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

[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

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