Advertisement

Integrability and Chaos in Figure Skating

  • Vaughn Gzenda
  • Vakhtang PutkaradzeEmail author
Article
  • 2 Downloads

Abstract

We derive and analyze a three-dimensional model of a figure skater. We model the skater as a three-dimensional body moving in space subject to a non-holonomic constraint enforcing movement along the skate’s direction and holonomic constraints of continuous contact with ice and pitch constancy of the skate. For a static (non-articulated) skater, we show that the system is integrable if and only if the projection of the center of mass on skate’s direction coincides with the contact point with ice and some mild (and realistic) assumptions on the directions of inertia’s axes. The integrability is proved by showing the existence of two new constants of motion linear in momenta, providing a new and highly non-trivial example of an integrable non-holonomic mechanical system. We also consider the case when the projection of the center of mass on skate’s direction does not coincide with the contact point and show that this non-integrable case exhibits apparent chaotic behavior, by studying the divergence of nearby trajectories. We also demonstrate the intricate behavior during the transition from the integrable to chaotic case. Our model shows many features of real-life skating, especially figure skating, and we conjecture that real-life skaters may intuitively use the discovered mechanical properties of the system for the control of the performance on ice.

Keywords

Non-holonomic dynamics Integrable systems Mechanics of sports 

Mathematics Subject Classification

70H06 70H33 70H45 70K50 70K55 

Notes

Acknowledgements

We are grateful to P. Balseiro, A. M. Bloch, F. Fasso, H. Dullin, I. Gabitov, L. Garcia-Naranjo, D. D. Holm, T. Ohsawa, P. Olver, T. S. Ratiu, S. Venkataramani and D. Volchenkov for enlightening scientific discussions. We want to especially thank Prof. D. V. Zenkov for his interest, availability and patience in answering our questions. We would like to thank Dr. S. M. Rogers for careful reading of the manuscript and finding errors and inaccuracies. We are also grateful to M. Hall and J. Hocher for teaching us the intricacies of skating techniques and the differences between hockey and figure skating. We are thankful to M. Hall for providing skating expertise and C. Hansen’s graphics processing for Fig. 1. The research of VP was partially supported by the University of Alberta and NSERC Discovery grant, which also partially supported VG through NSERC USRA program.

References

  1. Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics, Volume 3 of Encyclopadia of Math. Sciences. Springer, Berlin (1989)Google Scholar
  2. Balseiro, P., Sansonetto, N.: A geometric characterization of certain first integrals for nonholonomic systems with symmetries. SIGMA 12, 018 (2016)MathSciNetzbMATHGoogle Scholar
  3. Bates, L., Cushman, R.: What is a completely integrable nonholonomic dynamical system? Rep. Math. Phys. 44, 29–35 (1999)MathSciNetCrossRefGoogle Scholar
  4. Bates, L., Graumann, H., MacDonnel, C.: Examples of gauge conservation laws in nonholonomic systems. Rep. Math. Phys. 37, 295–308 (1996)MathSciNetCrossRefGoogle Scholar
  5. Bizyaev, I.A., Borisov, A.V., Mamaev, I.S.: The chaplygin sleigh with parametric excitation: chaotic dynamics and nonholonomic acceleration. Regul. Chaotic Dyn. 22, 955–975 (2017)MathSciNetCrossRefGoogle Scholar
  6. Bizyaev, I.A., Borisov, A.V., Kozlov, V.V., Mamaev, I.S.: Fermi-like acceleration and power-law energy growth in nonholonomic systems. Nonlinearity 32, 3209 (2018)MathSciNetCrossRefGoogle Scholar
  7. Bloch, A.M.: Nonholonomic Mechanics and Control, vol. 24. Springer, Berlin (2003)CrossRefGoogle Scholar
  8. Bloch, A.M., Krishnaprasad, P.S., Marsden, J.E., Murray, R.M.: Nonholonomic mechanical systems with symmetry. Arch. Ration. Mech. Anal. 136, 21–99 (1996)MathSciNetCrossRefGoogle Scholar
  9. Bloch, A.M., Marsden, J.E., Zenkov, D.V.: Quasivelocities and symmetries in nonholonomic systems. Dyn. Syst. 24, 187–222 (2009)MathSciNetCrossRefGoogle Scholar
  10. Borisov, A.V., Mamaev, S.: Symmetries and reduction in nonholonomic mechanics. Regul. Chaotic Dyn. 20, 553–604 (2015)MathSciNetCrossRefGoogle Scholar
  11. Borisov, A.V., Mamaev, I.S., Bizyaev, I.A.: Historical and critical review of the development of nonholonomic mechanics: the classical period. Regul. Chaotic Dyn. 21(4), 455–476 (2016)MathSciNetCrossRefGoogle Scholar
  12. Cvitanović, P., Christiansen, F., Putkaradze, V.: Hopf’s last hope: spatiotemporal chaos in terms of unstable recurrent patterns. Nonlinearity 10, 55 (1997)MathSciNetCrossRefGoogle Scholar
  13. de León, M.: A historical review on nonholomic mechanics. RACSAM 106, 191–224 (2012)MathSciNetCrossRefGoogle Scholar
  14. Ding, X., Chaté, H., Cvitanović, P., Siminos, E., Takeuchi, K.A.: Estimating dimension of inertial manifold from unstable periodic orbits. Phys. Rev. Lett. 117, 024101 (2017)MathSciNetCrossRefGoogle Scholar
  15. Fasso, F., Sansonetto, N.: An elemental overview of the nonholonomic Noether theorem. Int. J. Geom. Methods Mod. Phys. 06, 1343–1355 (2009)MathSciNetCrossRefGoogle Scholar
  16. Fasso, F., Giacobbe, A., Sansonetto, N.: Gauge conservation laws and the momentum equation in nonholonomic mechanics. Rep. Math. Phys. 62, 345–367 (2008)MathSciNetCrossRefGoogle Scholar
  17. Fasso, F., Giacobbe, A., Sansonetto, N.: Linear weakly noetherian constants of motion are horizontal gauge momenta. J. Geom. Mech. 4, 129–136 (2012)MathSciNetCrossRefGoogle Scholar
  18. Fedorov, Y.N., Kozlov, V.V.: Various aspects of n-dimensional rigid body dynamics. Am. Math. Soc. Transl. 168, 141–171 (1995)MathSciNetzbMATHGoogle Scholar
  19. Garcia-Naranjo, L.C., Montaldi, J.: Gauge momenta as casimir functions of nonholonomic systems. Arch. Ration. Mech. Anal. 228, 563–602 (2018)MathSciNetCrossRefGoogle Scholar
  20. Holm, D.D.: Geometric Mechanics: Rotating, Translating, and Rolling. Geometric Mechanics. Imperial College Press, London (2011)CrossRefGoogle Scholar
  21. Holm, D.D., Schmah, T., Stoica, C.: Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions. Oxford University Press, Oxford (2009)zbMATHGoogle Scholar
  22. Kozlov, V.V.: On the integration theory of the equations in nonholonomic mechanics. Adv. Mech. 8, 86–107 (1985)Google Scholar
  23. Kozlov, V.V.: Invariant measures of the Euler-Poincaré equations on lie algebras. Funct. Anal. Appl. 22, 69–70 (1988)MathSciNetzbMATHGoogle Scholar
  24. Kozlov, V.V.: On the integration theory of equations of nonholonomic mechanics. Regul. Chaotic Dyn. 7(2), 161–176 (2002)MathSciNetCrossRefGoogle Scholar
  25. Le Berre, M., Pomeau, Y.: Theory of ice-skating. Int. J. Non-linear Mech. 75, 77–86 (2015)CrossRefGoogle Scholar
  26. Lozowski, E., Szilder, K., Maw, S.: A model of ice friction for a speed skate blade. Sports Eng. 16, 239–253 (2013)CrossRefGoogle Scholar
  27. Marsden, J.E., Ratiu, T.: Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, vol. 17. Springer, Berlin (2013)zbMATHGoogle Scholar
  28. Neimark, J.I., Fufaev, N.A.: Dynamics of Nonholonomic Systems. AMS, Rhode Island (1972)zbMATHGoogle Scholar
  29. Rosenberg, R.: Why is ice so slippery. Phys. Today 58, 50–55 (2005)CrossRefGoogle Scholar
  30. Sniaticki, J.: Nonholonomic Noether theorem and reduction of symmetries. Rep. Math. Phys. 42, 5–23 (1998)MathSciNetCrossRefGoogle Scholar
  31. Veselov, A.P., Veselova, L.E.: Integrable nonholonomic systems on lie groups. Math. Notes 44, 810–819 (1988)MathSciNetCrossRefGoogle Scholar
  32. Volchenkov, D., Bläsing, B.E., Schack, T.: Spatio-temporal kinematic decomposition of movements. Engineering 6, 385–398 (2014)CrossRefGoogle Scholar
  33. Zenkov, D.V.: Linear conservation laws of nonholonomic systems with symmetry. In: Discrete and Continuous Dynamical Systems (Extended Volume), pp. 963–972 (2003)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada
  2. 2.ATCO SpaceLabCalgaryCanada

Personalised recommendations