American Institute of Mathematical Sciences

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

On the virial theorem for nonholonomic Lagrangian systems

José F. Cariñena 1, , Irina Gheorghiu 1, , Eduardo Martínez 2, and  Patrícia Santos 3,

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.
Keywords: Virial theorem, symplectic manifold., quasivelocity, nonholonomic Lagrangian system.
Mathematics Subject Classification: 37J05, 70H05, 70G4.
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

[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

[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

[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)

[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

[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

[7]

Collins GW, The Virial Theorem in Stellar Astrophysics, Astronomy and Astrophysics Series, 7, Tucson, AZ: Pachart Publication House, 1978

[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

[9]

Cushman R and Śniatycki J, Nonholonomic reduction for free and proper actions, Reg. Chaotic Dyn., 7, (2002) 61-72

[10]

de León M and Martín de Diego D, On the geometry of non-holonomic Lagrangian systems, J. Math. Phys., 37, (1996) 3389-3414

[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

[12]

Papastavridis J, On energy rate Theorems for linear first-order nonholonomic systems, J. Appl. Mech., 58, (1991) 536-544

[13]

Seeger RJ, The virial theorem for nonholonomic systems, Journal of the Washington Academy of Sciences, 24 (11), (1934) 461-464

[14]

Śniatycki J, Nonholonomic Noether theorem and reduction of symmetries, Rep. Math. Phys., 42, (1998) 5-23

[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

show all references

References:
[1]

Bates L and Śniatycki J, Nonholonomic reduction, Rep. Math. Phys., 32, (1992) 99-115

[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

[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

[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)

[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

[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

[7]

Collins GW, The Virial Theorem in Stellar Astrophysics, Astronomy and Astrophysics Series, 7, Tucson, AZ: Pachart Publication House, 1978

[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

[9]

Cushman R and Śniatycki J, Nonholonomic reduction for free and proper actions, Reg. Chaotic Dyn., 7, (2002) 61-72

[10]

de León M and Martín de Diego D, On the geometry of non-holonomic Lagrangian systems, J. Math. Phys., 37, (1996) 3389-3414

[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

[12]

Papastavridis J, On energy rate Theorems for linear first-order nonholonomic systems, J. Appl. Mech., 58, (1991) 536-544

[13]

Seeger RJ, The virial theorem for nonholonomic systems, Journal of the Washington Academy of Sciences, 24 (11), (1934) 461-464

[14]

Śniatycki J, Nonholonomic Noether theorem and reduction of symmetries, Rep. Math. Phys., 42, (1998) 5-23

[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

[1]

Jorge Cortés, Manuel de León, Juan Carlos Marrero, Eduardo Martínez. Nonholonomic Lagrangian systems on Lie algebroids. Discrete and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Related Models, 2018, 11 (6) : 1277-1299. doi: 10.3934/krm.2018050

 Impact Factor: 

Tools

Metrics

  • PDF downloads (191)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy