# American Institute of Mathematical Sciences

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

## Nonlinear constraints in nonholonomic mechanics

 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.
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
##### References:

show all references

##### References:
 [1] Nassif Ghoussoub. A variational principle for nonlinear transport equations. Communications on Pure & Applied Analysis, 2005, 4 (4) : 735-742. doi: 10.3934/cpaa.2005.4.735 [2] H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77 [3] Md. Haider Ali Biswas, Maria do Rosário de Pinho. A nonsmooth maximum principle for optimal control problems with state and mixed constraints - convex case. Conference Publications, 2011, 2011 (Special) : 174-183. doi: 10.3934/proc.2011.2011.174 [4] Andrei V. Dmitruk, Nikolai P. Osmolovskii. Proof of the maximum principle for a problem with state constraints by the v-change of time variable. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2189-2204. doi: 10.3934/dcdsb.2019090 [5] Mikhail Gusev. On reachability analysis for nonlinear control systems with state constraints. Conference Publications, 2015, 2015 (special) : 579-587. doi: 10.3934/proc.2015.0579 [6] Maria Francesca Betta, Rosaria Di Nardo, Anna Mercaldo, Adamaria Perrotta. Gradient estimates and comparison principle for some nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2015, 14 (3) : 897-922. doi: 10.3934/cpaa.2015.14.897 [7] Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395 [8] Huseyin Coskun. Nonlinear decomposition principle and fundamental matrix solutions for dynamic compartmental systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-53. doi: 10.3934/dcdsb.2019155 [9] Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control & Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61 [10] Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017 [11] J.-P. Raymond. Nonlinear boundary control of semilinear parabolic problems with pointwise state constraints. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 341-370. doi: 10.3934/dcds.1997.3.341 [12] Chunlin Hao, Xinwei Liu. Global convergence of an SQP algorithm for nonlinear optimization with overdetermined constraints. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 19-29. doi: 10.3934/naco.2012.2.19 [13] Gaohang Yu, Shanzhou Niu, Jianhua Ma. Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints. Journal of Industrial & Management Optimization, 2013, 9 (1) : 117-129. doi: 10.3934/jimo.2013.9.117 [14] Jorge Cortés. Energy conserving nonholonomic integrators. Conference Publications, 2003, 2003 (Special) : 189-199. doi: 10.3934/proc.2003.2003.189 [15] Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897 [16] Chuanqiang Chen. On the microscopic spacetime convexity principle for fully nonlinear parabolic equations II: Spacetime quasiconcave solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4761-4811. doi: 10.3934/dcds.2016007 [17] Chuanqiang Chen. On the microscopic spacetime convexity principle of fully nonlinear parabolic equations I: Spacetime convex solutions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3383-3402. doi: 10.3934/dcds.2014.34.3383 [18] Marin Kobilarov, Jerrold E. Marsden, Gaurav S. Sukhatme. Geometric discretization of nonholonomic systems with symmetries. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 61-84. doi: 10.3934/dcdss.2010.3.61 [19] Oscar E. Fernandez, Anthony M. Bloch, P. J. Olver. Variational Integrators for Hamiltonizable Nonholonomic Systems. Journal of Geometric Mechanics, 2012, 4 (2) : 137-163. doi: 10.3934/jgm.2012.4.137 [20] Jorge Cortés, Manuel de León, Juan Carlos Marrero, Eduardo Martínez. Nonholonomic Lagrangian systems on Lie algebroids. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 213-271. doi: 10.3934/dcds.2009.24.213

2018 Impact Factor: 0.525