Geodesic
Equivalent Integrable Metrics on the Sphere with Quartic Invariants
Symmetry, integrability and geometry: methods and applications, Tome 18 (2022) Cet article a éte moissonné depuis la source Math-Net.Ru

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We discuss canonical transformations relating well-known geodesic flows on the cotangent bundle of the sphere with a set of geodesic flows with quartic invariants. By adding various potentials to the corresponding geodesic Hamiltonians, we can construct new integrable systems on the sphere with quartic invariants.
Keywords: integrable metrics, canonical transformations, two-dimensional sphere. 
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     author = {Andrey V. Tsiganov},
     title = {Equivalent {Integrable} {Metrics} on the {Sphere} with {Quartic} {Invariants}},
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     url = {http://geodesic.mathdoc.fr/item/SIGMA_2022_18_a93/}
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Andrey V. Tsiganov. Equivalent Integrable Metrics on the Sphere with Quartic Invariants. Symmetry, integrability and geometry: methods and applications, Tome 18 (2022). http://geodesic.mathdoc.fr/item/SIGMA_2022_18_a93/

[1] Arnold V.I., Mathematical methods of classical mechanics, Grad. Texts in Math., 60, 2nd ed., Springer, New York, 1989 | DOI

[2] Bialy M., Mironov A., “New semi-Hamiltonian hierarchy related to integrable magnetic flows on surfaces”, Cent. Eur. J. Math., 10 (2012), 1596–1604, arXiv: 1112.1232 | DOI

[3] Bishop R.L., Goldberg S.I., Tensor analysis on manifolds, Dover Publications, Inc., New York, 1980

[4] Błaszak M., Marciniak K., “On reciprocal equivalence of Stäckel systems”, Stud. Appl. Math., 129 (2012), 26–50, arXiv: 1201.0446 | DOI

[5] Bolsinov A.V., Fomenko A.T., Integrable geodesic flows on two-dimensional surfaces, Monogr. Contemp. Math., Consultants Bureau, New York, 2000 | DOI

[6] Bolsinov A.V., Kozlov V.V., Fomenko A.T., “The Maupertuis principle and geodesic flow on the sphere arising from integrable cases in the dynamic of a rigid body”, Russian Math. Surv., 50 (1995), 473–501 | DOI

[7] D'Ambra G., Gromov M., “Lectures on transformation groups: geometry and dynamics”, Surveys in Differential Geometry (Cambridge, MA, 1990), Lehigh University, Bethlehem, PA, 1991, 19–111

[8] Dorizzi B., Grammaticos B., Ramani A., Winternitz P., “Integrable Hamiltonian systems with velocity-dependent potentials”, J. Math. Phys., 26 (1985), 3070–3079 | DOI

[9] Kiyohara K., Topalov P., “On Liouville integrability of $h$-projectively equivalent Kähler metrics”, Proc. Amer. Math. Soc., 139 (2011), 231–242 | DOI

[10] Matveev V.S., “Quantum integrability for the Beltrami–Laplace operators of projectively equivalent metrics of arbitrary signatures”, Chebyshevskii Sb., 21 (2020), 275–289, arXiv: 1906.06757 | DOI

[11] McSween E., Winternitz P., “Integrable and superintegrable Hamiltonian systems in magnetic fields”, J. Math. Phys., 41 (2000), 2957–2967 | DOI

[12] Taber W., “Projectively equivalent metrics subject to constraints”, Trans. Amer. Math. Soc., 282 (1984), 711–737 | DOI

[13] Tsiganov A.V., “Duality between integrable Stäckel systems”, J. Phys. A, 32 (1999), 7965–7982, arXiv: solv-int/9812001 | DOI

[14] Tsiganov A.V., “The Maupertuis principle and canonical transformations of the extended phase space”, J. Nonlinear Math. Phys., 8 (2001), 157–182, arXiv: nlin.SI/0101061 | DOI

[15] Tsiganov A.V., “On natural Poisson bivectors on the sphere”, J. Phys. A, 44 (2011), 105203, 21 pp., arXiv: 1010.3492 | DOI

[16] Tsiganov A.V., “On auto and hetero Bäcklund transformations for the Hénon–Heiles systems”, Phys. Lett. A, 379 (2015), 2903–2907, arXiv: 1501.06695 | DOI

[17] Tsiganov A.V., “On the Chaplygin system on the sphere with velocity dependent potential”, J. Geom. Phys., 92 (2015), 94–99 | DOI

[18] Tsiganov A.V., “Simultaneous separation for the Neumann and Chaplygin systems”, Regul. Chaotic Dyn., 20 (2015), 74–93 | DOI

[19] Tsiganov A.V., “Bäcklund transformations for the Jacobi system on an ellipsoid”, Theoret. and Math. Phys., 192 (2017), 1350–1364 | DOI

[20] Tsiganov A.V., “Bäcklund transformations for the nonholonomic Veselova system”, Regul. Chaotic Dyn., 22 (2017), 163–179, arXiv: 1703.04251 | DOI

[21] Tsiganov A.V., “Integrable discretization and deformation of the nonholonomic Chaplygin ball”, Regul. Chaotic Dyn., 22 (2017), 353–367, arXiv: 1705.01866 | DOI

[22] Tsiganov A.V., “New bi-Hamiltonian systems on the plane”, J. Math. Phys., 58 (2017), 062901, 14 pp., arXiv: 1701.05716 | DOI

[23] Tsiganov A.V., “On discretization of the Euler top”, Regul. Chaotic Dyn., 23 (2018), 785–796, arXiv: 1803.06511 | DOI

[24] Tsiganov A.V., “On exact discretization of cubic-quintic Duffing oscillator”, J. Math. Phys., 59 (2018), 072703, 15 pp., arXiv: 1805.05693 | DOI

[25] Tsyganov A.V., “Discretization of Hamiltonian systems and intersection theory”, Theoret. and Math. Phys., 197 (2018), 1806–1822 | DOI

[26] Vershilov A.V., Tsiganov A.V., “On bi-Hamiltonian geometry of some integrable systems on the sphere with cubic integral of motion”, J. Phys. A, 42 (2009), 105203, 12 pp., arXiv: 0812.0217 | DOI

[27] Vinogradov A.M., Kupershmidt B.A., “The structure of Hamiltonian mechanics”, Russian Math. Surveys, 32:4 (1977), 177–243 | DOI