American Institute of Mathematical Sciences

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December  2014, 6(4): 527-547. doi: 10.3934/jgm.2014.6.527

Nonlinear constraints in nonholonomic mechanics

Paul Popescu 1, and  Cristian Ida 2,

1. 

Department of Applied Mathematics, University of Craiova, Craiova 200585, Str. A.I. Cuza 13, Romania
2. 

Department of Mathematics and Informatics, University Transilvania of Braşov, Braşov 500091, Str. Iuliu Maniu 50, Romania

Received  December 2013 Revised  June 2014 Published  December 2014

In this paper we have obtained some dynamics equations, in the presence of nonlinear nonholonomic constraints and according to a lagrangian and some Chetaev-like conditions. Using some natural regular conditions, a simple form of these equations is given. In the particular cases of linear and affine constraints, one recovers the classical equations in the forms known previously, for example, by Bloch and all [3,4]. The case of time-dependent constraints is also considered. Examples of linear constraints, time independent and time depenndent nonlinear constraints are considered, as well as their dynamics given by suitable lagrangians. All examples are based on classical ones, such as those given by Appell's machine.
Keywords: Nonlinear, nonholonomic, Chetaev principle., constraints.
Mathematics Subject Classification: Primary: 70F25, 37J60, 70H45; Secondary: 37C9.
Citation: Paul Popescu, Cristian Ida. Nonlinear constraints in nonholonomic mechanics. Journal of Geometric Mechanics, 2014, 6 (4) : 527-547. doi: 10.3934/jgm.2014.6.527
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show all references

References:
[1]

Journal of Nonlinear Science, 22 (2012), 213-233. doi: 10.1007/s00332-011-9114-1.  Google Scholar

[2]

Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 203-212.  Google Scholar

[3]

Vol. 24, Springer, 2003. doi: 10.1007/b97376.  Google Scholar

[4]

Archive for Rational Mechanics and Analysis, 136 (1996), 21-99. doi: 10.1007/BF02199365.  Google Scholar

[5]

Editura Academiei Romane, Bucuresti, 2007.  Google Scholar

[6]

J. Math. Phys., 45 (2004), 2785-2801. doi: 10.1063/1.1763245.  Google Scholar

[7]

arXiv preprint math-ph/0512003 (2005). Google Scholar

[8]

Illinois Journal of Mathematics, 38 (1994), 148-175.  Google Scholar

[9]

Journal of Physics A: Mathematical and Theoretical, 41 (2008), 175204, 25pp. doi: 10.1088/1751-8113/41/17/175204.  Google Scholar

[10]

International Journal of Geometric Methods in Modern Physics, 3 (2006), 559-575. doi: 10.1142/S0219887806001259.  Google Scholar

[11]

Chinese Physics, 10 (2001), p181. Google Scholar

[12]

Rend. Semin. Mat., Torino, 54 (1996), 295-317.  Google Scholar

[13]

Qualitative Theory of Dynamical Systems, 4 (2004), 383-411. doi: 10.1007/BF02970866.  Google Scholar

[14]

Journal of Mathematical Physics, 38 (1997), 5098-5126. doi: 10.1063/1.532196.  Google Scholar

[15]

Communications in Mathematics, 18 (2010), 51-77.  Google Scholar

[16]

3-th ed., Springer Verlag, New York, 1995. doi: 10.1007/978-1-4612-4182-9.  Google Scholar

[17]

Revista de la Real Academia de Ciencias Exactas, Fisicas Y Naturales (Serie A: Matematicas) 105, 2011. Google Scholar

[18]

International Journal of Theoretical Physics, 36 (1997), 979-995. doi: 10.1007/BF02435796.  Google Scholar

[19]

Journal of Geometric Mechanics, 6 (2014), 121-140. doi: 10.3934/jgm.2014.6.121.  Google Scholar

[20]

Reports on Mathematical Physics, 38 (1996), 11-28. doi: 10.1016/0034-4877(96)87675-0.  Google Scholar

[21]

Reports on Mathematical Physics, 63 (2009), 179-189. doi: 10.1016/S0034-4877(09)00012-3.  Google Scholar

[22]

Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 353-364.  Google Scholar

[23]

Reports on Mathematical Physics, 42 (1998), 211-229. doi: 10.1016/S0034-4877(98)80011-6.  Google Scholar

[24]

Journal of Physics A: Mathematical and General, 38 (2005), 1097-1111. doi: 10.1088/0305-4470/38/5/011.  Google Scholar

[25]

Birkhäuser, Progr. Math. 73, 1988. doi: 10.1007/978-1-4684-8670-4.  Google Scholar

[26]

Differential Geom. Appl., 27 (2009), 171-178. doi: 10.1016/j.difgeo.2008.06.017.  Google Scholar

[27]

J. Phys. A, 28 (1995), 3253-3268. doi: 10.1088/0305-4470/28/11/022.  Google Scholar

[28]

Communications in Mathematics, 19 (2011), 27-56.  Google Scholar

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