# American Institute of Mathematical Sciences

December  2015, 7(4): 483-515. doi: 10.3934/jgm.2015.7.483

## 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

Received  July 2014 Revised  July 2015 Published  October 2015

We give a characterization of linear canonoid transformations on symplectic manifolds and we use it to generate biHamiltonian structures for some mechanical systems. Using this characterization we also study the behavior of the harmonic oscillator under canonoid transformations. We present a description of canonoid transformations due to E.T. Whittaker, and we show that it leads, in a natural way, to the modern, coordinate-independent definition of canonoid transformations. We also generalize canonoid transformations to Poisson manifolds by introducing Poissonoid transformations. We give examples of such transformations for Euler's equations of the rigid body (on $\mathfrak{ so}^\ast (3)$ and $\mathfrak{ so}^\ast (4)$) and for an integrable case of Kirchhoff's equations for the motion of a rigid body immersed in an ideal fluid. We study the relationship between biHamiltonian structures and Poissonoid transformations for these examples. We analyze the link between Poissonoid transformations, constants of motion, and symmetries.
Citation: Giovanni Rastelli, Manuele Santoprete. Canonoid and Poissonoid transformations, symmetries and biHamiltonian structures. Journal of Geometric Mechanics, 2015, 7 (4) : 483-515. doi: 10.3934/jgm.2015.7.483
##### 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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##### 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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