Geodesic
Completely integrable one-dimensional classical and relativistic time-dependent hamiltonians
Teoretičeskaâ i matematičeskaâ fizika, Tome 99 (1994) no. 3, pp. 355-363 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper, we look for first integrals $I(q;p;t)$ of time-dependent one-dimensional Hamiltonians $H(q;p;t)$. We first present a formalism based on the use of canonical transformations, and it is seen that $I(q;p;t)$ can always be written in terms of two variables $I=P(u;v)$, whereu andv are functions of $q$, $p$ andt, without loss of generality. Moreover, it is shown that any Hamiltonian with first integral $I(q;p;t)$ can be made autonomous in the space $(u,v,T)$, where $T$ is a new time. On the other hand, the cases of a particle moving classically and relativistically in a time-dependent potential $V(q;t)$ are studied. In both cases, completely integrable potentials, together with the corresponding first integrals, are derived. 
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S. Bouquet. Completely integrable one-dimensional classical and relativistic time-dependent hamiltonians. Teoretičeskaâ i matematičeskaâ fizika, Tome 99 (1994) no. 3, pp. 355-363. http://geodesic.mathdoc.fr/item/TMF_1994_99_3_a0/

[1] H. R. Lewis, P. G. L. Leach, S. Bouquet, M. R. Feix, J. Math. Phys., 33 (1992), 591 | DOI | MR | Zbl

[2] H. R. Lewis, P. G. L. Leach, S. Bouquet, M. R. Feix, “Representation of time-dependent one-dimensional Hamiltonians in terms of their first integrals”, Nonlinear Evolution Equations and Dynamical Systems, eds. V. G. Makhankov, I. Puzynin, O. Pashaev, World Scientific, Singapore, 1993, 42 | MR

[3] E. Noether, Transp. Theory Stat. phys., 1 (1971), 186 | DOI | MR | Zbl

[4] W. Sarlet, F. Cantrijn, SIAM review, 23 (1983), 467 | DOI | MR

[5] P. J. Olver, Applications of Lie Groups to Differential Equations, GTM 107, Springer-Verlag, Berlin, 1986 | MR

[6] H. R. Lewis, P. G. L. Leach, J. Math. Phys., 23 (1982), 2371 | DOI | MR | Zbl

[7] M. R. Feix, S. Bouquet, H. R. Lewis, Physica D, 28 (1987), 80 | DOI | MR | Zbl

[8] J. Goedert, H. R. Lewis, J. Math. Phys., 28 (1987), 728 | DOI | MR | Zbl

[9] L. G. Pereira, J. Goedert, J. Math. Phys., 33 (1992), 2682 | DOI | MR | Zbl

[10] P. G. L. Leach, H. R. Lewis, W. Sarlet, J. Math. Phys., 25 (1984), 486 | DOI | MR | Zbl

[11] L. D. Landau, E. M. Lifshitz, Mecanique, 3-rd ed., Mir, Moscou, 1969 | MR

[12] H. Giacomini, J. Phys. A: Math. Gen, 23 (1990), 587 | DOI | MR

[13] H. Giacomini, J. Phys. A: Math. Gen., 23 (1990), 865 | DOI | MR

[14] A. Dewisme, S. Bouquet, J. Math. Phys., 34 (1993), 997 | DOI | MR | Zbl

[15] S. Bouquet, A. Dewisme, Nonlinear Evolution Equations and Dynamical Systems, World Scientific, Singapore, 1992 | MR | Zbl

[16] J. Martin, S. Bouquet, J. Math. Phys., 35:1 (1994), 181–189 | DOI | MR | Zbl

[17] G. Murphy, Ordinary Differential Equation and Their Solutions, D. Van Nostrand Company, New York, 1960 | MR | Zbl

[18] S. Bouquet, H. R. Lewis, J. Math. Phys., 37:11 (1996), 5496–5517 | DOI | MR

[19] M. R. Feix, private communication, 1992

[20] V. P. Ermakov, Univ. Izv. Kiev{, sér. 3}, 1980, no. 9, 1