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June  2018, 10(2): 173-187. doi: 10.3934/jgm.2018006

Symmetries of line bundles and Noether theorem for time-dependent nonholonomic systems

Božzidar Jovanović

Mathematical Institute SANU, Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11000 Belgrade, Serbia

Received  September 2016 Revised  March 2018 Published  May 2018

We consider Noether symmetries of the equations defined by the sections of characteristic line bundles of nondegenerate 1-forms and of the associated perturbed systems. It appears that this framework can be used for time-dependent systems with constraints and nonconservative forces, allowing a quite simple and transparent formulation of the momentum equation and the Noether theorem in their general forms.

Keywords: Noether's integrals, Poincaré–Cartan form, time-dependent affine constraints, momentum equation, gauge symmetries, moving energy.
Mathematics Subject Classification: Primary: 37J15, 37J60, 70F25; Secondary: 70H25, 70H33.
Citation: 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
References:
[1]

V. I. Arnold, V. V. Kozlov and A. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Dynamical Systems, Ⅲ. Third Edition. Encyclopaedia Math. Sci. 3. Springer, Berlin, 2006.  Google Scholar

[2]

P. Balseiro and N. Sansonetto, A Geometric Characterization of Certain First Integrals for Nonholonomic Systems with Symmetries, SIGMA, 12 (2016), Paper No. 018, 14 pages, arXiv: 1510.08314. doi: 10.3842/SIGMA.2016.018.  Google Scholar

[3]

L. Bates, H. Graumann and C. MacDonnell, Examples of gauge conservation laws in nonholonomic systems, Rep. Math. Phys., 37 (1996), 295-308.  doi: 10.1016/0034-4877(96)84069-9.  Google Scholar

[4]

A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and R. M. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal., 136 (1996), 21-99.  doi: 10.1007/BF02199365.  Google Scholar

[5]

A. M. Bloch, J. E. Marsden and D. V. Zenkov, Quasivelocities and symmetries in nonholonomic systems, Dynamical Systems, 24 (2009), 187-222.  doi: 10.1080/14689360802609344.  Google Scholar

[6]

A. V. Borisov and I. S. Mamaev, Symmetries and reduction in nonholonomic mechanics, Regular and Chaotic Dynamics, 20 (2015), 553–604 doi: 10.1134/S1560354715050044.  Google Scholar

[7]

A. V. Borisov, I. S. Mamaev and I. A. Bizyaev, The jacobi integral in nonholonomic mechanics, Regular and Chaotic Dynamics, 20 (2015), 383-400.  doi: 10.1134/S1560354715030107.  Google Scholar

[8]

F. Cantrjin and W. Sarlet, Generalizations of Noether's theorem in classical mechanics, SIAM Review, 23 (1981), 467-494.  doi: 10.1137/1023098.  Google Scholar

[9]

F. Cantrjn, M. de Leon, M. de Diego and J. Marrero, Reduction of nonholonomic mechanical systems with symmetries, Rep. Math. Phys, 42 (1998), 25-45.  doi: 10.1016/S0034-4877(98)80003-7.  Google Scholar

[10]

M. Crampin, Constants of the motion in Lagrangian mechanics, Int. J. Theor. Phys., 16 (1977), 741-754.  doi: 10.1007/BF01807231.  Google Scholar

[11]

M. Crampin and T. Mestdag, The Cartan form for constrained Lagrangian systems and the nonholonomic Noether theorem, Int. J. Geom. Methods. Mod. Phys., 8 (2011), 897-923, arXiv: 1101.3153. doi: 10.1142/S0219887811005452.  Google Scholar

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Dj. S. Djukić, Conservation laws in classical mechanics for quasi-coordinates, Arch. Ration. Mech. Anal., 56 (1974), 79-98.  doi: 10.1007/BF00279822.  Google Scholar

[13]

Dj. S. Djukić and B. D. Vujanović, Noether's theory in the classical nonconservative mechanics, Acta Mech., 23 (1975), 17-27.  doi: 10.1007/BF01177666.  Google Scholar

[14]

F. Fassò, A. Ramos and N. Sansonetto, The reaction-annihilator distribution and the nonholonomic Noether theorem for lifted actions, Regul. Chaotic Dyn., 12 (2007), 579-588.  doi: 10.1134/S1560354707060019.  Google Scholar

[15]

F. Fassò, A. Giacobbe and N. Sansonetto, Gauge conservation laws and the momentum equation in nonholonomic mechanics, Reports in Mathematical Physics, 62 (2008), 345-367.  doi: 10.1016/S0034-4877(09)00005-6.  Google Scholar

[16]

F. Fassò and N. Sansonetto, Conservation of energy and momenta in nonholonomic systems with affine constraints, Regular and Chaotic Dynamics, 20 (2015), 449-462, arXiv: 1505.01172. doi: 10.1134/S1560354715040048.  Google Scholar

[17]

F. Fassò and N. Sansonetto, Conservation of 'moving' energy in nonholonomic systems with affine constraints and integrability of spheres on rotating surfaces, J. Nonlinear Sci., 26 (2016), 519-544, arXiv: 1503.06661. doi: 10.1007/s00332-015-9283-4.  Google Scholar

[18]

E. Fiorani and A. Spiro, Lie algebras of conservation laws of variational ordinary differential equations, J. Geom. Phys., 88 (2015), 56-75, arXiv: 1411.6097. doi: 10.1016/j.geomphys.2014.11.005.  Google Scholar

[19]

G. Giachetta, First integrals of non-holonomic systems and their generators, J. Phys. A: Math. Gen., 33 (2000), 5369-5389.  doi: 10.1088/0305-4470/33/30/308.  Google Scholar

[20]

S. Hochgerner and L. C. Garcia-Naranjo, G-Chaplygin systems with internal symmetries, truncation, and an (almost) symplectic view of Chaplygin's ball, J. Geom. Mech., 1 (2009), 35-53, arXiv: 0810.5454. doi: 10.3934/jgm.2009.1.35.  Google Scholar

[21]

B. Jovanović, Hamiltonization and integrability of the Chaplygin sphere in Rn, J. Nonlinear. Sci., 20 (2010), 569-593, arXiv: 0902.4397. doi: 10.1007/s00332-010-9067-9.  Google Scholar

[22]

B. Jovanović, Noether symmetries and integrability in Hamiltonian time-dependent mechanics, Theoretical and Applied Mechanics, 43 (2016), 255-273, arXiv: 1608.07788. Google Scholar

[23]

B. Jovanović, Invariant measures of modified LR and L+R systems, Regular and Chaotic Dynamics, 20 (2015), 542-552, arXiv: 1508.04913. doi: 10.1134/S1560354715050032.  Google Scholar

[24]

Y. Kosmann-Schwarzbach, The Noether Theorems, Invariance and Conservation Laws in the Twentieth Century, Springer, New York, 2011. doi: 10.1007/978-0-387-87868-3.  Google Scholar

[25]

V. V. Kozlov and N. N. Kolesnikov, On theorems of dynamics, (Russian), Prikl. Mat. Mekh., 42 (1978), 28-33.   Google Scholar

[26]

P. Libermann and C. Marle, Symplectic Geometry, Analytical Mechanics, Riedel, Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.  Google Scholar

[27]

C.-M. Marle, On symmetries and constants of motion in Hamiltonian systems with nonholonomic constraints, In Classical and Quantum Integrability (Warsaw, 2001), 223-242, Banach Center Publ. 59, Polish Acad. Sci. Warsaw, 2003. doi: 10.4064/bc59-0-12.  Google Scholar

[28]

Dj. Mušicki, Noether's theorem for nonconservative systems in quasicoordinates, Theoretical and Applied Mechanics, 43 (2016), 1-17.   Google Scholar

[29]

E. Noether, Invariante variationsprobleme, Nachrichten von der Königlich Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-physikalische Klasse, (1918), 235-257.   Google Scholar

[30]

S. Simić, On Noetherian approach to integrable cases of the motion of heavy top, Bull. Cl. Sci. Math. Nat. Sci. Math., 25 (2000), 133-156.   Google Scholar

show all references

References:
[1]

V. I. Arnold, V. V. Kozlov and A. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Dynamical Systems, Ⅲ. Third Edition. Encyclopaedia Math. Sci. 3. Springer, Berlin, 2006.  Google Scholar

[2]

P. Balseiro and N. Sansonetto, A Geometric Characterization of Certain First Integrals for Nonholonomic Systems with Symmetries, SIGMA, 12 (2016), Paper No. 018, 14 pages, arXiv: 1510.08314. doi: 10.3842/SIGMA.2016.018.  Google Scholar

[3]

L. Bates, H. Graumann and C. MacDonnell, Examples of gauge conservation laws in nonholonomic systems, Rep. Math. Phys., 37 (1996), 295-308.  doi: 10.1016/0034-4877(96)84069-9.  Google Scholar

[4]

A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and R. M. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal., 136 (1996), 21-99.  doi: 10.1007/BF02199365.  Google Scholar

[5]

A. M. Bloch, J. E. Marsden and D. V. Zenkov, Quasivelocities and symmetries in nonholonomic systems, Dynamical Systems, 24 (2009), 187-222.  doi: 10.1080/14689360802609344.  Google Scholar

[6]

A. V. Borisov and I. S. Mamaev, Symmetries and reduction in nonholonomic mechanics, Regular and Chaotic Dynamics, 20 (2015), 553–604 doi: 10.1134/S1560354715050044.  Google Scholar

[7]

A. V. Borisov, I. S. Mamaev and I. A. Bizyaev, The jacobi integral in nonholonomic mechanics, Regular and Chaotic Dynamics, 20 (2015), 383-400.  doi: 10.1134/S1560354715030107.  Google Scholar

[8]

F. Cantrjin and W. Sarlet, Generalizations of Noether's theorem in classical mechanics, SIAM Review, 23 (1981), 467-494.  doi: 10.1137/1023098.  Google Scholar

[9]

F. Cantrjn, M. de Leon, M. de Diego and J. Marrero, Reduction of nonholonomic mechanical systems with symmetries, Rep. Math. Phys, 42 (1998), 25-45.  doi: 10.1016/S0034-4877(98)80003-7.  Google Scholar

[10]

M. Crampin, Constants of the motion in Lagrangian mechanics, Int. J. Theor. Phys., 16 (1977), 741-754.  doi: 10.1007/BF01807231.  Google Scholar

[11]

M. Crampin and T. Mestdag, The Cartan form for constrained Lagrangian systems and the nonholonomic Noether theorem, Int. J. Geom. Methods. Mod. Phys., 8 (2011), 897-923, arXiv: 1101.3153. doi: 10.1142/S0219887811005452.  Google Scholar

[12]

Dj. S. Djukić, Conservation laws in classical mechanics for quasi-coordinates, Arch. Ration. Mech. Anal., 56 (1974), 79-98.  doi: 10.1007/BF00279822.  Google Scholar

[13]

Dj. S. Djukić and B. D. Vujanović, Noether's theory in the classical nonconservative mechanics, Acta Mech., 23 (1975), 17-27.  doi: 10.1007/BF01177666.  Google Scholar

[14]

F. Fassò, A. Ramos and N. Sansonetto, The reaction-annihilator distribution and the nonholonomic Noether theorem for lifted actions, Regul. Chaotic Dyn., 12 (2007), 579-588.  doi: 10.1134/S1560354707060019.  Google Scholar

[15]

F. Fassò, A. Giacobbe and N. Sansonetto, Gauge conservation laws and the momentum equation in nonholonomic mechanics, Reports in Mathematical Physics, 62 (2008), 345-367.  doi: 10.1016/S0034-4877(09)00005-6.  Google Scholar

[16]

F. Fassò and N. Sansonetto, Conservation of energy and momenta in nonholonomic systems with affine constraints, Regular and Chaotic Dynamics, 20 (2015), 449-462, arXiv: 1505.01172. doi: 10.1134/S1560354715040048.  Google Scholar

[17]

F. Fassò and N. Sansonetto, Conservation of 'moving' energy in nonholonomic systems with affine constraints and integrability of spheres on rotating surfaces, J. Nonlinear Sci., 26 (2016), 519-544, arXiv: 1503.06661. doi: 10.1007/s00332-015-9283-4.  Google Scholar

[18]

E. Fiorani and A. Spiro, Lie algebras of conservation laws of variational ordinary differential equations, J. Geom. Phys., 88 (2015), 56-75, arXiv: 1411.6097. doi: 10.1016/j.geomphys.2014.11.005.  Google Scholar

[19]

G. Giachetta, First integrals of non-holonomic systems and their generators, J. Phys. A: Math. Gen., 33 (2000), 5369-5389.  doi: 10.1088/0305-4470/33/30/308.  Google Scholar

[20]

S. Hochgerner and L. C. Garcia-Naranjo, G-Chaplygin systems with internal symmetries, truncation, and an (almost) symplectic view of Chaplygin's ball, J. Geom. Mech., 1 (2009), 35-53, arXiv: 0810.5454. doi: 10.3934/jgm.2009.1.35.  Google Scholar

[21]

B. Jovanović, Hamiltonization and integrability of the Chaplygin sphere in Rn, J. Nonlinear. Sci., 20 (2010), 569-593, arXiv: 0902.4397. doi: 10.1007/s00332-010-9067-9.  Google Scholar

[22]

B. Jovanović, Noether symmetries and integrability in Hamiltonian time-dependent mechanics, Theoretical and Applied Mechanics, 43 (2016), 255-273, arXiv: 1608.07788. Google Scholar

[23]

B. Jovanović, Invariant measures of modified LR and L+R systems, Regular and Chaotic Dynamics, 20 (2015), 542-552, arXiv: 1508.04913. doi: 10.1134/S1560354715050032.  Google Scholar

[24]

Y. Kosmann-Schwarzbach, The Noether Theorems, Invariance and Conservation Laws in the Twentieth Century, Springer, New York, 2011. doi: 10.1007/978-0-387-87868-3.  Google Scholar

[25]

V. V. Kozlov and N. N. Kolesnikov, On theorems of dynamics, (Russian), Prikl. Mat. Mekh., 42 (1978), 28-33.   Google Scholar

[26]

P. Libermann and C. Marle, Symplectic Geometry, Analytical Mechanics, Riedel, Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.  Google Scholar

[27]

C.-M. Marle, On symmetries and constants of motion in Hamiltonian systems with nonholonomic constraints, In Classical and Quantum Integrability (Warsaw, 2001), 223-242, Banach Center Publ. 59, Polish Acad. Sci. Warsaw, 2003. doi: 10.4064/bc59-0-12.  Google Scholar

[28]

Dj. Mušicki, Noether's theorem for nonconservative systems in quasicoordinates, Theoretical and Applied Mechanics, 43 (2016), 1-17.   Google Scholar

[29]

E. Noether, Invariante variationsprobleme, Nachrichten von der Königlich Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-physikalische Klasse, (1918), 235-257.   Google Scholar

[30]

S. Simić, On Noetherian approach to integrable cases of the motion of heavy top, Bull. Cl. Sci. Math. Nat. Sci. Math., 25 (2000), 133-156.   Google Scholar

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