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A note on time-optimal paths on perturbed spheroid
Symmetries of line bundles and Noether theorem for time-dependent nonholonomic systems
Mathematical Institute SANU, Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11000 Belgrade, Serbia |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
F. Cantrjin and W. Sarlet,
Generalizations of Noether's theorem in classical mechanics, SIAM Review, 23 (1981), 467-494.
doi: 10.1137/1023098. |
[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. |
[10] |
M. Crampin,
Constants of the motion in Lagrangian mechanics, Int. J. Theor. Phys., 16 (1977), 741-754.
doi: 10.1007/BF01807231. |
[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. |
[12] |
Dj. S. Djukić,
Conservation laws in classical mechanics for quasi-coordinates, Arch. Ration. Mech. Anal., 56 (1974), 79-98.
doi: 10.1007/BF00279822. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
B. Jovanović, Noether symmetries and integrability in Hamiltonian time-dependent mechanics, Theoretical and Applied Mechanics, 43 (2016), 255-273, arXiv: 1608.07788. |
[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. |
[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. |
[25] |
V. V. Kozlov and N. N. Kolesnikov,
On theorems of dynamics, (Russian), Prikl. Mat. Mekh., 42 (1978), 28-33.
|
[26] |
P. Libermann and C. Marle,
Symplectic Geometry, Analytical Mechanics, Riedel, Dordrecht, 1987.
doi: 10.1007/978-94-009-3807-6. |
[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. |
[28] |
Dj. Mušicki,
Noether's theorem for nonconservative systems in quasicoordinates, Theoretical and Applied Mechanics, 43 (2016), 1-17.
|
[29] |
E. Noether,
Invariante variationsprobleme, Nachrichten von der Königlich Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-physikalische Klasse, (1918), 235-257.
|
[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.
|
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
F. Cantrjin and W. Sarlet,
Generalizations of Noether's theorem in classical mechanics, SIAM Review, 23 (1981), 467-494.
doi: 10.1137/1023098. |
[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. |
[10] |
M. Crampin,
Constants of the motion in Lagrangian mechanics, Int. J. Theor. Phys., 16 (1977), 741-754.
doi: 10.1007/BF01807231. |
[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. |
[12] |
Dj. S. Djukić,
Conservation laws in classical mechanics for quasi-coordinates, Arch. Ration. Mech. Anal., 56 (1974), 79-98.
doi: 10.1007/BF00279822. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
B. Jovanović, Noether symmetries and integrability in Hamiltonian time-dependent mechanics, Theoretical and Applied Mechanics, 43 (2016), 255-273, arXiv: 1608.07788. |
[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. |
[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. |
[25] |
V. V. Kozlov and N. N. Kolesnikov,
On theorems of dynamics, (Russian), Prikl. Mat. Mekh., 42 (1978), 28-33.
|
[26] |
P. Libermann and C. Marle,
Symplectic Geometry, Analytical Mechanics, Riedel, Dordrecht, 1987.
doi: 10.1007/978-94-009-3807-6. |
[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. |
[28] |
Dj. Mušicki,
Noether's theorem for nonconservative systems in quasicoordinates, Theoretical and Applied Mechanics, 43 (2016), 1-17.
|
[29] |
E. Noether,
Invariante variationsprobleme, Nachrichten von der Königlich Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-physikalische Klasse, (1918), 235-257.
|
[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.
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