American Institute of Mathematical Sciences

Advanced
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

[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

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

[1]

Stephen Anco, Maria Rosa, Maria Luz Gandarias. Conservation laws and symmetries of time-dependent generalized KdV equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 607-615. doi: 10.3934/dcdss.2018035
[2]

Masahiro Kubo, Noriaki Yamazaki. Elliptic-parabolic variational inequalities with time-dependent constraints. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 335-359. doi: 10.3934/dcds.2007.19.335
[3]

Francesco Di Plinio, Gregory S. Duane, Roger Temam. Time-dependent attractor for the Oscillon equation. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 141-167. doi: 10.3934/dcds.2011.29.141
[4]

Jin Takahashi, Eiji Yanagida. Time-dependent singularities in the heat equation. Communications on Pure & Applied Analysis, 2015, 14 (3) : 969-979. doi: 10.3934/cpaa.2015.14.969
[5]

Nicola Guglielmi, László Hatvani. On small oscillations of mechanical systems with time-dependent kinetic and potential energy. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 911-926. doi: 10.3934/dcds.2008.20.911
[6]

Colin J. Cotter, Michael John Priestley Cullen. Particle relabelling symmetries and Noether's theorem for vertical slice models. Journal of Geometric Mechanics, 2019, 11 (2) : 139-151. doi: 10.3934/jgm.2019007
[7]

Rehana Naz, Imran Naeem. Exact solutions of a Black-Scholes model with time-dependent parameters by utilizing potential symmetries. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2841-2851. doi: 10.3934/dcdss.2020122
[8]

Zhidong Zhang. An undetermined time-dependent coefficient in a fractional diffusion equation. Inverse Problems & Imaging, 2017, 11 (5) : 875-900. doi: 10.3934/ipi.2017041
[9]

Jiayun Lin, Kenji Nishihara, Jian Zhai. Critical exponent for the semilinear wave equation with time-dependent damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4307-4320. doi: 10.3934/dcds.2012.32.4307
[10]

Holger Teismann. The Schrödinger equation with singular time-dependent potentials. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 705-722. doi: 10.3934/dcds.2000.6.705
[11]

Jungkwon Kim, Hyeongjin Lee, Ihyeok Seo, Jihyeon Seok. On Morawetz estimates with time-dependent weights for the Klein-Gordon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6275-6288. doi: 10.3934/dcds.2020279
[12]

Gastão S. F. Frederico, Delfim F. M. Torres. Noether's symmetry Theorem for variational and optimal control problems with time delay. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 619-630. doi: 10.3934/naco.2012.2.619
[13]

Zhijie Cao, Lijun Zhang. Symmetries and conservation laws of a time dependent nonlinear reaction-convection-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2703-2717. doi: 10.3934/dcdss.2020218
[14]

Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic & Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639
[15]

Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113
[16]

Yavar Kian, Alexander Tetlow. Hölder-stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic Schrödinger equation. Inverse Problems & Imaging, 2020, 14 (5) : 819-839. doi: 10.3934/ipi.2020038
[17]

Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903
[18]

Mourad Bellassoued, Oumaima Ben Fraj. Stability estimates for time-dependent coefficients appearing in the magnetic Schrödinger equation from arbitrary boundary measurements. Inverse Problems & Imaging, 2020, 14 (5) : 841-865. doi: 10.3934/ipi.2020039
[19]

Qingquan Chang, Dandan Li, Chunyou Sun. Random attractors for stochastic time-dependent damped wave equation with critical exponents. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2793-2824. doi: 10.3934/dcdsb.2020033
[20]

Olivier Brahic. Infinitesimal gauge symmetries of closed forms. Journal of Geometric Mechanics, 2011, 3 (3) : 277-312. doi: 10.3934/jgm.2011.3.277

2019 Impact Factor: 0.649

Tools

Metrics

  • PDF downloads (106)
  • HTML views (187)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences