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Symmetries of line bundles and Noether theorem for time-dependent nonholonomic systems
Vortex pairs on a triaxial ellipsoid and Kimura's conjecture
1. | Departamento de Matemática, Universidade Federal Rural de Pernambuco, Recife, PE CEP 52171-900 Brazil |
2. | Departamento de Matemática, Universidade Federal de Pernambuco, Recife, PE CEP 50740-540 Brazil |
3. | Departamento de Matemática, Universidade Federal de Juiz de Fora, Juiz de Fora, MG CEP 36036-900 Brazil |
We consider the problem of point vortices moving on the surface of a triaxial ellipsoid. Following Hally's approach, we obtain the equations of motion by constructing a conformal map from the ellipsoid into the sphere and composing with stereographic projection. We focus on the case of a pair of opposite vortices. Our approach is validated by testing a prediction by Kimura that a (infinitesimally close) vortex dipole follows the geodesic flow. Poincaré sections suggest that the global flow is non-integrable.
References:
[1] |
S. Boatto and J. Koiller, Vortices on closed surfaces, Geometry, Mechanics and Dynamics: The Legacy of Jerry Marsden, 185–237, Fields Inst. Commun., 73, Springer, New York, 2015.
doi: 10.1007/978-1-4939-2441-7_10. |
[2] |
V. A. Bogomolov,
The dynamics of vorticity on a sphere, (Russian), Izv. Akad. Nauk SSSR Ser. Meh. Zidk. Gaza, 6 (1977), 57-65.
|
[3] |
A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems. Geometry, Topology, Classification, Chapman & Hall/CRC, Boca Raton, FL, 2004.
doi: 10.1201/9780203643426. |
[4] |
A. V. Bolsinov, V. S. Matveev and A. T. Fomenko,
Two-dimensional Riemannian metrics with integrable geodesic flows. Local and global geometry, Sb. Math., 189 (1998), 1441-1466.
doi: 10.1070/SM1998v189n10ABEH000346. |
[5] |
P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, Berlin: Springer-Verlag, 1954. |
[6] |
B. C. Carlson,
Computing elliptic integrals by duplication, Numerische Mathematik, 33 (1979), 1-16.
doi: 10.1007/BF01396491. |
[7] |
C. Castilho and H. Machado, The N-vortex problem on a symmetric ellipsoid: A perturbation approach, J. Math. Phys., 49 (2008), 022703, 12pp.
doi: 10.1063/1.2863515. |
[8] |
T. Craig,
Orthomophic Projection of an Ellipsoid upon a sphere, Amer. J. Math., 3 (1880), 114-127.
doi: 10.2307/2369466. |
[9] |
D. G. Crowdy, E. H. Kropf, C. C. Green and M. M. S. Nasser,
The Schottky-Klein prime function: A theoretical and computational tool for applications, IMA J. Applied Math., 81 (2016), 589-628.
doi: 10.1093/imamat/hxw028. |
[10] |
D. G. Dritschel and S. Boatto, The motion of point vortices on closed surfaces, Proc. R. Soc. A, 471 (2015), 20140890, 25pp.
doi: 10.1098/rspa.2014.0890. |
[11] |
I. S. Gradshteyn, J. M. Ryzhik and A. Jeffrey, Table of Integrals, Series, and Products, 4. ed. New York: Academic Press, 1965. |
[12] |
F. Goes, L. Beibei, M. Budninskiy, Y. Tong and M. Desbrun, Discrete 2-Tensor Fields on Triangulations, Eurographics Symposium on Geometry Processing 33:5, editors Thomas Funkhouser and Shi-Min Hu, 2014. |
[13] |
I. Gromeka, Sobranie Socinenii (Russian) (Collected works), Izdat. Akad. Nauk SSSR, Moscow, 1952. |
[14] |
D. Hally,
Stability of streets of vortices on surfaces of revolution with a reflection symmetry, J. Math. Phys., 21 (1980), 211-217.
doi: 10.1063/1.524322. |
[15] |
H. Helmholtz,
Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen, (German), J. Reine Angew. Math., 55 (1858), 25-55.
doi: 10.1515/crll.1858.55.25. |
[16] |
M. Henon,
On the numerical computation of Poincaré maps, Physica D, 5 (1982), 412-414.
doi: 10.1016/0167-2789(82)90034-3. |
[17] |
C. G. J. Jacobi,
Note von der geodätischen Linie auf einem Ellipsoid und den verschiedenen Anwendungen einer merkwürdigen analytischen Substitution [The geodesic on an ellipsoid and various applications of a remarkable analytical substitution], Jour. Crelle, 19 (1839), 309-313.
doi: 10.1515/crll.1839.19.309. |
[18] |
C. G. J. Jacobi, Vorlesungen über Dynamik [Lectures on Dynamics], edited by Clebsch, Reimer, Berlin, 1866; second edition edited by Weierstrass, 1884; English translation by K. Balagangadharan (Hindustan Book Agency, 2009). |
[19] |
C. F. Karney, https://geographiclib.sourceforge.io/html/jacobi.html. |
[20] |
S.-C. Kim,
Latitudinal point vortex rings on the spheroid, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 1749-1768.
doi: 10.1098/rspa.2009.0597. |
[21] |
R. Kidambi and P. K. Newton,
Motion of three point vortices on a sphere, Physica D: Nonlinear Phenomena, 116 (1998), 143-175.
doi: 10.1016/S0167-2789(97)00236-4. |
[22] |
Y. Kimura,
Vortex motion on surfaces with constant curvature, Proc. R. Soc. Lond. A, 455 (1999), 245-259.
doi: 10.1098/rspa.1999.0311. |
[23] |
G. R. Kirchhoff, Vorlesungen über Mathematische Physik, Mechanic, Teubner, Leipzig, 1876. |
[24] |
F. Klein, On Riemann Theory of Algebraic Functions and their Integrals, Cambridge: Macmillan and Bowes (1893). Available online www.gutenberg.org/ebooks/36959 |
[25] |
J. Koiller and S. Boatto, Vortex pairs on surfaces,
AIP Conference Proceedings, , 77 (2009), p1130; (Geometry and Physics: ⅩⅦ International Fall Workshop, edited by F. Etayo, M. Fioravanti, and R. Santamaria). |
[26] |
J. Koiller and K. Ehlers,
Rubber rolling over a sphere, Reg. Chaotic Dyn., 12 (2007), 127-152.
doi: 10.1134/S1560354707020025. |
[27] |
C. C. Lin,
On the Motion of Vortices in Two Dimensions: Ⅰ. Existence of the Kirchhoff-Routh Function; Ⅱ. Some Further Investigations on the Kirchhoff-Routh Function, Proc. Natl.Acad. Sci. USA, 27 (1941), 575-577.
|
[28] |
A. S. Miguel, Numerical description of the motion of a point vortex pair on ovaloids, J. Phys. A: Math. Theor., 46 (2013), 115502, 21pp.
doi: 10.1088/1751-8113/46/11/115502. |
[29] |
J. Montaldi and C. Nava-Gaxiola, Point vortices on the hyperbolic plane, J. Math. Phys., 55 (2014), 102702, 14pp.
doi: 10.1063/1.4897210. |
[30] |
J. Moser, Integrable Hamiltonian Systems and Spectral Theory, Lezioni Fermiane. [Fermi Lectures] Scuola Normale Superiore, Pisa, 1983. |
[31] |
M. V. Nyrtsov, M. E. Flies, M. M. Borisov and P. J. Stooke,
Jacobi conformal projection of the triaxial ellipsoid: New projection for mapping of small celestial bodies, Cartography from Pole to Pole, (2014), 235-246.
|
[32] |
C. Ragazzo, The motion of a vortex on a closed surface: An algorithm and its application to the Bolza surface, Proc. Royal Soc. London A: Mathematical, Physical and Engineering Science, 473 (2017), 20170447, 17 pp.
doi: 10.1098/rspa.2017.0447. |
[33] |
A. Regis, Dinâmica de Vórtices Pontuais Sobre o Elipsóide Triaxial (Portuguese), Ph. D. thesis, Departamento de Matemática, Universidade Federal de Pernambuco, Brazil, 2011. |
[34] |
T. Sakajo,
The motion of three point vortices on a sphere, Japan Jounal of Industrial and Applied Mathematics, 16 (1999), 321-347.
doi: 10.1007/BF03167361. |
[35] |
E. Schering, Über Die Conforme Abbildung Des Ellipsoids Auf Der Ebene, Gesammelte Mathematische Werke, ch. Ⅲ, Mayer and Muller, Berlin (1902) (available in http://name.umdl.umich.edu/AAT1702.0001.001). |
[36] |
J. Steiner,
A geometrical mass and its extremal properties for metrics on $S^2$, Duke Math. J., 129 (2005), 63-86.
doi: 10.1215/S0012-7094-04-12913-6. |
[37] |
J. Vankerschaver and M. Leok,
A novel formulation of point vortex dynamics on the sphere: geometrical and numerical aspects, J. Nonlinear Science, 24 (2014), 1-37.
doi: 10.1007/s00332-013-9182-5. |
[38] |
E. Zermelo,
Hydrodynamische Untersuchungen über die Wirbelbewegungen in einer Kugelfläche, Z. Math. Phys., 47 (1902), 201-237.
|
show all references
References:
[1] |
S. Boatto and J. Koiller, Vortices on closed surfaces, Geometry, Mechanics and Dynamics: The Legacy of Jerry Marsden, 185–237, Fields Inst. Commun., 73, Springer, New York, 2015.
doi: 10.1007/978-1-4939-2441-7_10. |
[2] |
V. A. Bogomolov,
The dynamics of vorticity on a sphere, (Russian), Izv. Akad. Nauk SSSR Ser. Meh. Zidk. Gaza, 6 (1977), 57-65.
|
[3] |
A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems. Geometry, Topology, Classification, Chapman & Hall/CRC, Boca Raton, FL, 2004.
doi: 10.1201/9780203643426. |
[4] |
A. V. Bolsinov, V. S. Matveev and A. T. Fomenko,
Two-dimensional Riemannian metrics with integrable geodesic flows. Local and global geometry, Sb. Math., 189 (1998), 1441-1466.
doi: 10.1070/SM1998v189n10ABEH000346. |
[5] |
P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, Berlin: Springer-Verlag, 1954. |
[6] |
B. C. Carlson,
Computing elliptic integrals by duplication, Numerische Mathematik, 33 (1979), 1-16.
doi: 10.1007/BF01396491. |
[7] |
C. Castilho and H. Machado, The N-vortex problem on a symmetric ellipsoid: A perturbation approach, J. Math. Phys., 49 (2008), 022703, 12pp.
doi: 10.1063/1.2863515. |
[8] |
T. Craig,
Orthomophic Projection of an Ellipsoid upon a sphere, Amer. J. Math., 3 (1880), 114-127.
doi: 10.2307/2369466. |
[9] |
D. G. Crowdy, E. H. Kropf, C. C. Green and M. M. S. Nasser,
The Schottky-Klein prime function: A theoretical and computational tool for applications, IMA J. Applied Math., 81 (2016), 589-628.
doi: 10.1093/imamat/hxw028. |
[10] |
D. G. Dritschel and S. Boatto, The motion of point vortices on closed surfaces, Proc. R. Soc. A, 471 (2015), 20140890, 25pp.
doi: 10.1098/rspa.2014.0890. |
[11] |
I. S. Gradshteyn, J. M. Ryzhik and A. Jeffrey, Table of Integrals, Series, and Products, 4. ed. New York: Academic Press, 1965. |
[12] |
F. Goes, L. Beibei, M. Budninskiy, Y. Tong and M. Desbrun, Discrete 2-Tensor Fields on Triangulations, Eurographics Symposium on Geometry Processing 33:5, editors Thomas Funkhouser and Shi-Min Hu, 2014. |
[13] |
I. Gromeka, Sobranie Socinenii (Russian) (Collected works), Izdat. Akad. Nauk SSSR, Moscow, 1952. |
[14] |
D. Hally,
Stability of streets of vortices on surfaces of revolution with a reflection symmetry, J. Math. Phys., 21 (1980), 211-217.
doi: 10.1063/1.524322. |
[15] |
H. Helmholtz,
Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen, (German), J. Reine Angew. Math., 55 (1858), 25-55.
doi: 10.1515/crll.1858.55.25. |
[16] |
M. Henon,
On the numerical computation of Poincaré maps, Physica D, 5 (1982), 412-414.
doi: 10.1016/0167-2789(82)90034-3. |
[17] |
C. G. J. Jacobi,
Note von der geodätischen Linie auf einem Ellipsoid und den verschiedenen Anwendungen einer merkwürdigen analytischen Substitution [The geodesic on an ellipsoid and various applications of a remarkable analytical substitution], Jour. Crelle, 19 (1839), 309-313.
doi: 10.1515/crll.1839.19.309. |
[18] |
C. G. J. Jacobi, Vorlesungen über Dynamik [Lectures on Dynamics], edited by Clebsch, Reimer, Berlin, 1866; second edition edited by Weierstrass, 1884; English translation by K. Balagangadharan (Hindustan Book Agency, 2009). |
[19] |
C. F. Karney, https://geographiclib.sourceforge.io/html/jacobi.html. |
[20] |
S.-C. Kim,
Latitudinal point vortex rings on the spheroid, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 1749-1768.
doi: 10.1098/rspa.2009.0597. |
[21] |
R. Kidambi and P. K. Newton,
Motion of three point vortices on a sphere, Physica D: Nonlinear Phenomena, 116 (1998), 143-175.
doi: 10.1016/S0167-2789(97)00236-4. |
[22] |
Y. Kimura,
Vortex motion on surfaces with constant curvature, Proc. R. Soc. Lond. A, 455 (1999), 245-259.
doi: 10.1098/rspa.1999.0311. |
[23] |
G. R. Kirchhoff, Vorlesungen über Mathematische Physik, Mechanic, Teubner, Leipzig, 1876. |
[24] |
F. Klein, On Riemann Theory of Algebraic Functions and their Integrals, Cambridge: Macmillan and Bowes (1893). Available online www.gutenberg.org/ebooks/36959 |
[25] |
J. Koiller and S. Boatto, Vortex pairs on surfaces,
AIP Conference Proceedings, , 77 (2009), p1130; (Geometry and Physics: ⅩⅦ International Fall Workshop, edited by F. Etayo, M. Fioravanti, and R. Santamaria). |
[26] |
J. Koiller and K. Ehlers,
Rubber rolling over a sphere, Reg. Chaotic Dyn., 12 (2007), 127-152.
doi: 10.1134/S1560354707020025. |
[27] |
C. C. Lin,
On the Motion of Vortices in Two Dimensions: Ⅰ. Existence of the Kirchhoff-Routh Function; Ⅱ. Some Further Investigations on the Kirchhoff-Routh Function, Proc. Natl.Acad. Sci. USA, 27 (1941), 575-577.
|
[28] |
A. S. Miguel, Numerical description of the motion of a point vortex pair on ovaloids, J. Phys. A: Math. Theor., 46 (2013), 115502, 21pp.
doi: 10.1088/1751-8113/46/11/115502. |
[29] |
J. Montaldi and C. Nava-Gaxiola, Point vortices on the hyperbolic plane, J. Math. Phys., 55 (2014), 102702, 14pp.
doi: 10.1063/1.4897210. |
[30] |
J. Moser, Integrable Hamiltonian Systems and Spectral Theory, Lezioni Fermiane. [Fermi Lectures] Scuola Normale Superiore, Pisa, 1983. |
[31] |
M. V. Nyrtsov, M. E. Flies, M. M. Borisov and P. J. Stooke,
Jacobi conformal projection of the triaxial ellipsoid: New projection for mapping of small celestial bodies, Cartography from Pole to Pole, (2014), 235-246.
|
[32] |
C. Ragazzo, The motion of a vortex on a closed surface: An algorithm and its application to the Bolza surface, Proc. Royal Soc. London A: Mathematical, Physical and Engineering Science, 473 (2017), 20170447, 17 pp.
doi: 10.1098/rspa.2017.0447. |
[33] |
A. Regis, Dinâmica de Vórtices Pontuais Sobre o Elipsóide Triaxial (Portuguese), Ph. D. thesis, Departamento de Matemática, Universidade Federal de Pernambuco, Brazil, 2011. |
[34] |
T. Sakajo,
The motion of three point vortices on a sphere, Japan Jounal of Industrial and Applied Mathematics, 16 (1999), 321-347.
doi: 10.1007/BF03167361. |
[35] |
E. Schering, Über Die Conforme Abbildung Des Ellipsoids Auf Der Ebene, Gesammelte Mathematische Werke, ch. Ⅲ, Mayer and Muller, Berlin (1902) (available in http://name.umdl.umich.edu/AAT1702.0001.001). |
[36] |
J. Steiner,
A geometrical mass and its extremal properties for metrics on $S^2$, Duke Math. J., 129 (2005), 63-86.
doi: 10.1215/S0012-7094-04-12913-6. |
[37] |
J. Vankerschaver and M. Leok,
A novel formulation of point vortex dynamics on the sphere: geometrical and numerical aspects, J. Nonlinear Science, 24 (2014), 1-37.
doi: 10.1007/s00332-013-9182-5. |
[38] |
E. Zermelo,
Hydrodynamische Untersuchungen über die Wirbelbewegungen in einer Kugelfläche, Z. Math. Phys., 47 (1902), 201-237.
|







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