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A unifying mechanical equation with applications to non-holonomic constraints and dissipative phenomena
Canonoid and Poissonoid transformations, symmetries and biHamiltonian structures
1. | Dipartimento di Matematica, Università di Torino, Torino, via Carlo Alberto 10, Italy |
2. | Department of Mathematics, Wilfrid Laurier University, 75 University Avenue West, Waterloo, ON, Canada |
References:
[1] |
R. Abraham and J. E. Marsden, Foundations of Mechanics,, American Mathematical Soc., (1978). Google Scholar |
[2] |
A. V. Bolsinov, Compatible Poisson brackets on Lie algebras and completeness of families of functions in involution,, Mathematics of the USSR-Izvestiya, 38 (1992), 69.
|
[3] |
J. F. Cariñena, F. Falceto and M. F. Rañada, Canonoid transformations and master symmetries,, Journal of Geometric Mechanics, 5 (2013), 151.
doi: 10.3934/jgm.2013.5.151. |
[4] |
J. F. Cariñena and M. F. Rañada, Canonoid transformations from a geometric perspective,, Journal of Mathematical Physics, 29 (1988), 2181.
doi: 10.1063/1.528146. |
[5] |
J. F. Cariñena and M. F. Rañada, Generating functions, bi-Hamiltonian systems, and the quadratic-Hamiltonian theorem,, Journal of Mathematical Physics, 31 (1990), 801.
doi: 10.1063/1.529028. |
[6] |
Y. Choquet-Bruhat, Analysis, Manifolds and Physics, Part II - Revised and Enlarged Edition,, Elsevier, (2000). Google Scholar |
[7] |
D. G. Currie and E. J. Saletan, Canonical transformations and quadratic hamiltonians,, II Nuovo Cimento B Series 11, 9 (1972), 143.
doi: 10.1007/BF02735514. |
[8] |
C. Daskaloyannis and K. Ypsilantis, Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two-dimensional manifold,, Journal of Mathematical Physics, 47 (2006).
doi: 10.1063/1.2192967. |
[9] |
G. Falqui, A note on the rotationally symmetric SO(4) Euler rigid body,, Symmetry, 3 (2007).
doi: 10.3842/SIGMA.2007.032. |
[10] |
A. Fasano and S. Marmi, Analytical Mechanics : An Introduction,, Oxford University Press, (2006). Google Scholar |
[11] |
A. T. Fomenko, Integrability and Nonintegrability in Geometry and Mechanics,, Kluwer Academic Publishers, (1988).
doi: 10.1007/978-94-009-3069-8. |
[12] |
J. M. Jauch and E. L. Hill, On the problem of degeneracy} in quantum mechanics,, Physical Review, 57 (1940), 641.
doi: 10.1103/PhysRev.57.641. |
[13] |
E. G. Kalnins, J. M. Kress, G. S. Pogosyan and W. Miller, Jr, Completeness of superintegrability in two-dimensional constant-curvature spaces,, Journal of Physics A: Mathematical and General, 34 (2001), 4705.
doi: 10.1088/0305-4470/34/22/311. |
[14] |
G. Landolfi and G. Soliani, On certain canonoid transformations and invariants for the parametric oscillator,, Journal of Physics A: Mathematical and Theoretical, 40 (2007), 3413.
doi: 10.1088/1751-8113/40/13/009. |
[15] |
C. Laurent-Gengoux, A. Pichereau and P. Vanhaecke, Poisson Structures,, Grundlehren der mathematischen Wissenschaften, (2013).
doi: 10.1007/978-3-642-31090-4. |
[16] |
P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics,, Springer, (1987).
doi: 10.1007/978-94-009-3807-6. |
[17] |
F. Magri, P. Casati, G. Falqui and M. Pedroni, Eight lectures on integrable systems,, in Integrability of Nonlinear Systems (eds. Y. Kosmann-Schwarzbach, (2004), 209. Google Scholar |
[18] |
F. Magri, A simple model of the integrable Hamiltonian equation,, Journal of Mathematical Physics, 19 (1978), 1156.
doi: 10.1063/1.523777. |
[19] |
G. Marmo, Equivalent Lagrangians and quasicanonical transformations,, in Group Theoretical Methods in Physics (eds. A. Janner, 50 (1976), 568.
|
[20] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems,, Springer, (1999). Google Scholar |
[21] |
C. C. Moore and C. Schochet, Tangential cohomology,, in Global Analysis on Foliated Spaces, (1988), 68. Google Scholar |
[22] |
C. Morosi and L. Pizzocchero, On the Euler equation: Bi-Hamiltonian structure and integrals in involution,, Letters in Mathematical Physics, 37 (1996), 117.
doi: 10.1007/BF00416015. |
[23] |
L. J. Negri, L. C. Oliveira and J. M. Teixeira, Canonoid transformations and constants of motion,, Journal of Mathematical Physics, 28 (1987), 2369.
doi: 10.1063/1.527772. |
[24] |
H. Poincaré, Sur le probléme des trois corps et les équations de la dynamique,, Acta Mathematica, 13 (1890), 1. Google Scholar |
[25] |
G. Rudolph and M. Schmidt, Differential Geometry and Mathematical Physics: Part I. Manifolds, Lie Groups and Hamiltonian Systems,, Springer, (2013).
doi: 10.1007/978-94-007-5345-7. |
[26] |
E. J. Saletan and A. H. Cromer, Theoretical Mechanics,, Wiley, (1971). Google Scholar |
[27] |
P. Tempesta, E. Alfinito, R. A. Leo and G. Soliani, Quantum models related to fouled Hamiltonians of the harmonic oscillator,, Journal of Mathematical Physics, 43 (2002), 3538.
doi: 10.1063/1.1479300. |
[28] |
I. Vaisman, Lectures on the Geometry of Poisson Manifolds,, Springer, (1994).
doi: 10.1007/978-3-0348-8495-2. |
[29] |
F. J. D. Vegas, On the canonical transformation theorem of Currie and Saletan,, Journal of Physics A: Mathematical and General, 22 (1989), 1927.
doi: 10.1088/0305-4470/22/11/029. |
[30] |
E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies,, Cambridge University Press, (1988).
doi: 10.1017/CBO9780511608797. |
show all references
References:
[1] |
R. Abraham and J. E. Marsden, Foundations of Mechanics,, American Mathematical Soc., (1978). Google Scholar |
[2] |
A. V. Bolsinov, Compatible Poisson brackets on Lie algebras and completeness of families of functions in involution,, Mathematics of the USSR-Izvestiya, 38 (1992), 69.
|
[3] |
J. F. Cariñena, F. Falceto and M. F. Rañada, Canonoid transformations and master symmetries,, Journal of Geometric Mechanics, 5 (2013), 151.
doi: 10.3934/jgm.2013.5.151. |
[4] |
J. F. Cariñena and M. F. Rañada, Canonoid transformations from a geometric perspective,, Journal of Mathematical Physics, 29 (1988), 2181.
doi: 10.1063/1.528146. |
[5] |
J. F. Cariñena and M. F. Rañada, Generating functions, bi-Hamiltonian systems, and the quadratic-Hamiltonian theorem,, Journal of Mathematical Physics, 31 (1990), 801.
doi: 10.1063/1.529028. |
[6] |
Y. Choquet-Bruhat, Analysis, Manifolds and Physics, Part II - Revised and Enlarged Edition,, Elsevier, (2000). Google Scholar |
[7] |
D. G. Currie and E. J. Saletan, Canonical transformations and quadratic hamiltonians,, II Nuovo Cimento B Series 11, 9 (1972), 143.
doi: 10.1007/BF02735514. |
[8] |
C. Daskaloyannis and K. Ypsilantis, Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two-dimensional manifold,, Journal of Mathematical Physics, 47 (2006).
doi: 10.1063/1.2192967. |
[9] |
G. Falqui, A note on the rotationally symmetric SO(4) Euler rigid body,, Symmetry, 3 (2007).
doi: 10.3842/SIGMA.2007.032. |
[10] |
A. Fasano and S. Marmi, Analytical Mechanics : An Introduction,, Oxford University Press, (2006). Google Scholar |
[11] |
A. T. Fomenko, Integrability and Nonintegrability in Geometry and Mechanics,, Kluwer Academic Publishers, (1988).
doi: 10.1007/978-94-009-3069-8. |
[12] |
J. M. Jauch and E. L. Hill, On the problem of degeneracy} in quantum mechanics,, Physical Review, 57 (1940), 641.
doi: 10.1103/PhysRev.57.641. |
[13] |
E. G. Kalnins, J. M. Kress, G. S. Pogosyan and W. Miller, Jr, Completeness of superintegrability in two-dimensional constant-curvature spaces,, Journal of Physics A: Mathematical and General, 34 (2001), 4705.
doi: 10.1088/0305-4470/34/22/311. |
[14] |
G. Landolfi and G. Soliani, On certain canonoid transformations and invariants for the parametric oscillator,, Journal of Physics A: Mathematical and Theoretical, 40 (2007), 3413.
doi: 10.1088/1751-8113/40/13/009. |
[15] |
C. Laurent-Gengoux, A. Pichereau and P. Vanhaecke, Poisson Structures,, Grundlehren der mathematischen Wissenschaften, (2013).
doi: 10.1007/978-3-642-31090-4. |
[16] |
P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics,, Springer, (1987).
doi: 10.1007/978-94-009-3807-6. |
[17] |
F. Magri, P. Casati, G. Falqui and M. Pedroni, Eight lectures on integrable systems,, in Integrability of Nonlinear Systems (eds. Y. Kosmann-Schwarzbach, (2004), 209. Google Scholar |
[18] |
F. Magri, A simple model of the integrable Hamiltonian equation,, Journal of Mathematical Physics, 19 (1978), 1156.
doi: 10.1063/1.523777. |
[19] |
G. Marmo, Equivalent Lagrangians and quasicanonical transformations,, in Group Theoretical Methods in Physics (eds. A. Janner, 50 (1976), 568.
|
[20] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems,, Springer, (1999). Google Scholar |
[21] |
C. C. Moore and C. Schochet, Tangential cohomology,, in Global Analysis on Foliated Spaces, (1988), 68. Google Scholar |
[22] |
C. Morosi and L. Pizzocchero, On the Euler equation: Bi-Hamiltonian structure and integrals in involution,, Letters in Mathematical Physics, 37 (1996), 117.
doi: 10.1007/BF00416015. |
[23] |
L. J. Negri, L. C. Oliveira and J. M. Teixeira, Canonoid transformations and constants of motion,, Journal of Mathematical Physics, 28 (1987), 2369.
doi: 10.1063/1.527772. |
[24] |
H. Poincaré, Sur le probléme des trois corps et les équations de la dynamique,, Acta Mathematica, 13 (1890), 1. Google Scholar |
[25] |
G. Rudolph and M. Schmidt, Differential Geometry and Mathematical Physics: Part I. Manifolds, Lie Groups and Hamiltonian Systems,, Springer, (2013).
doi: 10.1007/978-94-007-5345-7. |
[26] |
E. J. Saletan and A. H. Cromer, Theoretical Mechanics,, Wiley, (1971). Google Scholar |
[27] |
P. Tempesta, E. Alfinito, R. A. Leo and G. Soliani, Quantum models related to fouled Hamiltonians of the harmonic oscillator,, Journal of Mathematical Physics, 43 (2002), 3538.
doi: 10.1063/1.1479300. |
[28] |
I. Vaisman, Lectures on the Geometry of Poisson Manifolds,, Springer, (1994).
doi: 10.1007/978-3-0348-8495-2. |
[29] |
F. J. D. Vegas, On the canonical transformation theorem of Currie and Saletan,, Journal of Physics A: Mathematical and General, 22 (1989), 1927.
doi: 10.1088/0305-4470/22/11/029. |
[30] |
E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies,, Cambridge University Press, (1988).
doi: 10.1017/CBO9780511608797. |
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