# American Institute of Mathematical Sciences

2015, 2015(special): 204-212. doi: 10.3934/proc.2015.0204

## On the virial theorem for nonholonomic Lagrangian systems

 1 Department of Theoretical Physics, University of Zaragoza, Spain, Spain 2 IUMA and Departamento de Matemática Aplicada, Universidad de Zaragoza, 50009 Zaragoza 3 CMUC, University of Coimbra, and Polytech. Inst. of Coimbra, ISEC, Portugal

Received  September 2014 Revised  January 2015 Published  November 2015

A generalization of the virial theorem to nonholonomic Lagrangian systems is given. We will first establish the theorem in terms of Lagrange multipliers and later on in terms of the nonholonomic bracket.
Citation: José F. Cariñena, Irina Gheorghiu, Eduardo Martínez, Patrícia Santos. On the virial theorem for nonholonomic Lagrangian systems. Conference Publications, 2015, 2015 (special) : 204-212. doi: 10.3934/proc.2015.0204
##### References:
 [1] Bates L and Śniatycki J, Nonholonomic reduction, Rep. Math. Phys., 32, (1992) 99-115 Google Scholar [2] Bocharov AV and Vinogradov AM, The Hamiltonian form of mechanics with friction, non-holonomic mechanics, invariant mechanics, the theory of refraction and impact, Appendix II in: A.M. Vinogradov and B.A. Kupershmidt, The structures of Hamiltonian mechanics, Russ. Math. Surveys 32 (1977) 177-243 Google Scholar [3] Cariñena JF, Falceto F and Rañada MF, A geometric approach to a generalized virial theorem, J. Phys. A: Math. Theor., 45, 395210 (2012) 19 Google Scholar [4] Cariñena JF, Gheorghiu I, Martínez E and Santos P, Virial theorem in quasi-coordinates and Lie algebroid formalism, Int. J. Geom. Methods Mod. Phys., 11, 1450055 (2014) Google Scholar [5] Cariñena JF, Gheorghiu I, Martínez E and Santos P, Conformal Killing vector fields and a virial theorem, J. Phys. A: Math. Theor., 47, 465206, (2014) 18 Google Scholar [6] Cariñena JF, Nunes da Costa J and Santos P, Quasi-coordinates from the point of view of Lie algebroid structures J. Phys. A: Math. Theor., 40, (2007)10031-10048 Google Scholar [7] Collins GW, The Virial Theorem in Stellar Astrophysics, Astronomy and Astrophysics Series, 7, Tucson, AZ: Pachart Publication House, 1978 Google Scholar [8] Cortés J, de León M, Marrero JC, and Martínez E, Nonholonomic Lagrangian systems on Lie algebroids, Discrete and Continuous Dynamical Systems A, 24 (2), (2009) 213-271 Google Scholar [9] Cushman R and Śniatycki J, Nonholonomic reduction for free and proper actions, Reg. Chaotic Dyn., 7, (2002) 61-72 Google Scholar [10] de León M and Martín de Diego D, On the geometry of non-holonomic Lagrangian systems, J. Math. Phys., 37, (1996) 3389-3414 Google Scholar [11] Papastavridis J, Time-integral theorems for nonholonomic systems, Int. J. Engng. Sci., 25 (7), (1987) 833-854, DOI:10.1016/0020-7225(87)90120-0 Google Scholar [12] Papastavridis J, On energy rate Theorems for linear first-order nonholonomic systems, J. Appl. Mech., 58, (1991) 536-544 Google Scholar [13] Seeger RJ, The virial theorem for nonholonomic systems, Journal of the Washington Academy of Sciences, 24 (11), (1934) 461-464 Google Scholar [14] Śniatycki J, Nonholonomic Noether theorem and reduction of symmetries, Rep. Math. Phys., 42, (1998) 5-23 Google Scholar [15] Vinogradov AM and Kupershmidt BA, The structure of Hamiltonian mechanics, London Math. Soc. Lect. Notes Ser., 60, Cambridge Univ. Press, London, (1981) 173-239 Google Scholar

show all references

##### References:
 [1] Bates L and Śniatycki J, Nonholonomic reduction, Rep. Math. Phys., 32, (1992) 99-115 Google Scholar [2] Bocharov AV and Vinogradov AM, The Hamiltonian form of mechanics with friction, non-holonomic mechanics, invariant mechanics, the theory of refraction and impact, Appendix II in: A.M. Vinogradov and B.A. Kupershmidt, The structures of Hamiltonian mechanics, Russ. Math. Surveys 32 (1977) 177-243 Google Scholar [3] Cariñena JF, Falceto F and Rañada MF, A geometric approach to a generalized virial theorem, J. Phys. A: Math. Theor., 45, 395210 (2012) 19 Google Scholar [4] Cariñena JF, Gheorghiu I, Martínez E and Santos P, Virial theorem in quasi-coordinates and Lie algebroid formalism, Int. J. Geom. Methods Mod. Phys., 11, 1450055 (2014) Google Scholar [5] Cariñena JF, Gheorghiu I, Martínez E and Santos P, Conformal Killing vector fields and a virial theorem, J. Phys. A: Math. Theor., 47, 465206, (2014) 18 Google Scholar [6] Cariñena JF, Nunes da Costa J and Santos P, Quasi-coordinates from the point of view of Lie algebroid structures J. Phys. A: Math. Theor., 40, (2007)10031-10048 Google Scholar [7] Collins GW, The Virial Theorem in Stellar Astrophysics, Astronomy and Astrophysics Series, 7, Tucson, AZ: Pachart Publication House, 1978 Google Scholar [8] Cortés J, de León M, Marrero JC, and Martínez E, Nonholonomic Lagrangian systems on Lie algebroids, Discrete and Continuous Dynamical Systems A, 24 (2), (2009) 213-271 Google Scholar [9] Cushman R and Śniatycki J, Nonholonomic reduction for free and proper actions, Reg. Chaotic Dyn., 7, (2002) 61-72 Google Scholar [10] de León M and Martín de Diego D, On the geometry of non-holonomic Lagrangian systems, J. Math. Phys., 37, (1996) 3389-3414 Google Scholar [11] Papastavridis J, Time-integral theorems for nonholonomic systems, Int. J. Engng. Sci., 25 (7), (1987) 833-854, DOI:10.1016/0020-7225(87)90120-0 Google Scholar [12] Papastavridis J, On energy rate Theorems for linear first-order nonholonomic systems, J. Appl. Mech., 58, (1991) 536-544 Google Scholar [13] Seeger RJ, The virial theorem for nonholonomic systems, Journal of the Washington Academy of Sciences, 24 (11), (1934) 461-464 Google Scholar [14] Śniatycki J, Nonholonomic Noether theorem and reduction of symmetries, Rep. Math. Phys., 42, (1998) 5-23 Google Scholar [15] Vinogradov AM and Kupershmidt BA, The structure of Hamiltonian mechanics, London Math. Soc. Lect. Notes Ser., 60, Cambridge Univ. Press, London, (1981) 173-239 Google Scholar
 [1] Jorge Cortés, Manuel de León, Juan Carlos Marrero, Eduardo Martínez. Nonholonomic Lagrangian systems on Lie algebroids. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 213-271. doi: 10.3934/dcds.2009.24.213 [2] Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems by stages. Journal of Geometric Mechanics, 2020, 12 (4) : 607-639. doi: 10.3934/jgm.2020029 [3] Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems. Journal of Geometric Mechanics, 2010, 2 (1) : 69-111. doi: 10.3934/jgm.2010.2.69 [4] Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2021, 13 (1) : 25-53. doi: 10.3934/jgm.2021001 [5] Marie-Claude Arnaud. When are the invariant submanifolds of symplectic dynamics Lagrangian?. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1811-1827. doi: 10.3934/dcds.2014.34.1811 [6] Juan Carlos Marrero, David Martín de Diego, Ari Stern. Symplectic groupoids and discrete constrained Lagrangian mechanics. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 367-397. doi: 10.3934/dcds.2015.35.367 [7] Mario Jorge Dias Carneiro, Rafael O. Ruggiero. On the graph theorem for Lagrangian minimizing tori. Discrete & Continuous Dynamical Systems, 2018, 38 (12) : 6029-6045. doi: 10.3934/dcds.2018260 [8] Božzidar Jovanović. Symmetries of line bundles and Noether theorem for time-dependent nonholonomic systems. Journal of Geometric Mechanics, 2018, 10 (2) : 173-187. doi: 10.3934/jgm.2018006 [9] Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure & Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839 [10] Jérome Lohéac, Jean-François Scheid. Time optimal control for a nonholonomic system with state constraint. Mathematical Control & Related Fields, 2013, 3 (2) : 185-208. doi: 10.3934/mcrf.2013.3.185 [11] C. D. Ahlbrandt, A. C. Peterson. A general reduction of order theorem for discrete linear symplectic systems. Conference Publications, 1998, 1998 (Special) : 7-18. doi: 10.3934/proc.1998.1998.7 [12] Rodolfo Ríos-Zertuche. Characterization of minimizable Lagrangian action functionals and a dual Mather theorem. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2615-2639. doi: 10.3934/dcds.2020143 [13] Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855 [14] Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067 [15] Ze Cheng, Genggeng Huang. A Liouville theorem for the subcritical Lane-Emden system. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1359-1377. doi: 10.3934/dcds.2019058 [16] Jingbo Dou, Ye Li. Liouville theorem for an integral system on the upper half space. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 155-171. doi: 10.3934/dcds.2015.35.155 [17] Jacques Féjoz. On "Arnold's theorem" on the stability of the solar system. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3555-3565. doi: 10.3934/dcds.2013.33.3555 [18] Xinjing Wang. Liouville type theorem for Fractional Laplacian system. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5253-5268. doi: 10.3934/cpaa.2020236 [19] Xian-gao Liu, Xiaotao Zhang. Liouville theorem for MHD system and its applications. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2329-2350. doi: 10.3934/cpaa.2018111 [20] Gianluca Crippa, Silvia Ligabue, Chiara Saffirio. Lagrangian solutions to the Vlasov-Poisson system with a point charge. Kinetic & Related Models, 2018, 11 (6) : 1277-1299. doi: 10.3934/krm.2018050

Impact Factor:

## Metrics

• PDF downloads (151)
• HTML views (0)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]