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September  2020, 12(3): 507-523. doi: 10.3934/jgm.2020018

Getting into the vortex: On the contributions of james montaldi

Departamento de Matemática da UFJF, Cidade Universitária, Juiz de Fora, 36036, Brazil

 

Received  June 2020 Revised  June 2020 Published  August 2020

Fund Project: Supported by a visiting contract at UFJF, Brazil

James Montaldi's expertises span many areas on pure and applied mathematics. I will discuss here just one, his contributions to the motion of point vortices, specially the role of symmetries in the bifurcations and stability of equilibrium configurations in surfaces of constant curvature. This approach leads, for instance, to a very elegant proof of a classical result, the nonlinear stability of Thompson's regular heptagon in the plane. Here the plane appears "in passing", just as the transition between positive and negative curvatures.

Citation: Jair Koiller. Getting into the vortex: On the contributions of james montaldi. Journal of Geometric Mechanics, 2020, 12 (3) : 507-523. doi: 10.3934/jgm.2020018
References:
[1]

H. ArefP. K. NewtonM. A. StremlerT. Tokieda and D. L. Vainchtein, Vortex crystals, Advances in Applied Mechanics, 39 (2003), 1-79.  doi: 10.1016/S0065-2156(02)39001-X.  Google Scholar

[2]

H. Aref, On the equilibrium and stability of a row of point vortices, Journal of Fluid Mechanics, 290 (1995), 167-181.  doi: 10.1017/S002211209500245X.  Google Scholar

[3]

H. Aref, Point vortex dynamics: A classical mathematics playground, Journal of Mathematical Physics, 48 (2007), 065401. doi: 10.1063/1.2425103.  Google Scholar

[4]

H. Aref and M. A. Stremler, On the motion of three point vortices in a periodic strip, Journal of Fluid Mechanics, 314 (1996), 1-25.  doi: 10.1017/S0022112096000213.  Google Scholar

[5]

I. A. BizyaevA. V. Borisov and I. S. Mamaev, The dynamics of three vortex sources, Regul. Chaotic Dyn., 19 (2014), 694-701.  doi: 10.1134/S1560354714060070.  Google Scholar

[6]

I. A. BizyaevA. V. Borisov and I. S. Mamaev, The dynamics of vortex sources in a deformation flow, Regul. Chaotic Dyn., 21 (2016), 367-376.  doi: 10.1134/S1560354716030084.  Google Scholar

[7]

S. Boatto and J. Koiller (2015), Vortices on closed surfaces, in Geometry, Mechanics, and Dynamics: The Legacy of Jerry Marsden (eds. D. E. Chang, D. Holm, G.Patrick and T.Ratiu), Fields Institute Communications book series, 73, Springer-Verlag, New York, 2015,185–237.  Google Scholar

[8]

A. V. Borisov and I. S. Mamaev, On the problem of motion of vortex sources on a plane, Regul. Chaotic Dyn., 11 (2006), 455-466.  doi: 10.1070/RD2006v011n04ABEH000363.  Google Scholar

[9]

H. W. Broer, Bernoulli's light ray solution of the brachistochrone problem through {H}amilton's eyes, International Journal of Bifurcation and Chaos, 24 (2014), 1440009. doi: 10.1142/S0218127414400094.  Google Scholar

[10]

P. L. Buono, F. Laurent-Polz and J. Montaldi, Symmetric Hamiltonian bifurcations, in Geometric Mechanics and Symmetry: The Peyresq Lectures (eds. J. Montaldi and T. Ratiu), London Mathematical Society Lecture Note Series, 306, Cambridge University Press, 2005, 357–402. doi: 10.1017/CBO9780511526367.007.  Google Scholar

[11]

H. Cabral and D. Schmidt, Stability of relative equilibria in the problem of $n+1$ vortices, SIAM Journal on Mathematical Analysis, 31 (2000), 231-250.  doi: 10.1137/S0036141098302124.  Google Scholar

[12]

C. J. CotterJ. ElderingD. D. HolmH. O. Jacobs and D. M. Meier, Weak dual pairs and jetlet methods for ideal incompressible fluid models in $n \ge 2$ dimensions, J. Nonlinear Sci., 26 (2016), 1723-1765.  doi: 10.1007/s00332-016-9317-6.  Google Scholar

[13]

A. C. DeVoria and K. Mohsen, Vortex sheet roll-up revisited, Journal of Fluid Mechanics, 855 (2018), 299-321.  doi: 10.1017/jfm.2018.663.  Google Scholar

[14]

A. A. Fridman and P. Y. Polubarinova, On moving singularities of a flat motion of an incompressible fluid, (In Russian) Geofizicheskii Sbornik, (1928), 9–23. Google Scholar

[15]

C. García-Azpeitia, Relative periodic solutions of the $n$-vortex problem on the sphere, Journal of Geometric Mechanics, 11 (2019), 427-438.  doi: 10.3934/jgm.2019021.  Google Scholar

[16]

R. Gibbs, The search continues for Kármán's St Christopher, Nature, 406 (2000), 122-122.  doi: 10.1038/35018274.  Google Scholar

[17]

R. N. Govardhan and O. N. Ramesh, A stroll down Kármán street, Resonance, 10 (2005), 25-37.  doi: 10.1007/BF02866744.  Google Scholar

[18]

B. Gustafsson, Vortex motion and geometric function theory: The role of connections, Philosophical Transactions of the Royal Society A, 377 (2019), 20180341. doi: 10.1098/rsta.2018.0341.  Google Scholar

[19]

A. K. HindsE. R. Jonhson and N. R. McDonald, Vortex scattering by step topography, Journal of Fluid Mechanics, 571 (2007), 495-505.  doi: 10.1017/S002211200600348X.  Google Scholar

[20]

D. D. Holm and H. O. Jacobs, Multipole vortex blobs (MVB): Symplectic geometry and dynamics, J. Nonlinear Sci., 27 (2017), 973-1006.  doi: 10.1007/s00332-017-9367-4.  Google Scholar

[21]

S. Hwang and S. C. Kim, Relative equilibria of point vortices on the hyperbolic sphere, Journal of Mathematical Physics, 54 (2013), 063101. doi: 10.1063/1.4811454.  Google Scholar

[22]

A. Johann, H. P. Kruse, F. Rupp, and S. Schmitz, Recent Trends in Dynamical Systems, Proceedings of a Conference in Honor of Jürgen Scheurle, Springer Proceedings in Mathematics & Statistics, 35, Springer, Basel, 2013. doi: 10.1007/978-3-0348-0451-6.  Google Scholar

[23]

Y. Kimura, Vortex motion on surfaces with constant curvature, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 455 (1999), 245-259.  doi: 10.1098/rspa.1999.0311.  Google Scholar

[24]

H. Kragh, The vortex atom: A Victorian theory of everything, Centaurus, 44 (2002), 32-114.  doi: 10.1034/j.1600-0498.2002.440102.x.  Google Scholar

[25]

L. G. Kurakin and V. I. Yudovich, The stability of stationary rotation of a regular vortex polygon, Chaos: An Interdisciplinary Journal of Nonlinear Science, 12 (2002), 574-595.  doi: 10.1063/1.1482175.  Google Scholar

[26]

E. A. Lacomba, Interaction of point sources and vortices for incompressible planar fluids, Qual. Theory Dyn. Syst., 8 (2009), 371-379.  doi: 10.1007/s12346-010-0015-8.  Google Scholar

[27]

H. Lamb, Hydrodynamics, 6th edition, Cambridge Univ. Press, Cambridge, 1993.  Google Scholar

[28]

F. Laurent-Poltz, Étude Géométrique de la Dynamique de $N$ Tourbillons Ponctuels sur une Sphère., Ph.D. thesis, Université de Nice, 2002. Google Scholar

[29]

F. Laurent-Poltz, Point vortics on a rotating sphere, Regular and Chaotic Dynamics, 10 (2005), 39-58.  doi: 10.1070/RD2005v010n01ABEH000299.  Google Scholar

[30]

F. Laurent-Polz, Point vortices on the sphere: A case with opposite vorticities, Nonlinearity, 15 (2001), 143-171.  doi: 10.1088/0951-7715/15/1/307.  Google Scholar

[31]

F. Laurent-Polz, Relative periodic orbits in point vortex systems, Nonlinearity, 17 (2004), 1989-2013.  doi: 10.1088/0951-7715/17/6/001.  Google Scholar

[32]

J. C. LiaoD. N. BealG. V. Lauder and M. S. Triantafyllou, The Kármán gait: Novel body kinematics of rainbow trout swimming in a vortex street, Journal of Experimental Biology, 206 (2003), 1059-1073.  doi: 10.1242/jeb.00209.  Google Scholar

[33]

S. G. Llewellyn Smith, How do singularities move in potential flow?, Physica D: Nonlinear Phenomena, 240 (2011), 1644-1651.  doi: 10.1016/j.physd.2011.06.010.  Google Scholar

[34]

N. R. McDonald, The motion of geophysical vortices, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 357 (1999), 3427-3444.  doi: 10.1098/rsta.1999.0501.  Google Scholar

[35]

V. V. Meleshko and H. Aref, A bibliography of vortex dynamics 1858-1956, Advances in Applied Mechanics, 41 (2007), 197-292.   Google Scholar

[36]

G. J. Mertz, Stability of body-centered polygonal configurations of ideal vortices, The Physics of Fluids, 21 (1978), 1092-1095.  doi: 10.1063/1.862347.  Google Scholar

[37]

T. MizotaM. ZdravkovichK. U. Graw and A. Leder, Science in culture, Nature, 404 (2000), 226-226.  doi: 10.1038/35005158.  Google Scholar

[38]

J. Montaldi and R. M. Roberts, Relative equilibria of molecules, J. Nonlinear Sci., 9 (1999), 53-88.  doi: 10.1007/s003329900064.  Google Scholar

[39]

J. Montaldi, Persistence and stability of relative equilibria, Nonlinearity, 10 (1997), 449-466.  doi: 10.1088/0951-7715/10/2/009.  Google Scholar

[40]

J. Montaldi and C. Nava-Gaxiola, Point vortices on the hyperbolic plane, Journal of Mathematical Physic, 55 (2014), 102702. doi: 10.1063/1.4897210.  Google Scholar

[41]

J. Montaldi and A. Shaddad, Generalized point vortex dynamics on $\mathbb{CP}^2$, Journal of Geometric Mechanics, 11 (2019), 601-619.  doi: 10.3934/jgm.2019030.  Google Scholar

[42]

J. Montaldi and A. Shaddad, {Non-A}belian momentum polytopes for products of $\mathbb{CP}^2$, Journal of Geometric Mechanics, 11 (2019), 575-599.  doi: 10.3934/jgm.2019029.  Google Scholar

[43]

J. Montaldi and T. Tokieda, Dynamics of poles with position-dependent strengths and its optical analogues, Physica D: Nonlinear Phenomena, 240 (2011), 1636-1643.  doi: 10.1016/j.physd.2011.07.004.  Google Scholar

[44]

J. Montaldi and T. Tokieda, Deformation of geometry and bifurcations of vortex rings, in Recent trends in dynamical systems (eds. A. Johann, H. P. Kruse, F. Rupp and S. Schmitz), Springer Proc. Math. Stat., 35, Springer, Basel, 2013,335–370.  Google Scholar

[45]

P. K. Newton, The N-Vortex Problem: Analytical Techniques, Applied Mathematical Sciences, 145, Springer-Verlag New York, 2001. doi: 10.1007/978-1-4684-9290-3.  Google Scholar

[46]

P. K. Newton, Point vortex dynamics in the post-Aref era, Fluid Dynamics Research, 46 (2014), 031401. doi: 10.1088/0169-5983/46/3/031401.  Google Scholar

[47]

L. Rosenhead, The formation of vortices from a surface of discontinuity, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 134 (1931), 170-192.   Google Scholar

[48]

V. P. Ruban, Dynamics of quantum vortices in a quasi-two-dimensional Bose-Einstein condensate with two "holes", JETP Letters, 105 (2017), 458-463.  doi: 10.1134/S0021364017070128.  Google Scholar

[49]

V. P. Ruban, Dynamics of straight vortex filaments in a Bose-Einstein condensate with the Gaussian density profile, Journal of Experimental and Theoretical Physics, 124 (2017), 932-942.  doi: 10.1134/S1063776117050041.  Google Scholar

[50]

A. Soulière, Dynamique de N pôles à intensités variables, Ph.D. thesis, Département de mathématiques et de statistique, University of Montréal, 2007 (http://hdl.handle.net/1866/18142).  Google Scholar

[51]

I. Starace, Il Grande Libro Degli Haiku, Castelvecchi, 2018. Google Scholar

[52]

L. Tophøj and H. Aref, Instability of vortex pair leapfrogging, Physics of Fluids, 25 (2013), 014107. Google Scholar

[53]

T. von Kármán, Über den Mechanismus des Wistandes, den ein bewegter Körper in einer Flüssigkeit erfärt (On the mechanism of drag generation on the body moving in fluid), Nachrichten Gesellschaft Wissenschaffen, mathematisch-physikalische Klasse, Gottingen, (1911), 509–517 (part 1), (1912), 547–556 (part 2). Google Scholar

[54]

T. von Kármán, Aerodynamics, Paperbacks Science, Mathematics and Engineering, McGraw-Hill, 1963. Google Scholar

show all references

References:
[1]

H. ArefP. K. NewtonM. A. StremlerT. Tokieda and D. L. Vainchtein, Vortex crystals, Advances in Applied Mechanics, 39 (2003), 1-79.  doi: 10.1016/S0065-2156(02)39001-X.  Google Scholar

[2]

H. Aref, On the equilibrium and stability of a row of point vortices, Journal of Fluid Mechanics, 290 (1995), 167-181.  doi: 10.1017/S002211209500245X.  Google Scholar

[3]

H. Aref, Point vortex dynamics: A classical mathematics playground, Journal of Mathematical Physics, 48 (2007), 065401. doi: 10.1063/1.2425103.  Google Scholar

[4]

H. Aref and M. A. Stremler, On the motion of three point vortices in a periodic strip, Journal of Fluid Mechanics, 314 (1996), 1-25.  doi: 10.1017/S0022112096000213.  Google Scholar

[5]

I. A. BizyaevA. V. Borisov and I. S. Mamaev, The dynamics of three vortex sources, Regul. Chaotic Dyn., 19 (2014), 694-701.  doi: 10.1134/S1560354714060070.  Google Scholar

[6]

I. A. BizyaevA. V. Borisov and I. S. Mamaev, The dynamics of vortex sources in a deformation flow, Regul. Chaotic Dyn., 21 (2016), 367-376.  doi: 10.1134/S1560354716030084.  Google Scholar

[7]

S. Boatto and J. Koiller (2015), Vortices on closed surfaces, in Geometry, Mechanics, and Dynamics: The Legacy of Jerry Marsden (eds. D. E. Chang, D. Holm, G.Patrick and T.Ratiu), Fields Institute Communications book series, 73, Springer-Verlag, New York, 2015,185–237.  Google Scholar

[8]

A. V. Borisov and I. S. Mamaev, On the problem of motion of vortex sources on a plane, Regul. Chaotic Dyn., 11 (2006), 455-466.  doi: 10.1070/RD2006v011n04ABEH000363.  Google Scholar

[9]

H. W. Broer, Bernoulli's light ray solution of the brachistochrone problem through {H}amilton's eyes, International Journal of Bifurcation and Chaos, 24 (2014), 1440009. doi: 10.1142/S0218127414400094.  Google Scholar

[10]

P. L. Buono, F. Laurent-Polz and J. Montaldi, Symmetric Hamiltonian bifurcations, in Geometric Mechanics and Symmetry: The Peyresq Lectures (eds. J. Montaldi and T. Ratiu), London Mathematical Society Lecture Note Series, 306, Cambridge University Press, 2005, 357–402. doi: 10.1017/CBO9780511526367.007.  Google Scholar

[11]

H. Cabral and D. Schmidt, Stability of relative equilibria in the problem of $n+1$ vortices, SIAM Journal on Mathematical Analysis, 31 (2000), 231-250.  doi: 10.1137/S0036141098302124.  Google Scholar

[12]

C. J. CotterJ. ElderingD. D. HolmH. O. Jacobs and D. M. Meier, Weak dual pairs and jetlet methods for ideal incompressible fluid models in $n \ge 2$ dimensions, J. Nonlinear Sci., 26 (2016), 1723-1765.  doi: 10.1007/s00332-016-9317-6.  Google Scholar

[13]

A. C. DeVoria and K. Mohsen, Vortex sheet roll-up revisited, Journal of Fluid Mechanics, 855 (2018), 299-321.  doi: 10.1017/jfm.2018.663.  Google Scholar

[14]

A. A. Fridman and P. Y. Polubarinova, On moving singularities of a flat motion of an incompressible fluid, (In Russian) Geofizicheskii Sbornik, (1928), 9–23. Google Scholar

[15]

C. García-Azpeitia, Relative periodic solutions of the $n$-vortex problem on the sphere, Journal of Geometric Mechanics, 11 (2019), 427-438.  doi: 10.3934/jgm.2019021.  Google Scholar

[16]

R. Gibbs, The search continues for Kármán's St Christopher, Nature, 406 (2000), 122-122.  doi: 10.1038/35018274.  Google Scholar

[17]

R. N. Govardhan and O. N. Ramesh, A stroll down Kármán street, Resonance, 10 (2005), 25-37.  doi: 10.1007/BF02866744.  Google Scholar

[18]

B. Gustafsson, Vortex motion and geometric function theory: The role of connections, Philosophical Transactions of the Royal Society A, 377 (2019), 20180341. doi: 10.1098/rsta.2018.0341.  Google Scholar

[19]

A. K. HindsE. R. Jonhson and N. R. McDonald, Vortex scattering by step topography, Journal of Fluid Mechanics, 571 (2007), 495-505.  doi: 10.1017/S002211200600348X.  Google Scholar

[20]

D. D. Holm and H. O. Jacobs, Multipole vortex blobs (MVB): Symplectic geometry and dynamics, J. Nonlinear Sci., 27 (2017), 973-1006.  doi: 10.1007/s00332-017-9367-4.  Google Scholar

[21]

S. Hwang and S. C. Kim, Relative equilibria of point vortices on the hyperbolic sphere, Journal of Mathematical Physics, 54 (2013), 063101. doi: 10.1063/1.4811454.  Google Scholar

[22]

A. Johann, H. P. Kruse, F. Rupp, and S. Schmitz, Recent Trends in Dynamical Systems, Proceedings of a Conference in Honor of Jürgen Scheurle, Springer Proceedings in Mathematics & Statistics, 35, Springer, Basel, 2013. doi: 10.1007/978-3-0348-0451-6.  Google Scholar

[23]

Y. Kimura, Vortex motion on surfaces with constant curvature, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 455 (1999), 245-259.  doi: 10.1098/rspa.1999.0311.  Google Scholar

[24]

H. Kragh, The vortex atom: A Victorian theory of everything, Centaurus, 44 (2002), 32-114.  doi: 10.1034/j.1600-0498.2002.440102.x.  Google Scholar

[25]

L. G. Kurakin and V. I. Yudovich, The stability of stationary rotation of a regular vortex polygon, Chaos: An Interdisciplinary Journal of Nonlinear Science, 12 (2002), 574-595.  doi: 10.1063/1.1482175.  Google Scholar

[26]

E. A. Lacomba, Interaction of point sources and vortices for incompressible planar fluids, Qual. Theory Dyn. Syst., 8 (2009), 371-379.  doi: 10.1007/s12346-010-0015-8.  Google Scholar

[27]

H. Lamb, Hydrodynamics, 6th edition, Cambridge Univ. Press, Cambridge, 1993.  Google Scholar

[28]

F. Laurent-Poltz, Étude Géométrique de la Dynamique de $N$ Tourbillons Ponctuels sur une Sphère., Ph.D. thesis, Université de Nice, 2002. Google Scholar

[29]

F. Laurent-Poltz, Point vortics on a rotating sphere, Regular and Chaotic Dynamics, 10 (2005), 39-58.  doi: 10.1070/RD2005v010n01ABEH000299.  Google Scholar

[30]

F. Laurent-Polz, Point vortices on the sphere: A case with opposite vorticities, Nonlinearity, 15 (2001), 143-171.  doi: 10.1088/0951-7715/15/1/307.  Google Scholar

[31]

F. Laurent-Polz, Relative periodic orbits in point vortex systems, Nonlinearity, 17 (2004), 1989-2013.  doi: 10.1088/0951-7715/17/6/001.  Google Scholar

[32]

J. C. LiaoD. N. BealG. V. Lauder and M. S. Triantafyllou, The Kármán gait: Novel body kinematics of rainbow trout swimming in a vortex street, Journal of Experimental Biology, 206 (2003), 1059-1073.  doi: 10.1242/jeb.00209.  Google Scholar

[33]

S. G. Llewellyn Smith, How do singularities move in potential flow?, Physica D: Nonlinear Phenomena, 240 (2011), 1644-1651.  doi: 10.1016/j.physd.2011.06.010.  Google Scholar

[34]

N. R. McDonald, The motion of geophysical vortices, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 357 (1999), 3427-3444.  doi: 10.1098/rsta.1999.0501.  Google Scholar

[35]

V. V. Meleshko and H. Aref, A bibliography of vortex dynamics 1858-1956, Advances in Applied Mechanics, 41 (2007), 197-292.   Google Scholar

[36]

G. J. Mertz, Stability of body-centered polygonal configurations of ideal vortices, The Physics of Fluids, 21 (1978), 1092-1095.  doi: 10.1063/1.862347.  Google Scholar

[37]

T. MizotaM. ZdravkovichK. U. Graw and A. Leder, Science in culture, Nature, 404 (2000), 226-226.  doi: 10.1038/35005158.  Google Scholar

[38]

J. Montaldi and R. M. Roberts, Relative equilibria of molecules, J. Nonlinear Sci., 9 (1999), 53-88.  doi: 10.1007/s003329900064.  Google Scholar

[39]

J. Montaldi, Persistence and stability of relative equilibria, Nonlinearity, 10 (1997), 449-466.  doi: 10.1088/0951-7715/10/2/009.  Google Scholar

[40]

J. Montaldi and C. Nava-Gaxiola, Point vortices on the hyperbolic plane, Journal of Mathematical Physic, 55 (2014), 102702. doi: 10.1063/1.4897210.  Google Scholar

[41]

J. Montaldi and A. Shaddad, Generalized point vortex dynamics on $\mathbb{CP}^2$, Journal of Geometric Mechanics, 11 (2019), 601-619.  doi: 10.3934/jgm.2019030.  Google Scholar

[42]

J. Montaldi and A. Shaddad, {Non-A}belian momentum polytopes for products of $\mathbb{CP}^2$, Journal of Geometric Mechanics, 11 (2019), 575-599.  doi: 10.3934/jgm.2019029.  Google Scholar

[43]

J. Montaldi and T. Tokieda, Dynamics of poles with position-dependent strengths and its optical analogues, Physica D: Nonlinear Phenomena, 240 (2011), 1636-1643.  doi: 10.1016/j.physd.2011.07.004.  Google Scholar

[44]

J. Montaldi and T. Tokieda, Deformation of geometry and bifurcations of vortex rings, in Recent trends in dynamical systems (eds. A. Johann, H. P. Kruse, F. Rupp and S. Schmitz), Springer Proc. Math. Stat., 35, Springer, Basel, 2013,335–370.  Google Scholar

[45]

P. K. Newton, The N-Vortex Problem: Analytical Techniques, Applied Mathematical Sciences, 145, Springer-Verlag New York, 2001. doi: 10.1007/978-1-4684-9290-3.  Google Scholar

[46]

P. K. Newton, Point vortex dynamics in the post-Aref era, Fluid Dynamics Research, 46 (2014), 031401. doi: 10.1088/0169-5983/46/3/031401.  Google Scholar

[47]

L. Rosenhead, The formation of vortices from a surface of discontinuity, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 134 (1931), 170-192.   Google Scholar

[48]

V. P. Ruban, Dynamics of quantum vortices in a quasi-two-dimensional Bose-Einstein condensate with two "holes", JETP Letters, 105 (2017), 458-463.  doi: 10.1134/S0021364017070128.  Google Scholar

[49]

V. P. Ruban, Dynamics of straight vortex filaments in a Bose-Einstein condensate with the Gaussian density profile, Journal of Experimental and Theoretical Physics, 124 (2017), 932-942.  doi: 10.1134/S1063776117050041.  Google Scholar

[50]

A. Soulière, Dynamique de N pôles à intensités variables, Ph.D. thesis, Département de mathématiques et de statistique, University of Montréal, 2007 (http://hdl.handle.net/1866/18142).  Google Scholar

[51]

I. Starace, Il Grande Libro Degli Haiku, Castelvecchi, 2018. Google Scholar

[52]

L. Tophøj and H. Aref, Instability of vortex pair leapfrogging, Physics of Fluids, 25 (2013), 014107. Google Scholar

[53]

T. von Kármán, Über den Mechanismus des Wistandes, den ein bewegter Körper in einer Flüssigkeit erfärt (On the mechanism of drag generation on the body moving in fluid), Nachrichten Gesellschaft Wissenschaffen, mathematisch-physikalische Klasse, Gottingen, (1911), 509–517 (part 1), (1912), 547–556 (part 2). Google Scholar

[54]

T. von Kármán, Aerodynamics, Paperbacks Science, Mathematics and Engineering, McGraw-Hill, 1963. Google Scholar

Figure 1.  James' interests and eight Erdös paths
Figure 2.  An ninth Erdös path
Figure 3.  Types of relative equilibria bifurcating from equilibria. Table 5 from [Lim, Montaldi and Roberts (2001)]
Figure 4.  Also Fig. 4 in the paper. The caption says: "Orbit type lattice for the action of $ O(3) \times S_N $ for $ N = 3-5 $ identical vortices. The underlined strata are those which we prove contain relative equilibria for any $ G $-invariant Hamiltonian. The strata marked with a dagger ($ \dagger $) do not contain any relative equilibria for the point-vortex Hamiltonian. See Section 5 [of the paper)] for summaries of these existence and non-existence results"
Figure 5.  Image from NASA
Figure 6.  Double rows of alternating vortices; the symmetric one (upper) is unstable; the alternating configuration (lower) is stable. Cylinder moves to the left. This figure is from the delightful book [54] von Kármán (1963)
Figure 7.  Snell's law
Figure 8.  Law of reflection
Figure 9.  Thomson's problem
Figure 10.  Thomson's mistake
Figure 11.  Stereographic projection on the complex plane
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