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

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.
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. doi: 10.1103/PhysRev.38.815. Google Scholar

[2]

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

[3]

A. M. Bloch, J. Baillieul, P. Crouch and J. Marsden, Nonholonomic Mechanics and Control,, Springer, (2003). doi: 10.1007/b97376. Google Scholar

[4]

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

[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. doi: 10.1007/s00009-009-0020-9. Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

J. D. Jackson, Classical Electrodynamics,, John Wiley & Sons, (1975). Google Scholar

[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. doi: 10.1016/j.physleta.2013.10.001. Google Scholar

[13]

T. Levi-Civita, Sul moto di un sistema di punti materiali soggetti a resistenze proporzionali alle rispettive velocità,, Atti Ist. Ven., 54 (): 1004. Google Scholar

[14]

A. Lurie, Analytical Mechanics,, Springer, (2002). doi: 10.1007/978-3-540-45677-3. Google Scholar

[15]

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

[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). doi: 10.1016/j.difgeo.2011.04.020. Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

H. Rund, The Hamilton-Jacobi Theory in the Calculus of Variations,, Van Nostrand, (1966). Google Scholar

[22]

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

[23]

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

show all references

References:
[1]

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

[2]

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

[3]

A. M. Bloch, J. Baillieul, P. Crouch and J. Marsden, Nonholonomic Mechanics and Control,, Springer, (2003). doi: 10.1007/b97376. Google Scholar

[4]

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

[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. doi: 10.1007/s00009-009-0020-9. Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

J. D. Jackson, Classical Electrodynamics,, John Wiley & Sons, (1975). Google Scholar

[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. doi: 10.1016/j.physleta.2013.10.001. Google Scholar

[13]

T. Levi-Civita, Sul moto di un sistema di punti materiali soggetti a resistenze proporzionali alle rispettive velocità,, Atti Ist. Ven., 54 (): 1004. Google Scholar

[14]

A. Lurie, Analytical Mechanics,, Springer, (2002). doi: 10.1007/978-3-540-45677-3. Google Scholar

[15]

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

[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). doi: 10.1016/j.difgeo.2011.04.020. Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

H. Rund, The Hamilton-Jacobi Theory in the Calculus of Variations,, Van Nostrand, (1966). Google Scholar

[22]

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

[23]

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

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