SIGMA 17 (2021), 096, 17 pages      arXiv:2105.11123      https://doi.org/10.3842/SIGMA.2021.096

Generically, Arnold-Liouville Systems Cannot be Bi-Hamiltonian

Hassan Boualem ^a and Robert Brouzet ^b
a) IMAG, Université de Montpellier, France
^b) LAMPS, EA 4217, Université Perpignan Via Domitia, France

Received May 24, 2021, in final form October 22, 2021; Published online October 29, 2021

Abstract
We state and prove that a certain class of smooth functions said to be BH-separable is a meagre subset for the Fréchet topology. Because these functions are the only admissible Hamiltonians for Arnold-Liouville systems admitting a bi-Hamiltonian structure, we get that, generically, Arnold-Liouville systems cannot be bi-Hamiltonian. At the end of the paper, we determine, both as a concrete representation of our general result and as an illustrative list, which polynomial Hamiltonians $H$ of the form $H(x,y)=xy+ax^3+bx^2y+cxy^2+dy^3$ are BH-separable.

Key words: completely integrable Hamiltonian system; Arnold-Liouville theorem; action-angle coordinates; bi-Hamiltonian system; separability of functions; change of coordinates; Fréchet topology; meagre set.

pdf (484 kb)   tex (23 kb)  

References

1. Abraham R., Marsden J.E., Foundations of mechanics, 2nd ed., Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978.
2. Arnold V., Les méthodes mathématiques de la mécanique classique, Mir, Moscow, 1976.
3. Boualem H., Brouzet R., Topology of the space of locally separable functions, Topology Appl. 299 (2021), 107728, 16 pages.
4. Brouzet R., About the existence of recursion operators for completely integrable Hamiltonian systems near a Liouville torus, J. Math. Phys. 34 (1993), 1309-1313.
5. Brouzet R., Molino P., Turiel F.J., Géométrie des systèmes bihamiltoniens, Indag. Math. (N.S.) 4 (1993), 269-296.
6. Craik A.D.D., Prehistory of Faà di Bruno's formula, Amer. Math. Monthly 112 (2005), 119-130.
7. Ehresmann C., Les connexions infinitésimales dans un espace fibré différentiable, in Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson & Cie, Paris, 1951, 29-55.
8. Falqui G., Pedroni M., Separation of variables for bi-Hamiltonian systems, Math. Phys. Anal. Geom. 6 (2003), 139-179, arXiv:nlin.SI/0204029.
9. Fernandes R.L., Completely integrable bi-Hamiltonian systems, J. Dynam. Differential Equations 6 (1994), 53-69.
10. Gel'fand I.M., Dorfman I.Ja., Hamiltonian operators and algebraic structures related to them, Funct. Anal. Appl. 13 (1979), 248-262.
11. Hardy M., Combinatorics of partial derivatives, Electron. J. Combin. 13 (2006), 1, 13 pages, arXiv:math.CO/0601149.
12. Hirsch M.W., Differential topology, Graduate Texts in Mathematics, Vol. 33, Springer-Verlag, New York - Heidelberg, 1976.
13. Ibort A., Magri F., Marmo G., Bihamiltonian structures and Stäckel separability, J. Geom. Phys. 33 (2000), 210-228.
14. Johnson W.P., The curious history of Faà di Bruno's formula, Amer. Math. Monthly 109 (2002), 217-234.
15. Landi G., Marmo G., Vilasi G., Recursion operators: meaning and existence for completely integrable systems, J. Math. Phys. 35 (1994), 808-815.
16. Libermann P., Marle C.-M., Géométrie symplectique, bases théoriques de la mécanique. Tome I, Publications Mathématiques de l'Université Paris VII, Vol. 21, Université de Paris VII, U.E.R. de Mathématiques, Paris, 1986.
17. Ma T.-W., Higher chain formula proved by combinatorics, Electron. J. Combin. 16 (2009), 21, 7 pages.
18. Magri F., A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), 1156-1162.
19. Magri F., Morosi C., A geometrical characterization of integrable Hamiltonian systems through the theory of Poisson-Nijenhuis manifolds, Department of Mathematics, University of Milan, 1984, available at https://boa.unimib.it/retrieve/handle/10281/17656/21540/quaderno3-2008.pdf.
20. Nijenhuis A., Jacobi-type identities for bilinear differential concomitants of certain tensor fields, Indag. Math. 58 (1955), 390-403.
21. Praught J., Smirnov R.G., Andrew Lenard: a mystery unraveled, SIGMA 1 (2005), 005, 7 pages, arXiv:nlin.SI/0510055.
22. Tempesta P., Tondo G., Haantjes algebras of classical integrable systems, Ann. Mat. Pura Appl., to appear, arXiv:1405.5118.