Google Scholar
  16. Chow, W. L.: Uber Systeme van Linearen partiellen Differentialgleichungen erster Ordnung. Math. Ann117, 98–105 (1939)CrossRefGoogle Scholar
  17. Courant, R., Hilbert, D.: Methods of mathematical physics vol. I, New York: Interscience 1953Google Scholar
  18. Faibusovich, L. E.: Explicitly solvable nonlinear optimal controls. Int'l J. Control48, 2507–2526 (1988)Google Scholar
  19. Gunther, N. L.: Hamoltonian mechanics and optimal control. Harvard thesis 1982Google Scholar
  20. Ge Zhong: On a constrained variation problem and the space of horizontal paths. M.S.R.I. preprint#04224-89 (1989)Google Scholar
  21. Hamenstädt, U.: Über Theorie von Carnot-Caratheodory-Metriken und ihren Anwendungen. Doktorarbeit, Bonn 1986Google Scholar
  22. Hamenstädt, U.: Some regularity, theorems for Carnot-Caratheodory metrics. Preprint, Cal. Tech. 1988Google Scholar
  23. Hermann, R.: Some differential geometric aspects of the lagrante variational problem. Indiana Math. J. 634–673 (1962)Google Scholar
  24. Hermann, R.: Geodesics of singular Riemannian metrics. Bull. AMS79, 780–782 (1973)Google Scholar
  25. Iwai, T.: A gauge theory for the quantum planar three-body system. J. Math. Phys.28, 1315–1326 (1987a)CrossRefGoogle Scholar
  26. Iwai, T.: A geometric setting for internal motions of the quantum three-body system. J. Math. Phys.28, 1315–1326 (1987b)CrossRefGoogle Scholar
  27. Iwai, T.: A geometric setting for classical molecular dynamics. Ann. Inst. Henri Poincairé, Phys. Th.,47, 199–219 (1987c)Google Scholar
  28. Kane, T. R., Scher, M. P.: A dynamical explanation of the falling cat phenomenon. Intl. J. Solids Structures,5, 663–670 (1969)CrossRefGoogle Scholar
  29. Koenig, M., Mueller, C., Zwanziger, J.: private conversations (1989)Google Scholar
  30. Montgomery, R.: Canonical formulations of a classical particle in a Yang-Mills field and Wong's equations. Lett. Math. Phys.8, 59–67 (1984)CrossRefGoogle Scholar
  31. Montgomery, R.: Shortest loops with a fixed holonomy. MSRI preprint series#01224-89 (1988)Google Scholar
  32. Montgomery, R.: Optimal control of deformable bodies, isoholonomic problems, and sub-Riemannian geometry. MSRI preprint series#05324-89 (1989)Google Scholar
  33. Shapere, A.: Gauge mechanics of deformable bodies. PhD. thesis, Physics, Princeton (1989)Google Scholar
  34. Shapere, A., Wilczek, F.: Self-propulsion at low Reynolds number. Phys. Rev. Lett.58, 2051–2054 (1987)CrossRefGoogle Scholar
  35. Simon, B.: Holonomy, the quantum adiabatic theorem, and Berry's phase. Phys. Rev. Lett.51, 2167–2170 (1983)CrossRefGoogle Scholar
  36. Strichartz, R.: Sub-Riemannian geometry. J. Diff. Geom.24, 221–263 (1983)Google Scholar
  37. Suter, D., Mueller, K. T., Pines, A.: Study of the Aharonov-Anandan quantum phase by NMR interferometry. Phys. Rev. Lett.60, 1218–1220 (1988)CrossRefGoogle Scholar
  38. Taylor, T. J. S.: Some aspects of differential geometry associated with hypoelliptic second order operators. Pac. J. Math.136, 355–378 (1989)Google Scholar
  39. Tomita, A., Chiao, R. Y.: Observation of Berry's topological phase by use of an optical fiber. Phys. Rev. Lett.57, 937–940 (1986)CrossRefGoogle Scholar
  40. Tycko, R.: Adiabatic rotational splittings and Berry's phase in nuclear quadraplole resonance. Phys. Rev. Lett.58, 2281–2284 (1987)CrossRefGoogle Scholar
  41. Vershik, A. M., Ya Gershkovich, V.: Non-holonomic Riemannian manifolds. In: Dynamical systems vol. 7, part of the new Mathematical Encyclopaedia series vol. 16. In Russian, MIR pub. Berlin, Heidelberg, New York: Springer 1988Google Scholar
  42. Weinstein, A.: Fat bundles and symplectic manifolds. Adv. Math.37, 239–250 (1980)CrossRefGoogle Scholar
  43. Wilczek, F.: Gauge theory of deformable bodies. Inst. Adv. Studies preprint#-88/41 (1988)Google Scholar
  44. Wilczek, F., Zee, A.: Appearence of gauge structure in simple dynamical systems. Phys. Rev. Lett.52, 2111–2114 (1984)CrossRefGoogle Scholar
  45. Wong, S. K.: Field and particle equations for the classical Yang-Mills field and particles with isotopic spin. Nuovo Cimento65A, 689–693 (1970)Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • R. Montgomery
    • 1
  1. 1.M.S.R.I.BerkeleyUSA

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