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Structure preserving discretization of time-reparametrized Hamiltonian systems with application to nonholonomic mechanics
1. | Departamento de Matemáticas y Mecánica, IIMAS, UNAM, Apdo. Postal 20-126, 01000 Mexico City, Mexico, Current address: Dipartimento di Matematica "Tullio Levi-Civita", Università di Padova, Via Trieste 63, 35121 Padova, Italy |
2. | School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK |
We propose a discretization of vector fields that are Hamiltonian up to multiplication by a positive function on the phase space that may be interpreted as a time reparametrization. We construct a family of maps, labeled by an arbitrary $ \ell \in \mathbb{N} $ indicating the desired order of accuracy, and prove that our method is structure preserving in the sense that the discrete flow is interpolated to order $ \ell $ by the flow of a continuous system possessing the same structure as the vector field that is being discretized. In particular, our discretization preserves a smooth measure on the phase space to the arbitrary order $ \ell $. We present applications to a remarkable class of nonholonomic mechanical systems that allow Hamiltonization. To our best knowledge, these results provide the first instance of a measure preserving discretization (to arbitrary order) of measure preserving nonholonomic systems.
References:
[1] |
A. Anahory Simoes, J. C. Marrero and D. Martín de Diego, Exact discrete Lagrangian mechanics for nonholonomic mechanics, preprint, 2020, arXiv: 2003.11362. |
[2] |
L. Bates and J. Śniatycki,
Nonholonomic reduction, Reports on Mathematical Physics, 32 (1993), 99-115.
doi: 10.1016/0034-4877(93)90073-N. |
[3] |
G. Benettin and A. Giorgilli,
On the Hamiltonian interpolation of near-to-the identity symplectic mappings with application to symplectic integration algorithms, Journal of Statistical Physics, 74 (1994), 1117-1143.
doi: 10.1007/BF02188219. |
[4] |
A. M. Bloch, P. Krishnaprasad, J. E. Marsden and R. M. Murray,
Nonholonomic mechanical systems with symmetry, Archive for Rational Mechanics and Analysis, 136 (1996), 21-99.
doi: 10.1007/BF02199365. |
[5] |
A. V. Borisov and I. S. Mamaev,
Isomorphism and Hamilton representation of some nonholonomic systems, Siberian Mathematical Journal, 48 (2007), 26-36.
doi: 10.1007/s11202-007-0004-6. |
[6] |
A. V. Borisov and I. S. Mamaev,
Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems, Regular and Chaotic Dynamics, 13 (2008), 443-490.
doi: 10.1134/S1560354708050079. |
[7] |
A. V. Borisov, I. S. Mamaev and I. A. Bizyaev,
The hierarchy of dynamics of a rigid body rolling without slipping and spinning on a plane and a sphere, Regular and Chaotic Dynamics, 18 (2013), 277-328.
doi: 10.1134/S1560354713030064. |
[8] |
R. C. Calleja, A. Celletti and R. de la Llave,
A KAM theory for conformally symplectic systems: Efficient algorithms and their validation, Journal of Differential Equations, 255 (2013), 978-1049.
doi: 10.1016/j.jde.2013.05.001. |
[9] |
F. Cantrijn, J. Cortés, M. De León and D. Martín de Diego,
On the geometry of generalized Chaplygin systems, Mathematical Proceedings of the Cambridge Philosophical Society, 132 (2002), 323-351.
doi: 10.1017/S0305004101005679. |
[10] |
S. A. Chaplygin,
On the theory of motion of nonholonomic systems. The reducing-multiplier theorem, Regular and Chaotic Dynamics, 13 (2008), 369-376.
doi: 10.1134/S1560354708040102. |
[11] |
J. Cortés and S. Martínez,
Non-holonomic integrators, Nonlinearity, 14 (2001), 1365-1392.
doi: 10.1088/0951-7715/14/5/322. |
[12] |
K. Ehlers, J. Koiller, R. Montgomery and P. M. Rios, Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization, in The Breadth of Symplectic and Poisson Geometry Birkhäuser Boston, Boston, MA, (2005), 75–120.
doi: 10.1007/0-8176-4419-9_4. |
[13] |
Y. N. Fedorov and B. Jovanovic,
Nonholonomic LR systems as generalized Chaplygin systems with an invariant measure and flows on homogeneous spaces, Journal of Nonlinear Science, 14 (2004), 341-381.
doi: 10.1007/s00332-004-0603-3. |
[14] |
Y. N. Fedorov and B. Jovanović,
Hamiltonization of the generalized Veselova LR system, Regular and Chaotic Dynamics, 14 (2009), 495-505.
doi: 10.1134/S1560354709040066. |
[15] |
Y. N. Fedorov, L. C. García-Naranjo and J. C. Marrero,
Unimodularity and preservation of volumes in nonholonomic mechanics, Journal of Nonlinear Science, 25 (2015), 203-246.
doi: 10.1007/s00332-014-9227-4. |
[16] |
O. E. Fernandez, T. Mestdag and A. M. Bloch,
A generalization of Chaplygin's reducibility theorem, Regular and Chaotic Dynamics, 14 (2009), 635-655.
doi: 10.1134/S1560354709060033. |
[17] |
O. E. Fernandez, A. M. Bloch and P. J. Olver,
Variational integrators for Hamiltonizable nonholonomic systems, Journal of Geometric Mechanics, 4 (2012), 137-163.
doi: 10.3934/jgm.2012.4.137. |
[18] |
O. E. Fernandez,
Poincaré transformations in nonholonomic mechanics, Applied Mathematics Letters, 43 (2015), 96-100.
doi: 10.1016/j.aml.2014.12.004. |
[19] |
S. Ferraro, D. Iglesias and D. Martín de Diego,
Momentum and energy preserving integrators for nonholonomic dynamics, Nonlinearity, 21 (2008), 1911-1928.
doi: 10.1088/0951-7715/21/8/009. |
[20] |
S. Ferraro, F. Jiménez and D. Martín de Diego,
New developments on the geometric nonholonomic integrator, Nonlinearity, 28 (2015), 871-900.
doi: 10.1088/0951-7715/28/4/871. |
[21] |
L. C. García-Naranjo, Generalisation of Chaplygin's reducing multiplier theorem with an application to multi-dimensional nonholonomic dynamics, Journal of Physics A: Mathematical and Theoretical, 52 (2019), 205203, 16 pp.
doi: 10.1088/1751-8121/ab15f8. |
[22] |
L. C. García-Naranjo and J. C. Marrero,
The geometry of nonholonomic Chaplygin systems revisited, Nonlinearity, 33 (2020), 1297-1341.
doi: 10.1088/1361-6544/ab5c0a. |
[23] |
E. Hairer,
Variable time step integration with symplectic methods, Applied Numerical Mathematics, 25 (1997), 219-227.
doi: 10.1016/S0168-9274(97)00061-5. |
[24] |
E. Hairer and C. Lubich,
The life-span of backward error analysis for numerical integrators, Numerische Mathematik, 76 (1997), 441-462.
doi: 10.1007/s002110050271. |
[25] |
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edition, Springer-Verlag, Berlin, 2006.
doi: 10.1007/3-540-30666-8. |
[26] |
D. Iglesias, J. C. Marrero, D. Martín de Diego and E. Martínez,
Discrete nonholonomic lagrangian systems on lie groupoids, Journal of Nonlinear Science, 18 (2008), 221-276.
doi: 10.1007/s00332-007-9012-8. |
[27] |
B. Jovanović,
Note on a ball rolling over a sphere: Integrable Chaplygin system with an invariant measure without Chaplygin Hamiltonization, Theoretical and Applied Mechanics, 46 (2019), 97-108.
doi: 10.2298/TAM190322003J. |
[28] |
M. Kobilarov, J. E. Marsden and G. S. Sukhatme,
Geometric discretization of nonholonomic systems with symmetries, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 61-84.
doi: 10.3934/dcdss.2010.3.61. |
[29] |
J. Koiller,
Reduction of some classical non-holonomic systems with symmetry, Archive for Rational Mechanics and Analysis, 118 (1992), 113-148.
doi: 10.1007/BF00375092. |
[30] |
B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics Cambridge Monographs on Applied and Computational Mathematics, vol. 14, Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511614118. |
[31] |
T. Levi-Civita,
Sur la résolution qualitative du probleme restreint des trois corps, Acta Mathematica, 30 (1906), 305-327.
doi: 10.1007/BF02418577. |
[32] |
C.-M. Marle,
A property of conformally Hamiltonian vector fields; Application to the Kepler problem, Journal of Geometric Mechanics, 4 (2012), 181-206.
doi: 10.3934/jgm.2012.4.181. |
[33] |
J. E. Marsden and M. West,
Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.
doi: 10.1017/S096249290100006X. |
[34] |
R. McLachlan and M. Perlmutter,
Conformal Hamiltonian systems, Journal of Geometry and Physics, 39 (2001), 276-300.
doi: 10.1016/S0393-0440(01)00020-1. |
[35] |
R. McLachlan and M. Perlmutter,
Integrators for nonholonomic mechanical systems, J. Nonlinear Sci., 16 (2006), 283-328.
doi: 10.1007/s00332-005-0698-1. |
[36] |
K. Modin and O. Verdier,
What makes nonholonomic integrators work?, Numerische Mathematik, 145 (2020), 405-435.
doi: 10.1007/s00211-020-01126-y. |
[37] |
S. Reich,
Backward error analysis for numerical integrators, SIAM Journal on Numerical Analysis, 36 (1999), 1549-1570.
doi: 10.1137/S0036142997329797. |
[38] |
S. Stanchenko,
Non-holonomic Chaplygin systems, Journal of Applied Mathematics and Mechanics, 53 (1989), 11-17.
doi: 10.1016/0021-8928(89)90126-3. |
[39] |
M. Vermeeren,
Modified equations for variational integrators, Numerische Mathematik, 137 (2017), 1001-1037.
doi: 10.1007/s00211-017-0896-4. |
[40] |
M. Vermeeren, Support code for "Structure preserving discretization of time-reparametrized Hamiltonian systems with application to nonholonomic mechanics", https://github.com/mvermeeren/conf-ham-sys-2020.
doi: 10.5281/zenodo.3988087. |
[41] |
A. P. Veselov and L. Veselova,
Integrable nonholonomic systems on Lie groups, Mathematical Notes of the Academy of Sciences of the USSR, 44 (1988), 810-819.
doi: 10.1007/BF01158420. |
[42] |
P. Woronetz,
Über die Bewegung eines starren Körpers, der ohne Gleitung auf einer beliebigen Fläche rollt, Mathematische Annalen, 70 (1911), 410-453.
doi: 10.1007/BF01564505. |
show all references
References:
[1] |
A. Anahory Simoes, J. C. Marrero and D. Martín de Diego, Exact discrete Lagrangian mechanics for nonholonomic mechanics, preprint, 2020, arXiv: 2003.11362. |
[2] |
L. Bates and J. Śniatycki,
Nonholonomic reduction, Reports on Mathematical Physics, 32 (1993), 99-115.
doi: 10.1016/0034-4877(93)90073-N. |
[3] |
G. Benettin and A. Giorgilli,
On the Hamiltonian interpolation of near-to-the identity symplectic mappings with application to symplectic integration algorithms, Journal of Statistical Physics, 74 (1994), 1117-1143.
doi: 10.1007/BF02188219. |
[4] |
A. M. Bloch, P. Krishnaprasad, J. E. Marsden and R. M. Murray,
Nonholonomic mechanical systems with symmetry, Archive for Rational Mechanics and Analysis, 136 (1996), 21-99.
doi: 10.1007/BF02199365. |
[5] |
A. V. Borisov and I. S. Mamaev,
Isomorphism and Hamilton representation of some nonholonomic systems, Siberian Mathematical Journal, 48 (2007), 26-36.
doi: 10.1007/s11202-007-0004-6. |
[6] |
A. V. Borisov and I. S. Mamaev,
Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems, Regular and Chaotic Dynamics, 13 (2008), 443-490.
doi: 10.1134/S1560354708050079. |
[7] |
A. V. Borisov, I. S. Mamaev and I. A. Bizyaev,
The hierarchy of dynamics of a rigid body rolling without slipping and spinning on a plane and a sphere, Regular and Chaotic Dynamics, 18 (2013), 277-328.
doi: 10.1134/S1560354713030064. |
[8] |
R. C. Calleja, A. Celletti and R. de la Llave,
A KAM theory for conformally symplectic systems: Efficient algorithms and their validation, Journal of Differential Equations, 255 (2013), 978-1049.
doi: 10.1016/j.jde.2013.05.001. |
[9] |
F. Cantrijn, J. Cortés, M. De León and D. Martín de Diego,
On the geometry of generalized Chaplygin systems, Mathematical Proceedings of the Cambridge Philosophical Society, 132 (2002), 323-351.
doi: 10.1017/S0305004101005679. |
[10] |
S. A. Chaplygin,
On the theory of motion of nonholonomic systems. The reducing-multiplier theorem, Regular and Chaotic Dynamics, 13 (2008), 369-376.
doi: 10.1134/S1560354708040102. |
[11] |
J. Cortés and S. Martínez,
Non-holonomic integrators, Nonlinearity, 14 (2001), 1365-1392.
doi: 10.1088/0951-7715/14/5/322. |
[12] |
K. Ehlers, J. Koiller, R. Montgomery and P. M. Rios, Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization, in The Breadth of Symplectic and Poisson Geometry Birkhäuser Boston, Boston, MA, (2005), 75–120.
doi: 10.1007/0-8176-4419-9_4. |
[13] |
Y. N. Fedorov and B. Jovanovic,
Nonholonomic LR systems as generalized Chaplygin systems with an invariant measure and flows on homogeneous spaces, Journal of Nonlinear Science, 14 (2004), 341-381.
doi: 10.1007/s00332-004-0603-3. |
[14] |
Y. N. Fedorov and B. Jovanović,
Hamiltonization of the generalized Veselova LR system, Regular and Chaotic Dynamics, 14 (2009), 495-505.
doi: 10.1134/S1560354709040066. |
[15] |
Y. N. Fedorov, L. C. García-Naranjo and J. C. Marrero,
Unimodularity and preservation of volumes in nonholonomic mechanics, Journal of Nonlinear Science, 25 (2015), 203-246.
doi: 10.1007/s00332-014-9227-4. |
[16] |
O. E. Fernandez, T. Mestdag and A. M. Bloch,
A generalization of Chaplygin's reducibility theorem, Regular and Chaotic Dynamics, 14 (2009), 635-655.
doi: 10.1134/S1560354709060033. |
[17] |
O. E. Fernandez, A. M. Bloch and P. J. Olver,
Variational integrators for Hamiltonizable nonholonomic systems, Journal of Geometric Mechanics, 4 (2012), 137-163.
doi: 10.3934/jgm.2012.4.137. |
[18] |
O. E. Fernandez,
Poincaré transformations in nonholonomic mechanics, Applied Mathematics Letters, 43 (2015), 96-100.
doi: 10.1016/j.aml.2014.12.004. |
[19] |
S. Ferraro, D. Iglesias and D. Martín de Diego,
Momentum and energy preserving integrators for nonholonomic dynamics, Nonlinearity, 21 (2008), 1911-1928.
doi: 10.1088/0951-7715/21/8/009. |
[20] |
S. Ferraro, F. Jiménez and D. Martín de Diego,
New developments on the geometric nonholonomic integrator, Nonlinearity, 28 (2015), 871-900.
doi: 10.1088/0951-7715/28/4/871. |
[21] |
L. C. García-Naranjo, Generalisation of Chaplygin's reducing multiplier theorem with an application to multi-dimensional nonholonomic dynamics, Journal of Physics A: Mathematical and Theoretical, 52 (2019), 205203, 16 pp.
doi: 10.1088/1751-8121/ab15f8. |
[22] |
L. C. García-Naranjo and J. C. Marrero,
The geometry of nonholonomic Chaplygin systems revisited, Nonlinearity, 33 (2020), 1297-1341.
doi: 10.1088/1361-6544/ab5c0a. |
[23] |
E. Hairer,
Variable time step integration with symplectic methods, Applied Numerical Mathematics, 25 (1997), 219-227.
doi: 10.1016/S0168-9274(97)00061-5. |
[24] |
E. Hairer and C. Lubich,
The life-span of backward error analysis for numerical integrators, Numerische Mathematik, 76 (1997), 441-462.
doi: 10.1007/s002110050271. |
[25] |
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edition, Springer-Verlag, Berlin, 2006.
doi: 10.1007/3-540-30666-8. |
[26] |
D. Iglesias, J. C. Marrero, D. Martín de Diego and E. Martínez,
Discrete nonholonomic lagrangian systems on lie groupoids, Journal of Nonlinear Science, 18 (2008), 221-276.
doi: 10.1007/s00332-007-9012-8. |
[27] |
B. Jovanović,
Note on a ball rolling over a sphere: Integrable Chaplygin system with an invariant measure without Chaplygin Hamiltonization, Theoretical and Applied Mechanics, 46 (2019), 97-108.
doi: 10.2298/TAM190322003J. |
[28] |
M. Kobilarov, J. E. Marsden and G. S. Sukhatme,
Geometric discretization of nonholonomic systems with symmetries, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 61-84.
doi: 10.3934/dcdss.2010.3.61. |
[29] |
J. Koiller,
Reduction of some classical non-holonomic systems with symmetry, Archive for Rational Mechanics and Analysis, 118 (1992), 113-148.
doi: 10.1007/BF00375092. |
[30] |
B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics Cambridge Monographs on Applied and Computational Mathematics, vol. 14, Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511614118. |
[31] |
T. Levi-Civita,
Sur la résolution qualitative du probleme restreint des trois corps, Acta Mathematica, 30 (1906), 305-327.
doi: 10.1007/BF02418577. |
[32] |
C.-M. Marle,
A property of conformally Hamiltonian vector fields; Application to the Kepler problem, Journal of Geometric Mechanics, 4 (2012), 181-206.
doi: 10.3934/jgm.2012.4.181. |
[33] |
J. E. Marsden and M. West,
Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.
doi: 10.1017/S096249290100006X. |
[34] |
R. McLachlan and M. Perlmutter,
Conformal Hamiltonian systems, Journal of Geometry and Physics, 39 (2001), 276-300.
doi: 10.1016/S0393-0440(01)00020-1. |
[35] |
R. McLachlan and M. Perlmutter,
Integrators for nonholonomic mechanical systems, J. Nonlinear Sci., 16 (2006), 283-328.
doi: 10.1007/s00332-005-0698-1. |
[36] |
K. Modin and O. Verdier,
What makes nonholonomic integrators work?, Numerische Mathematik, 145 (2020), 405-435.
doi: 10.1007/s00211-020-01126-y. |
[37] |
S. Reich,
Backward error analysis for numerical integrators, SIAM Journal on Numerical Analysis, 36 (1999), 1549-1570.
doi: 10.1137/S0036142997329797. |
[38] |
S. Stanchenko,
Non-holonomic Chaplygin systems, Journal of Applied Mathematics and Mechanics, 53 (1989), 11-17.
doi: 10.1016/0021-8928(89)90126-3. |
[39] |
M. Vermeeren,
Modified equations for variational integrators, Numerische Mathematik, 137 (2017), 1001-1037.
doi: 10.1007/s00211-017-0896-4. |
[40] |
M. Vermeeren, Support code for "Structure preserving discretization of time-reparametrized Hamiltonian systems with application to nonholonomic mechanics", https://github.com/mvermeeren/conf-ham-sys-2020.
doi: 10.5281/zenodo.3988087. |
[41] |
A. P. Veselov and L. Veselova,
Integrable nonholonomic systems on Lie groups, Mathematical Notes of the Academy of Sciences of the USSR, 44 (1988), 810-819.
doi: 10.1007/BF01158420. |
[42] |
P. Woronetz,
Über die Bewegung eines starren Körpers, der ohne Gleitung auf einer beliebigen Fläche rollt, Mathematische Annalen, 70 (1911), 410-453.
doi: 10.1007/BF01564505. |






Eqn. | Notation | Name | Equations of motion |
Original system | |||
(3) | Conformal Hamiltonian | ||
(5) | Altered Hamiltonian | on |
|
Modified system, interpolating numerical solutions | |||
(16) | Modified altered Hamiltonian | on |
|
Truncated modified altered Hamiltonian | |||
(18) | Modified conformal Hamiltonian | ||
Truncated modified conformal Hamiltonian |
Eqn. | Notation | Name | Equations of motion |
Original system | |||
(3) | Conformal Hamiltonian | ||
(5) | Altered Hamiltonian | on |
|
Modified system, interpolating numerical solutions | |||
(16) | Modified altered Hamiltonian | on |
|
Truncated modified altered Hamiltonian | |||
(18) | Modified conformal Hamiltonian | ||
Truncated modified conformal Hamiltonian |
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