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December  2015, 7(4): 473-482. doi: 10.3934/jgm.2015.7.473

A unifying mechanical equation with applications to non-holonomic constraints and dissipative phenomena

E. Minguzzi 1,

1. 

Dipartimento di Matematica e Informatica "U. Dini", Università degli Studi di Firenze, Via S. Marta 3, I-50139 Firenze, Italy

Received  October 2014 Revised  July 2015 Published  October 2015

A mechanical covariant equation is introduced which retains all the effectingness of the Lagrange equation while being able to describe, in a unified way, other phenomena including friction, non-holonomic constraints and energy radiation (Lorentz-Abraham-Dirac force equation). A quantization rule adapted to the dissipative degrees of freedom is proposed which does not pass through the variational formulation.
Keywords: non-holonomicity, Lorentz-Abraham-Dirac force., Friction.
Mathematics Subject Classification: Primary: 70G45, 37J60; Secondary: 70F4.
Citation: E. Minguzzi. A unifying mechanical equation with applications to non-holonomic constraints and dissipative phenomena. Journal of Geometric Mechanics, 2015, 7 (4) : 473-482. doi: 10.3934/jgm.2015.7.473
References:
[1]

H. Bateman, On dissipative systems and related variational principles, Physical Review, 38 (1931), 815-819. doi: 10.1103/PhysRev.38.815.

[2]

P. Bauer, Dissipative dynamical systems I, Proc. Natl. Acad. Sci. USA, 17 (1931), 311-314. doi: 10.1073/pnas.17.5.311.

[3]

A. M. Bloch, J. Baillieul, P. Crouch and J. Marsden, Nonholonomic Mechanics and Control, Springer, New York, 2003. doi: 10.1007/b97376.

[4]

A. M. Bloch and A. G. Rojo, Quantization of a nonholonomic system, Phys. Rev. Lett., 101 (2008), 030402, 4pp. doi: 10.1103/PhysRevLett.101.030402.

[5]

I. Bucataru and R. Miron, The geometry of systems of third order differential equations induced by second order regular Lagrangians, Mediterr. J. Math., 6 (2009), 483-500. doi: 10.1007/s00009-009-0020-9.

[6]

A. Carati, A Lagrangian Formulation for the Abraham-Lorentz-Dirac Equation, vol. Quaderno del GFNM n. 54, 1998.

[7]

H. V. Craig, On a generalized tangent vector, Amer. J. Math., 57 (1935), 457-462. doi: 10.2307/2371220.

[8]

H. Dekker, Classical and quantum mechanics of the damped harmonic oscillator, Phys. Rep., 80 (1981), 1-112. doi: 10.1016/0370-1573(81)90033-8.

[9]

H. H. Denman, On linear friction in Lagrange's equation, Am. J. Phys., 34 (1966), 1147-1149. doi: 10.1119/1.1972535.

[10]

C. R. Galley, Classical mechanics of nonconservative systems, Phys. Rev. Lett., 110 (2013), 174301. doi: 10.1103/PhysRevLett.110.174301.

[11]

J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, New York, 1975.

[12]

Y. Kuwahara, Y. Nakamura and Y. Yamanaka, From classical mechanics with doubled degrees of freedom to quantum field theory for nonconservative systems, Phys. Lett. A, 377 (2013), 3102-3105. doi: 10.1016/j.physleta.2013.10.001.

[13]

T. Levi-Civita, Sul moto di un sistema di punti materiali soggetti a resistenze proporzionali alle rispettive velocità, Atti Ist. Ven., 54 (1895/96), 1004-1008, Serie 7.

[14]

A. Lurie, Analytical Mechanics, Springer, Berlin, 2002. doi: 10.1007/978-3-540-45677-3.

[15]

C.-M. Marle, Various approaches to conservative and nonconservative nonholonomic systems, Rep. Math. Phys., 42 (1998), 211-229. doi: 10.1016/S0034-4877(98)80011-6.

[16]

R. Y. Matsyuk, Higher order variational origin of the Dixon's system and its relation to the quasi-classical 'zitterbewegung' in general relativity, Diff. Geom. Appl., 29 (2011), S149-S155. doi: 10.1016/j.difgeo.2011.04.020.

[17]

E. Minguzzi, Rayleigh's dissipation function at work, Eur. J. Phys., 36 (2015), 035014, arXiv:1409.4041. doi: 10.1088/0143-0807/36/3/035014.

[18]

J. I. Neimark and N. A. Fufaev, Dynamics of Nonholonomic Systems, vol. 33 of Translations of Mathematical Monographs, American Mathematical Society, 2004.

[19]

F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E, 53 (1996), 1890-1899. doi: 10.1103/PhysRevE.53.1890.

[20]

P. Riewe, Relativistic classical spinning-particle mechanics, Nuovo Cimento B, 8 (1972), 271-277. doi: 10.1007/BF02743522.

[21]

H. Rund, The Hamilton-Jacobi Theory in the Calculus of Variations, Van Nostrand, New York, 1966.

[22]

J. L. Synge, Some intrinsic and derived vectors in a Kawaguchi space, Amer. J. Math., 57 (1935), 679-691. doi: 10.2307/2371196.

[23]

C. Yuce, A. Kilic and A. Coruh, Inverted oscillator, Phys. Scr., 74 (2006), 114-116. doi: 10.1088/0031-8949/74/1/014.

show all references

References:
[1]

H. Bateman, On dissipative systems and related variational principles, Physical Review, 38 (1931), 815-819. doi: 10.1103/PhysRev.38.815.

[2]

P. Bauer, Dissipative dynamical systems I, Proc. Natl. Acad. Sci. USA, 17 (1931), 311-314. doi: 10.1073/pnas.17.5.311.

[3]

A. M. Bloch, J. Baillieul, P. Crouch and J. Marsden, Nonholonomic Mechanics and Control, Springer, New York, 2003. doi: 10.1007/b97376.

[4]

A. M. Bloch and A. G. Rojo, Quantization of a nonholonomic system, Phys. Rev. Lett., 101 (2008), 030402, 4pp. doi: 10.1103/PhysRevLett.101.030402.

[5]

I. Bucataru and R. Miron, The geometry of systems of third order differential equations induced by second order regular Lagrangians, Mediterr. J. Math., 6 (2009), 483-500. doi: 10.1007/s00009-009-0020-9.

[6]

A. Carati, A Lagrangian Formulation for the Abraham-Lorentz-Dirac Equation, vol. Quaderno del GFNM n. 54, 1998.

[7]

H. V. Craig, On a generalized tangent vector, Amer. J. Math., 57 (1935), 457-462. doi: 10.2307/2371220.

[8]

H. Dekker, Classical and quantum mechanics of the damped harmonic oscillator, Phys. Rep., 80 (1981), 1-112. doi: 10.1016/0370-1573(81)90033-8.

[9]

H. H. Denman, On linear friction in Lagrange's equation, Am. J. Phys., 34 (1966), 1147-1149. doi: 10.1119/1.1972535.

[10]

C. R. Galley, Classical mechanics of nonconservative systems, Phys. Rev. Lett., 110 (2013), 174301. doi: 10.1103/PhysRevLett.110.174301.

[11]

J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, New York, 1975.

[12]

Y. Kuwahara, Y. Nakamura and Y. Yamanaka, From classical mechanics with doubled degrees of freedom to quantum field theory for nonconservative systems, Phys. Lett. A, 377 (2013), 3102-3105. doi: 10.1016/j.physleta.2013.10.001.

[13]

T. Levi-Civita, Sul moto di un sistema di punti materiali soggetti a resistenze proporzionali alle rispettive velocità, Atti Ist. Ven., 54 (1895/96), 1004-1008, Serie 7.

[14]

A. Lurie, Analytical Mechanics, Springer, Berlin, 2002. doi: 10.1007/978-3-540-45677-3.

[15]

C.-M. Marle, Various approaches to conservative and nonconservative nonholonomic systems, Rep. Math. Phys., 42 (1998), 211-229. doi: 10.1016/S0034-4877(98)80011-6.

[16]

R. Y. Matsyuk, Higher order variational origin of the Dixon's system and its relation to the quasi-classical 'zitterbewegung' in general relativity, Diff. Geom. Appl., 29 (2011), S149-S155. doi: 10.1016/j.difgeo.2011.04.020.

[17]

E. Minguzzi, Rayleigh's dissipation function at work, Eur. J. Phys., 36 (2015), 035014, arXiv:1409.4041. doi: 10.1088/0143-0807/36/3/035014.

[18]

J. I. Neimark and N. A. Fufaev, Dynamics of Nonholonomic Systems, vol. 33 of Translations of Mathematical Monographs, American Mathematical Society, 2004.

[19]

F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E, 53 (1996), 1890-1899. doi: 10.1103/PhysRevE.53.1890.

[20]

P. Riewe, Relativistic classical spinning-particle mechanics, Nuovo Cimento B, 8 (1972), 271-277. doi: 10.1007/BF02743522.

[21]

H. Rund, The Hamilton-Jacobi Theory in the Calculus of Variations, Van Nostrand, New York, 1966.

[22]

J. L. Synge, Some intrinsic and derived vectors in a Kawaguchi space, Amer. J. Math., 57 (1935), 679-691. doi: 10.2307/2371196.

[23]

C. Yuce, A. Kilic and A. Coruh, Inverted oscillator, Phys. Scr., 74 (2006), 114-116. doi: 10.1088/0031-8949/74/1/014.

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