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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 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

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