March 2017, 9(1): 91-130. doi: 10.3934/jgm.2017004

Uniform motions in central fields

1. 

Dept. of Mathematics, Faculty of Science, University of Ostrava, 30. dubna 22,701 03, Ostrava, Czech Republic

2. 

Dept. of Mathematics and Descriptive Geometry, VŠSB -Technical University of Ostrava, 17. listopadu 15,708 33, Ostrava, Czech Republic

Received  March 2016 Revised  January 2017 Published  March 2017

Fund Project: Both authors appreciate support of their departments

We present a theoretical problem of uniform motions, i.e. motions with constant magnitude of the velocity in central fields as a nonholonomic system of one particle with a nonlinear constraint. The concept of the article is in analogy with the recent paper [21]. The problem is analysed from the kinematic and dynamic point of view. The corresponding reduced equation of motion in the Newtonian central gravitational field is solved numerically. Appropriate trajectories for suitable initial conditions are presented. Symmetries and conservation laws are investigated using the concept of constrained Noetherian symmetry [9] and the corresponding constrained Noetherian conservation law. Isotachytonic version of the conservation law of mechanical energy is found as one of the corresponding constraint Noetherian conservation law of this nonholonomic system.

Citation: Martin Swaczyna, Petr Volný. Uniform motions in central fields. Journal of Geometric Mechanics, 2017, 9 (1) : 91-130. doi: 10.3934/jgm.2017004
References:
[1] A. M. Bloch, Nonholonomic Mechanics and Control, Springer-Verlag, New York, 2003. doi: 10.1007/b97376_5.
[2]

M. Brdička and A. Hladĺk, Theoretical Mechanics, Academia, Praha, 1987 (in Czech).

[3] F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems, Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7.
[4]

Yu. F. Golubev, Motion with a constant velocity modulus in a central gravitational field, J. Appl. Math. Mech., 66 (2002), 1001-1013 (English translation), Prikl. Mat. Mekh., 66 (2002), 1052-1065 (in Russian). doi: 10.1016/s0021-8928(02)00141-7.

[5]

J. Janová and J. Musilová, Non-holonomic mechanics: A geometrical treatment of general coupled rolling motion, Int. J. Non-Linear Mechanics, 44 (2009), 98-105. doi: 10.1016/j.ijnonlinmec.2008.09.002.

[6]

W. S. Koon and J. E. Marsden, The Hamiltonian and Lagrangian approaches to the dynamics of nonholonomic system, Rep. Math. Phys., 40 (1997), 21-62. doi: 10.1016/S0034-4877(97)85617-0.

[7]

O. Krupková, Mechanical systems with nonholonomic constraints, J. Math. Phys., 38 (1997), 5098-5126. doi: 10.1063/1.532196.

[8]

O. Krupková, Higher order mechanical systems with constraints, J. Math. Phys., 41 (2000), 5304-5324. doi: 10.1063/1.533411.

[9]

O. Krupková, Noether Theorem, 90 years on, in Geometry and Physics: XVII International Fall Workshop (eds. F. Etayo, M. Fioravanti and R. Santamarĺa), AIP Conference Proceedings, 1130 (2009), 159-170. doi: 10.1063/1.3146232.

[10]

O. Krupková and J. Musilová, The relativistic particle as a mechanical system with nonlinear constraints, J. Phys. A: Math. Gen., 34 (2001), 3859-3875. doi: 10.1088/0305-4470/34/18/313.

[11]

O. Krupková and P. Volný, Differential equations with constraints in jet bundles: Lagrangian and Hamiltonian systems, Lobachevskii J. Math., 23 (2006), 95-150, http://www.mathnet.ru/links/09b54373abba73adc8935fc2403fc66d/ljm19.pdf.

[12]

M. de LeónJ. C. Marrero and D. M. de Diego, Mechanical systems with nonlinear constraints, Int. J. Theor. Phys., 36 (1997), 979-995. doi: 10.1007/BF02435796.

[13]

M. de LeónJ. C. Marrero and D. M. de Diego, Non-holonomic Lagrangian systems in jet manifolds, J. Phys. A: Math. Gen., 30 (1997), 1167-1190. doi: 10.1088/0305-4470/30/4/018.

[14]

J. E. Marsden and T. S. Ratio, Introduction to Mechanics and Symmetry, 2nded., Texts in Applied Mathematics 17, Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5.

[15]

J. C. Monforte, Geometric, Control and Numerical Aspects of Nonholonomic Systems, Lecture Notes in Mathematics 1793, Springer, Berlin, 2002. doi: 10.1007/b84020.

[16]

J. C. MonforteM. de LeónJ. C. Marrero and E. Martínez, Nonholonomic Lagrangian systems on Lie algebroids, Discrete Continuous Dynam. Systems -A, 24 (2009), 213-271. doi: 10.3934/dcds.2009.24.213.

[17]

Ju. I. Neimark and N. A. Fufaev, Dynamics of Nonholonomic Systems, Translations of Mathematical Monographs 33, American Mathematical Society, Rhode Island, 1972.

[18]

C. M. Roithmayr and D. H. Hodges, Forces associated with non-linear non-holonomic constraint equations, Int. J. Nonlinear Mech., 45 (2010), 357-369. doi: 10.1016/j.ijnonlinmec.2009.12.009.

[19]

M. Swaczyna, Mechanical systems with nonholonomic constraints of the second order, in Geometry and Physics: XIX International Fall Workshop (eds. C. Herdeiro, R. Picken), Melville, New York: American Institute of Physics, AIP Conference Proceedings, 1360 (2011), 164-169. doi: 10.1063/1.3599143.

[20]

M. Swaczyna, Several examples of nonholonomic mechanical systems, Communications in Math., 19 (2011), 27-56, http://cm.osu.cz/sites/default/files/contents/19-1/cm019-2011-1_27-56.pdf.

[21]

M. Swaczyna and P. Volný, Uniform projectile motion: Dynamics, symmetries and conservation laws, Rep. Math. Phys., 73 (2014), 177-200. doi: 10.1016/s0034-4877(14)60039-2.

show all references

References:
[1] A. M. Bloch, Nonholonomic Mechanics and Control, Springer-Verlag, New York, 2003. doi: 10.1007/b97376_5.
[2]

M. Brdička and A. Hladĺk, Theoretical Mechanics, Academia, Praha, 1987 (in Czech).

[3] F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems, Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7.
[4]

Yu. F. Golubev, Motion with a constant velocity modulus in a central gravitational field, J. Appl. Math. Mech., 66 (2002), 1001-1013 (English translation), Prikl. Mat. Mekh., 66 (2002), 1052-1065 (in Russian). doi: 10.1016/s0021-8928(02)00141-7.

[5]

J. Janová and J. Musilová, Non-holonomic mechanics: A geometrical treatment of general coupled rolling motion, Int. J. Non-Linear Mechanics, 44 (2009), 98-105. doi: 10.1016/j.ijnonlinmec.2008.09.002.

[6]

W. S. Koon and J. E. Marsden, The Hamiltonian and Lagrangian approaches to the dynamics of nonholonomic system, Rep. Math. Phys., 40 (1997), 21-62. doi: 10.1016/S0034-4877(97)85617-0.

[7]

O. Krupková, Mechanical systems with nonholonomic constraints, J. Math. Phys., 38 (1997), 5098-5126. doi: 10.1063/1.532196.

[8]

O. Krupková, Higher order mechanical systems with constraints, J. Math. Phys., 41 (2000), 5304-5324. doi: 10.1063/1.533411.

[9]

O. Krupková, Noether Theorem, 90 years on, in Geometry and Physics: XVII International Fall Workshop (eds. F. Etayo, M. Fioravanti and R. Santamarĺa), AIP Conference Proceedings, 1130 (2009), 159-170. doi: 10.1063/1.3146232.

[10]

O. Krupková and J. Musilová, The relativistic particle as a mechanical system with nonlinear constraints, J. Phys. A: Math. Gen., 34 (2001), 3859-3875. doi: 10.1088/0305-4470/34/18/313.

[11]

O. Krupková and P. Volný, Differential equations with constraints in jet bundles: Lagrangian and Hamiltonian systems, Lobachevskii J. Math., 23 (2006), 95-150, http://www.mathnet.ru/links/09b54373abba73adc8935fc2403fc66d/ljm19.pdf.

[12]

M. de LeónJ. C. Marrero and D. M. de Diego, Mechanical systems with nonlinear constraints, Int. J. Theor. Phys., 36 (1997), 979-995. doi: 10.1007/BF02435796.

[13]

M. de LeónJ. C. Marrero and D. M. de Diego, Non-holonomic Lagrangian systems in jet manifolds, J. Phys. A: Math. Gen., 30 (1997), 1167-1190. doi: 10.1088/0305-4470/30/4/018.

[14]

J. E. Marsden and T. S. Ratio, Introduction to Mechanics and Symmetry, 2nded., Texts in Applied Mathematics 17, Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5.

[15]

J. C. Monforte, Geometric, Control and Numerical Aspects of Nonholonomic Systems, Lecture Notes in Mathematics 1793, Springer, Berlin, 2002. doi: 10.1007/b84020.

[16]

J. C. MonforteM. de LeónJ. C. Marrero and E. Martínez, Nonholonomic Lagrangian systems on Lie algebroids, Discrete Continuous Dynam. Systems -A, 24 (2009), 213-271. doi: 10.3934/dcds.2009.24.213.

[17]

Ju. I. Neimark and N. A. Fufaev, Dynamics of Nonholonomic Systems, Translations of Mathematical Monographs 33, American Mathematical Society, Rhode Island, 1972.

[18]

C. M. Roithmayr and D. H. Hodges, Forces associated with non-linear non-holonomic constraint equations, Int. J. Nonlinear Mech., 45 (2010), 357-369. doi: 10.1016/j.ijnonlinmec.2009.12.009.

[19]

M. Swaczyna, Mechanical systems with nonholonomic constraints of the second order, in Geometry and Physics: XIX International Fall Workshop (eds. C. Herdeiro, R. Picken), Melville, New York: American Institute of Physics, AIP Conference Proceedings, 1360 (2011), 164-169. doi: 10.1063/1.3599143.

[20]

M. Swaczyna, Several examples of nonholonomic mechanical systems, Communications in Math., 19 (2011), 27-56, http://cm.osu.cz/sites/default/files/contents/19-1/cm019-2011-1_27-56.pdf.

[21]

M. Swaczyna and P. Volný, Uniform projectile motion: Dynamics, symmetries and conservation laws, Rep. Math. Phys., 73 (2014), 177-200. doi: 10.1016/s0034-4877(14)60039-2.

Figure 1.  Interaction of two bodies
Figure 2.  The initial conditions scheme
Figure 3.  The effective potential of the Newtonian gravitational field
Figure 4.  Infinite motions in the Newtonian gravitational field
Figure 5.  Finite motion in the interval of distances $r\in\langle r_0,r_1\rangle$
Figure 6.  Finite motion at the constant distance $r_0$
Figure 7.  Finite motion in the interval of distances $r\in\langle r_1,r_0\rangle$
Figure 8.  Comparison of trajectories: uniform vs. classical motions
Figure 9.  Modified effective potential of the Newtonian gr. field
Figure 10.  Classification of uniform motions in the Newtonian gr. field
Figure 11.  Perturbed circular motions
Figure 12.  Modified effective potential of the central field (171)
Figure 13.  Sinusoidal spirals
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