Motion of Nonholonomous Rheonomous Systems in the Lagrangian Formalism


The main purpose of the paper consists in illustrating a procedure for expressing the equations of motion for a general time-dependent constrained system. Constraints are both of geometrical and differential type. The use of quasi-velocities as variables of the mathematical problem opens the possibility of incorporating some remarkable and classic cases of equations of motion. Afterwards, the scheme of equations is implemented for a pair of substantial examples, which are presented in a double version, acting either as a scleronomic system and as a rheonomic system.

Share and Cite:

Talamucci, F. (2015) Motion of Nonholonomous Rheonomous Systems in the Lagrangian Formalism. Journal of Applied Mathematics and Physics, 3, 295-309. doi: 10.4236/jamp.2015.33043.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Poincare, H. (1901) Sur une forme nouvelle des èquations de la mechanique. Comptes Rendus de l’Academie des Sciences, 132, 369-371.
[2] Gantmacher, F.R. (1975) Lectures in Analytical Mechanics. MIR.
[3] Maruskin, J.M. and Bloch, A.M. (2011) The Boltzman-Hamel Equations for the Optimal Control of Mechanical Systems with Nonholonomic Constraints. International Journal of Robust and Nonlinear Control, 21, 373-386.
[4] Cameron, J.M. and Book, W.J. (1997) Modeling Mechanisms with Nonholonomic Joints Using the Boltzmann-Hamel Equations. Journal International Journal of Robotics Research, 16, 47-59.
[5] Talamucci, F. (2014) The Lagrangian Method for a Basic Bicycle. Journal of Applied Mathematics and Physics, 2, 46-60.
[6] Levi, M. (2014) Bike Tracks, Quasi-Magnetic Forces, and the Schrodinger Equation. SIAM News, 47.
[7] Zenkov, V., Bloch, A.M. and Mardsen, J.E. (2002) Stabilization of the Unicycle with Rider. Systems and Control Letters, 46, 293-302.
[8] Bloch, A.M., Krishnaprasad, P.S., Mardsen, J.E. and Murray, R. (1996) Nonholonomic Mechanical Systems with Symmetry. Archive for Rational Mechanics and Analysis, 136, 21-99.
[9] Bloch, A.M., Mardsen, J.E. and Zenkov, D.V. (2009) Quasivelocities and Symmetries in Non-Holonomic Systems. Dynamical Systems, 24, 187-222.

Copyright © 2023 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.