Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 036, 33 pages      math.DS/0703043
Contribution to the Vadim Kuznetsov Memorial Issue

A 'User-Friendly' Approach to the Dynamical Equations of Non-Holonomic Systems

Sergio Benenti
Department of Mathematics, University of Turin, Italy

Received November 29, 2006, in final form February 13, 2007; Published online March 01, 2007

Two effective methods for writing the dynamical equations for non-holonomic systems are illustrated. They are based on the two types of representation of the constraints: by parametric equations or by implicit equations. They can be applied to linear as well as to non-linear constraints. Only the basic notions of vector calculus on Euclidean 3-space and on tangent bundles are needed. Elementary examples are illustrated.

Key words: non-holonomic systems; dynamical systems.

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  1. Benenti S., Geometrical aspects of the dynamics of non-holonomic systems, Rend. Sem. Mat. Univ. Pol. Torino 54 (1996), 203-212.
  2. Bullo F., Lewis A.D., Geometric control of mechanical systems, Texts in Applied Mathematics, Vol. 49, Springer, Berlin, 2004.
  3. Carathéodory C., Sur les équations de la mécanique, Actes Congrès Interbalcanian Math. (1934, Athènes), 1935, 211-214.
  4. Cortés Monforte J., Geometrical, control and numerical aspects of nonholonomic systems, Lecture Notes in Mathematics, Vol. 1793, Springer, Berlin, 2002.
  5. Gantmacher F., Lectures in analytical mechanics, Mir, Moscow, 1970.
  6. Marle C.-M., Reduction of constrained mechanical systems and stability of relative equilibria, Comm. Math. Phys. 174 (1995), 295-318.
  7. Massa E., Pagani E., A new look at classical mechanics of constrained systems, Ann. Inst. H. Poincaré Phys. Théor. 66 (1997), 1-36.
  8. Massa E., Pagani E., Classical dynamics of non-holonomic systems: a geometric approach, Ann. Inst. H. Poincaré Phys. Théor. 55 (1991), 511-544.
  9. Neimark J.I., Fufaev N.A., Dynamics of nonholonomic systems, Translations of Mathematical Monographs, Vol. 33, American Mathematical Society, Providence, Rhode Island, 1972.
  10. Oliva W.M., Kobayashi M.H., A note on the conservation of energy and volume in the setting of nonholonomic mechanical systems, Qual. Theory Dyn. Syst. 4 (2004), 383-411.

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