March  2019, 11(1): 77-122. doi: 10.3934/jgm.2019005

A comparison of vakonomic and nonholonomic dynamics with applications to non-invariant Chaplygin systems

1. 

Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw, Poland

2. 

University of Warsaw, Faculty of Mathematics, Informatics and Mechanics, Banacha 2, 02-097 Warsaw, Poland

3. 

Normandie Université, INSA de Rouen, LMI, 685 Avenue de l'Université 76800 Saint-Etienne-du-Rouvray, France

* Corresponding author

Received  September 2018 Revised  November 2018 Published  January 2019

Fund Project: This research was supported by the National Science Center under the grant DEC-2011/02/A/ST1/00208 "Solvability, chaos and control in quantum systems"

We study relations between vakonomically and nonholonomically constrained Lagrangian dynamics for the same set of linear constraints. The basic idea is to compare both situations at the level of generalized variational principles, not equations of motion as has been done so far. The method seems to be quite powerful and effective. In particular, it allows to derive, interpret and generalize many known results on non-Abelian Chaplygin systems. We apply it also to a class of systems on Lie groups with a left-invariant constraints distribution. Concrete examples of the unicycle in a potential field, the two-wheeled carriage and the generalized Heisenberg system are discussed.

Citation: Michał Jóźwikowski, Witold Respondek. A comparison of vakonomic and nonholonomic dynamics with applications to non-invariant Chaplygin systems. Journal of Geometric Mechanics, 2019, 11 (1) : 77-122. doi: 10.3934/jgm.2019005
References:
[1]

V. I. Arnold, E. Khukhro, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, vol. 3 of Encyclopaedia of Mathematical Sciences, Springer, 2006.

[2]

A. M. Bloch and P. E. Crouch, Nonholonomic and vakonomic control systems on Riemannian manifolds, Fields Institute Communications, American Mathematical Soc., 1 (1993), 25-52.

[3]

F. CantrijnJ. CortésM. De León and D. De Diego, On the geometry of generalized Chaplygin systems, Math. Proc. Cambridge Philos. Soc., 132 (2002), 323-351. doi: 10.1017/S0305004101005679.

[4]

F. Cardin and M. Favretti, On nonholonomic and vakonomic dynamics of mechanical systems with nonintegrable constraints, J. Geom. Phys., 18 (1996), 295-325. doi: 10.1016/0393-0440(95)00016-X.

[5]

J. CortésM. de LeónD. de Diego and S. Martínez, Geometric description of vakonomic and nonholonomic dynamics. Comparison of solutions, SIAM J. Control Optim., 41 (2002), 1389-1412. doi: 10.1137/S036301290036817X.

[6]

M. Crampin and T. Mestdag, Anholonomic frames in constrained dynamics, Dyn. Syst., 25 (2010), 159-187. doi: 10.1080/14689360903360888.

[7]

M. Favretti, Equivalence of dynamics for nonholonomic systems with transverse constraints, J. Dyn. Diff. Equations, 10 (1998), 511-536. doi: 10.1023/A:1022667307485.

[8]

O. E. Fernandez and A. M. Bloch, Equivalence of the dynamics of nonholonomic and variational nonholonomic systems for certain initial data, J. Phys. A: Math. Theor., 41 (2008), 344005, 20pp. doi: 10.1088/1751-8113/41/34/344005.

[9]

K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids, J. Phys. A: Math. Theor., 41 (2008), 175204, 25pp. doi: 10.1088/1751-8113/41/17/175204.

[10]

K. GrabowskaJ. Grabowski and P. Urbański, Geometrical mechanics on algebroids, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 559-575. doi: 10.1142/S0219887806001259.

[11]

J. Grabowski, M. de León, J. C. Marrero and D. M. de Diego, Nonholonomic constraints: A new viewpoint, J. Math. Phys., 50 (2009), 013520, 17pp. doi: 10.1063/1.3049752.

[12]

X. GráciaJ. Marín-Solano and M.-C. Mũoz Lecanda, Some geometric aspects of variational calculus in constrained systems, Rep. Math. Phys., 51 (2003), 127-148. doi: 10.1016/S0034-4877(03)80006-X.

[13]

Y.-X. Guo, S.-X. Liu, C. Liu, S.-K. Luo and Y. Wang, Influence of nonholonomic constraints on variations, symplectic structure, and dynamics of mechanical systems, J. Math. Phys., 48 (2007), 082901, 11pp. doi: 10.1063/1.2762175.

[14] V. Jurdjevic, Geometric Control Theory, Cambridge University Press, 1997.
[15]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, vol. 1, Jonh Willeys & Sons, 1963.

[16]

J. Koiller, Reduction of some classical non-holonomic systems with symmetry, Arch. Rational Mech. Anal., 118 (1992), 113-148. doi: 10.1007/BF00375092.

[17]

I. Kolář, P. W. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer, 1993. doi: 10.1007/978-3-662-02950-3.

[18]

M. d. León, A historical review on nonholomic mechanics, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM, 106 (2012), 191-224. doi: 10.1007/s13398-011-0046-2.

[19]

A. D. Lewis and R. M. Murray, Variational principles for constrained systems: Theory and experiment, Int. J. Non-Linear Mechanics, 30 (1995), 793-815. doi: 10.1016/0020-7462(95)00024-0.

[20]

E. Martínez, Variational calculus on Lie algebroids, ESAIM Control Optim. Calc. Var., 14 (2008), 356-380. doi: 10.1051/cocv:2007056.

[21]

V. V. Rumiantsev, On Hamiltons principle for nonholonomic systems, P. M. M. USSR, 42 (1978), 387-399.

[22]

G. Terra, Vakonomic versus nonholonomic mechanics revisited, São Paulo J. Math. Sci., 12 (2018), 136-145. doi: 10.1007/s40863-017-0062-z.

[23]

W. M. Tulczyjew, The origin of variational principles, Banach Center Publications, 59 (2003), 41-75. doi: 10.4064/bc59-0-2.

show all references

References:
[1]

V. I. Arnold, E. Khukhro, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, vol. 3 of Encyclopaedia of Mathematical Sciences, Springer, 2006.

[2]

A. M. Bloch and P. E. Crouch, Nonholonomic and vakonomic control systems on Riemannian manifolds, Fields Institute Communications, American Mathematical Soc., 1 (1993), 25-52.

[3]

F. CantrijnJ. CortésM. De León and D. De Diego, On the geometry of generalized Chaplygin systems, Math. Proc. Cambridge Philos. Soc., 132 (2002), 323-351. doi: 10.1017/S0305004101005679.

[4]

F. Cardin and M. Favretti, On nonholonomic and vakonomic dynamics of mechanical systems with nonintegrable constraints, J. Geom. Phys., 18 (1996), 295-325. doi: 10.1016/0393-0440(95)00016-X.

[5]

J. CortésM. de LeónD. de Diego and S. Martínez, Geometric description of vakonomic and nonholonomic dynamics. Comparison of solutions, SIAM J. Control Optim., 41 (2002), 1389-1412. doi: 10.1137/S036301290036817X.

[6]

M. Crampin and T. Mestdag, Anholonomic frames in constrained dynamics, Dyn. Syst., 25 (2010), 159-187. doi: 10.1080/14689360903360888.

[7]

M. Favretti, Equivalence of dynamics for nonholonomic systems with transverse constraints, J. Dyn. Diff. Equations, 10 (1998), 511-536. doi: 10.1023/A:1022667307485.

[8]

O. E. Fernandez and A. M. Bloch, Equivalence of the dynamics of nonholonomic and variational nonholonomic systems for certain initial data, J. Phys. A: Math. Theor., 41 (2008), 344005, 20pp. doi: 10.1088/1751-8113/41/34/344005.

[9]

K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids, J. Phys. A: Math. Theor., 41 (2008), 175204, 25pp. doi: 10.1088/1751-8113/41/17/175204.

[10]

K. GrabowskaJ. Grabowski and P. Urbański, Geometrical mechanics on algebroids, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 559-575. doi: 10.1142/S0219887806001259.

[11]

J. Grabowski, M. de León, J. C. Marrero and D. M. de Diego, Nonholonomic constraints: A new viewpoint, J. Math. Phys., 50 (2009), 013520, 17pp. doi: 10.1063/1.3049752.

[12]

X. GráciaJ. Marín-Solano and M.-C. Mũoz Lecanda, Some geometric aspects of variational calculus in constrained systems, Rep. Math. Phys., 51 (2003), 127-148. doi: 10.1016/S0034-4877(03)80006-X.

[13]

Y.-X. Guo, S.-X. Liu, C. Liu, S.-K. Luo and Y. Wang, Influence of nonholonomic constraints on variations, symplectic structure, and dynamics of mechanical systems, J. Math. Phys., 48 (2007), 082901, 11pp. doi: 10.1063/1.2762175.

[14] V. Jurdjevic, Geometric Control Theory, Cambridge University Press, 1997.
[15]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, vol. 1, Jonh Willeys & Sons, 1963.

[16]

J. Koiller, Reduction of some classical non-holonomic systems with symmetry, Arch. Rational Mech. Anal., 118 (1992), 113-148. doi: 10.1007/BF00375092.

[17]

I. Kolář, P. W. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer, 1993. doi: 10.1007/978-3-662-02950-3.

[18]

M. d. León, A historical review on nonholomic mechanics, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM, 106 (2012), 191-224. doi: 10.1007/s13398-011-0046-2.

[19]

A. D. Lewis and R. M. Murray, Variational principles for constrained systems: Theory and experiment, Int. J. Non-Linear Mechanics, 30 (1995), 793-815. doi: 10.1016/0020-7462(95)00024-0.

[20]

E. Martínez, Variational calculus on Lie algebroids, ESAIM Control Optim. Calc. Var., 14 (2008), 356-380. doi: 10.1051/cocv:2007056.

[21]

V. V. Rumiantsev, On Hamiltons principle for nonholonomic systems, P. M. M. USSR, 42 (1978), 387-399.

[22]

G. Terra, Vakonomic versus nonholonomic mechanics revisited, São Paulo J. Math. Sci., 12 (2018), 136-145. doi: 10.1007/s40863-017-0062-z.

[23]

W. M. Tulczyjew, The origin of variational principles, Banach Center Publications, 59 (2003), 41-75. doi: 10.4064/bc59-0-2.

Figure 1.  A homotopy-generated variation and its generator $ \xi $. Note that the homotopy with a fixed end-point(s) corresponds to a generator vanishing at that end-point(s)
Figure 2.  A (non-invariant) Chaplygin system is a principal $G$-bundle $\pi:Q \to M$, equipped with a horizontal distribution ${\rm{H}} Q$ and a Lagrangian $L:{\rm{T}} Q\to\mathbb{R}$, none of which needs to be $G$-invariant.
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