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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. |
[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), 69-90. |
[3] |
J. F. Cariñena, F. Falceto and M. F. Rañada, Canonoid transformations and master symmetries, Journal of Geometric Mechanics, 5 (2013), 151-166.
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), 2181-2186.
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), 801-807.
doi: 10.1063/1.529028. |
[6] |
Y. Choquet-Bruhat, Analysis, Manifolds and Physics, Part II - Revised and Enlarged Edition, Elsevier, 2000. |
[7] |
D. G. Currie and E. J. Saletan, Canonical transformations and quadratic hamiltonians, II Nuovo Cimento B Series 11, 9 (1972), 143-153.
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), 042904, 38pp.
doi: 10.1063/1.2192967. |
[9] |
G. Falqui, A note on the rotationally symmetric SO(4) Euler rigid body, Symmetry, Integrability and Geometry: Methods and Applications, 3 (2007), Paper 032, 13 pp.
doi: 10.3842/SIGMA.2007.032. |
[10] |
A. Fasano and S. Marmi, Analytical Mechanics : An Introduction, Oxford University Press, 2006. |
[11] |
A. T. Fomenko, Integrability and Nonintegrability in Geometry and Mechanics, Kluwer Academic Publishers, Dordrecht, 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), 641-645.
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), 4705-4720.
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), 3413-3423.
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, K. M. Tamizhmani and B. Grammaticos), no. 638 in Lecture Notes in Physics, Springer Berlin Heidelberg, 2004, 209-250. |
[18] |
F. Magri, A simple model of the integrable Hamiltonian equation, Journal of Mathematical Physics, 19 (1978), 1156-1162.
doi: 10.1063/1.523777. |
[19] |
G. Marmo, Equivalent Lagrangians and quasicanonical transformations, in Group Theoretical Methods in Physics (eds. A. Janner, T. Janssen and M. Boon), Lecture Notes in Physics, Springer Berlin Heidelberg, 50 (1976), 568-572. |
[20] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Springer, 1999. |
[21] |
C. C. Moore and C. Schochet, Tangential cohomology, in Global Analysis on Foliated Spaces, no. 9 in Mathematical Sciences Research Institute Publications, Springer New York, 1988, 68-91. |
[22] |
C. Morosi and L. Pizzocchero, On the Euler equation: Bi-Hamiltonian structure and integrals in involution, Letters in Mathematical Physics, 37 (1996), 117-135.
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), 2369-2372.
doi: 10.1063/1.527772. |
[24] |
H. Poincaré, Sur le probléme des trois corps et les équations de la dynamique, Acta Mathematica, 13 (1890), 1-270. |
[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. |
[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), 3538-3553.
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), 1927-1931.
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. |
[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), 69-90. |
[3] |
J. F. Cariñena, F. Falceto and M. F. Rañada, Canonoid transformations and master symmetries, Journal of Geometric Mechanics, 5 (2013), 151-166.
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), 2181-2186.
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), 801-807.
doi: 10.1063/1.529028. |
[6] |
Y. Choquet-Bruhat, Analysis, Manifolds and Physics, Part II - Revised and Enlarged Edition, Elsevier, 2000. |
[7] |
D. G. Currie and E. J. Saletan, Canonical transformations and quadratic hamiltonians, II Nuovo Cimento B Series 11, 9 (1972), 143-153.
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), 042904, 38pp.
doi: 10.1063/1.2192967. |
[9] |
G. Falqui, A note on the rotationally symmetric SO(4) Euler rigid body, Symmetry, Integrability and Geometry: Methods and Applications, 3 (2007), Paper 032, 13 pp.
doi: 10.3842/SIGMA.2007.032. |
[10] |
A. Fasano and S. Marmi, Analytical Mechanics : An Introduction, Oxford University Press, 2006. |
[11] |
A. T. Fomenko, Integrability and Nonintegrability in Geometry and Mechanics, Kluwer Academic Publishers, Dordrecht, 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), 641-645.
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), 4705-4720.
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), 3413-3423.
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, K. M. Tamizhmani and B. Grammaticos), no. 638 in Lecture Notes in Physics, Springer Berlin Heidelberg, 2004, 209-250. |
[18] |
F. Magri, A simple model of the integrable Hamiltonian equation, Journal of Mathematical Physics, 19 (1978), 1156-1162.
doi: 10.1063/1.523777. |
[19] |
G. Marmo, Equivalent Lagrangians and quasicanonical transformations, in Group Theoretical Methods in Physics (eds. A. Janner, T. Janssen and M. Boon), Lecture Notes in Physics, Springer Berlin Heidelberg, 50 (1976), 568-572. |
[20] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Springer, 1999. |
[21] |
C. C. Moore and C. Schochet, Tangential cohomology, in Global Analysis on Foliated Spaces, no. 9 in Mathematical Sciences Research Institute Publications, Springer New York, 1988, 68-91. |
[22] |
C. Morosi and L. Pizzocchero, On the Euler equation: Bi-Hamiltonian structure and integrals in involution, Letters in Mathematical Physics, 37 (1996), 117-135.
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), 2369-2372.
doi: 10.1063/1.527772. |
[24] |
H. Poincaré, Sur le probléme des trois corps et les équations de la dynamique, Acta Mathematica, 13 (1890), 1-270. |
[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. |
[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), 3538-3553.
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), 1927-1931.
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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