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December  2019, 11(4): 601-619. doi: 10.3934/jgm.2019030

Generalized point vortex dynamics on $ \mathbb{CP} ^2 $

James Montaldi , and  Amna Shaddad ,

Dept of Mathematics, University of Manchester, Manchester M13 9PL, UK

* Corresponding author

Dedicated to Darryl Holm on the occasion of his 70th birthday

Received  April 2018 Revised  June 2019 Published  November 2019

This is the second of two companion papers. We describe a generalization of the point vortex system on surfaces to a Hamiltonian dynamical system consisting of two or three points on complex projective space $ \mathbb{CP} ^2 $ interacting via a Hamiltonian function depending only on the distance between the points. The system has symmetry group SU(3). The first paper describes all possible momentum values for such systems, and here we apply methods of symplectic reduction and geometric mechanics to analyze the possible relative equilibria of such interacting generalized vortices.

The different types of polytope depend on the values of the 'vortex strengths', which are manifested as coefficients of the symplectic forms on the copies of $ \mathbb{CP} ^2 $. We show that the reduced space for this Hamiltonian action for 3 vortices is generically a 2-sphere, and proceed to describe the reduced dynamics under simple hypotheses on the type of Hamiltonian interaction. The other non-trivial reduced spaces are topological spheres with isolated singular points. For 2 generalized vortices, the reduced spaces are just points, and the motion is governed by a collective Hamiltonian, whereas for 3 the reduced spaces are of dimension at most 2. In both cases the system will be completely integrable in the non-abelian sense.

Keywords: Hamiltonian systems, momentum map, symplectic reduction, non-abelian integrability.
Mathematics Subject Classification: 37J15, 53D20, 70H06.
Citation: James Montaldi, Amna Shaddad. Generalized point vortex dynamics on $ \mathbb{CP} ^2 $. Journal of Geometric Mechanics, 2019, 11 (4) : 601-619. doi: 10.3934/jgm.2019030
References:
[1]

H. Aref, Point vortex dynamics: A classical mathematics playground, J. Math. Phys., 48 (2007), 065401, 23 pp. doi: 10.1063/1.2425103.

[2]

H. Aref, P. K. Newton, M. A. Stremler, T. Tokieda and D. L. Vainchtein, Vortex crystals, Adv. in Appl. Mech., 39 (2003), 1-79. 

[3]

S. Boatto and J. Koiller, Vortices on closed surfaces, Geometry, Mechanics and Dynamics: The Legacy of Jerry Marsden, Fields Inst. Commun., Springer, 73 (2015), 185-237.  doi: 10.1007/978-1-4939-2441-7_10.

[4]

A. V. Bolsinov, A. V. Borisov and I. S. Mamaev, Lie algebras in vortex dynamics and celestial mechanics. IV, Regular and Chaotic Dynamics, 4 (1999), 23-50.  doi: 10.1070/rd1999v004n01ABEH000097.

[5]

P.-L. Buono, F. Laurent-Polz and J. Montaldi, Symmetric Hamiltonian bifurcations, London Math. Soc. Lecture Note Ser., Geometric mechanics and symmetry, Cambridge Univ. Press, Cambridge, 306 (2005), 357-402.  doi: 10.1017/CBO9780511526367.007.

[6]

D. G. Dritschel and S. Boatto, The motion of point vortices on closed surfaces, Proc. R. Soc. A, 471 (2015), 20140890, 25 pp, http://dx.doi.org/10.1098/rspa.2014.0890. doi: 10.1098/rspa.2014.0890.

[7]

J. J. Duistermaat and G. J. Heckman, On the variation in the cohomology of the sympleetic form of the reduced phase space, Invent. Math., 69 (1982), 259-268.  doi: 10.1007/BF01399506.

[8] V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, Cambridge, 1984. 
[9]

V. Guillemin and S. Sternberg, Birational equivalence in the symplectic category, Invent. Math., 97 (1989), 485-522.  doi: 10.1007/BF01388888.

[10]

R. Kidambi and P. K. Newton, Motion of three point vortices on a sphere, Physica D, 116 (1998), 143-175.  doi: 10.1016/S0167-2789(97)00236-4.

[11] F. C. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, Mathematical Notes, 31. Princeton University Press, Princeton, NJ, 1984.  doi: 10.1007/BF01145470.
[12]

F. C. Kirwan, The topology of reduced phase spaces of the motion of vortices on a sphere, Phy. D, 30 (1988), 99-123.  doi: 10.1016/0167-2789(88)90100-5.

[13]

F. Laurent-Polz, J. Montaldi and M. Roberts, Point vortices on the sphere: Stability of symmetric relative equilibria, J. Geom. Mech., 3 (2011), 439-486.  doi: 10.3934/jgm.2011.3.439.

[14]

C. C. Lim, Existence of Kolmogorov-Arnold-Moser tori in the phase-space of lattice vortex systems, Z. Angew. Math. Phys., 41 (1990), 227-244. 

[15]

C. Lim, J. Montaldi and M. Roberts, Relative equilibria of point vortices on the sphere, Phys. D., 148 (2001), 97-135.  doi: 10.1016/S0167-2789(00)00167-6.

[16]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Texts in Applied Mathematics, 17. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-2682-6.

[17]

J. Milnor, Morse Theory, Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton, N.J. 1963.

[18]

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

[19]

J. Montaldi, Relative equilibria and conserved quantities in symmetric Hamiltonian systems, Peyresq Lectures on Nonlinear Phenomena, World Sci. Publ., River Edge, NJ, (2000), 239–280. doi: 10.1142/9789812792778_0008.

[20]

J. Montaldi and C. Nava-Gaxiola, Point vortices on the hyperbolic plane, J. Math. Phys., 55 (2014), 102702, 14 pp, http://dx.doi.org/10.1063/1.4897210. doi: 10.1063/1.4897210.

[21]

J. Montaldi and M. Roberts, Stratification of the momentum map, in preparation.

[22]

J. Montaldi, A. Soulière and T. Tokieda, Vortex dynamics on cylinders, SIAM J. on Appl. Dyn. Sys., 2 (2003), 417-430.  doi: 10.1137/S1111111102415569.

[23]

J. Montaldi and A. Shaddad, Non-Abelian momentum polytopes for products of $\mathbb{CP}^2$, J. Geom. Mechanics, (this volume).

[24]

J. Montaldi and T. Tokieda, Openness of momentum maps and persistence of extremal relative equilibria, Topology, 42 (2003), 833-844.  doi: 10.1016/S0040-9383(02)00047-2.

[25]

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.

[26]

J.-P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Progress in Mathematics, 222. Birkhäuser Boston, Inc., Boston, MA, 2004. doi: 10.1007/978-1-4757-3811-7.

[27]

S. Pekarsky and J. E. Marsden, Point vortices on a sphere: Stability of relative equilibria, J. Math. Phys., 39 (1998), 5894-5907.  doi: 10.1063/1.532602.

[28]

A. R. Rodrigues, C. Castilho and J. Koiller, Vortex dynamics on a triaxial ellipsoid and Kimura's conjecture, J. Geom. Mech., 10 (2018), 189-208.  doi: 10.3934/jgm.2018007.

[29]

T. Sakajo and Y. Shimizu, Point vortex interactions on a toroidal surface, k Proc. R. Soc. A, 472 (2016), 20160271, 24 pp, http://dx.doi.org/10.1098/rspa.2016.0271. doi: 10.1098/rspa.2016.0271.

[30]

A. Shaddad, The Classification and Dynamics of the Momentum Polytopes of the SU(3) Action on Points in the Complex Projective Plane with an Application to Point Vortices, Ph.D. thesis, University of Manchester, 2018.

[31]

R. Sjamaar, Convexity properties of the moment mapping re-examined, Advances in Math., 138 (1998), 46-91.  doi: 10.1006/aima.1998.1739.

show all references

References:
[1]

H. Aref, Point vortex dynamics: A classical mathematics playground, J. Math. Phys., 48 (2007), 065401, 23 pp. doi: 10.1063/1.2425103.

[2]

H. Aref, P. K. Newton, M. A. Stremler, T. Tokieda and D. L. Vainchtein, Vortex crystals, Adv. in Appl. Mech., 39 (2003), 1-79. 

[3]

S. Boatto and J. Koiller, Vortices on closed surfaces, Geometry, Mechanics and Dynamics: The Legacy of Jerry Marsden, Fields Inst. Commun., Springer, 73 (2015), 185-237.  doi: 10.1007/978-1-4939-2441-7_10.

[4]

A. V. Bolsinov, A. V. Borisov and I. S. Mamaev, Lie algebras in vortex dynamics and celestial mechanics. IV, Regular and Chaotic Dynamics, 4 (1999), 23-50.  doi: 10.1070/rd1999v004n01ABEH000097.

[5]

P.-L. Buono, F. Laurent-Polz and J. Montaldi, Symmetric Hamiltonian bifurcations, London Math. Soc. Lecture Note Ser., Geometric mechanics and symmetry, Cambridge Univ. Press, Cambridge, 306 (2005), 357-402.  doi: 10.1017/CBO9780511526367.007.

[6]

D. G. Dritschel and S. Boatto, The motion of point vortices on closed surfaces, Proc. R. Soc. A, 471 (2015), 20140890, 25 pp, http://dx.doi.org/10.1098/rspa.2014.0890. doi: 10.1098/rspa.2014.0890.

[7]

J. J. Duistermaat and G. J. Heckman, On the variation in the cohomology of the sympleetic form of the reduced phase space, Invent. Math., 69 (1982), 259-268.  doi: 10.1007/BF01399506.

[8] V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, Cambridge, 1984. 
[9]

V. Guillemin and S. Sternberg, Birational equivalence in the symplectic category, Invent. Math., 97 (1989), 485-522.  doi: 10.1007/BF01388888.

[10]

R. Kidambi and P. K. Newton, Motion of three point vortices on a sphere, Physica D, 116 (1998), 143-175.  doi: 10.1016/S0167-2789(97)00236-4.

[11] F. C. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, Mathematical Notes, 31. Princeton University Press, Princeton, NJ, 1984.  doi: 10.1007/BF01145470.
[12]

F. C. Kirwan, The topology of reduced phase spaces of the motion of vortices on a sphere, Phy. D, 30 (1988), 99-123.  doi: 10.1016/0167-2789(88)90100-5.

[13]

F. Laurent-Polz, J. Montaldi and M. Roberts, Point vortices on the sphere: Stability of symmetric relative equilibria, J. Geom. Mech., 3 (2011), 439-486.  doi: 10.3934/jgm.2011.3.439.

[14]

C. C. Lim, Existence of Kolmogorov-Arnold-Moser tori in the phase-space of lattice vortex systems, Z. Angew. Math. Phys., 41 (1990), 227-244. 

[15]

C. Lim, J. Montaldi and M. Roberts, Relative equilibria of point vortices on the sphere, Phys. D., 148 (2001), 97-135.  doi: 10.1016/S0167-2789(00)00167-6.

[16]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Texts in Applied Mathematics, 17. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-2682-6.

[17]

J. Milnor, Morse Theory, Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton, N.J. 1963.

[18]

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

[19]

J. Montaldi, Relative equilibria and conserved quantities in symmetric Hamiltonian systems, Peyresq Lectures on Nonlinear Phenomena, World Sci. Publ., River Edge, NJ, (2000), 239–280. doi: 10.1142/9789812792778_0008.

[20]

J. Montaldi and C. Nava-Gaxiola, Point vortices on the hyperbolic plane, J. Math. Phys., 55 (2014), 102702, 14 pp, http://dx.doi.org/10.1063/1.4897210. doi: 10.1063/1.4897210.

[21]

J. Montaldi and M. Roberts, Stratification of the momentum map, in preparation.

[22]

J. Montaldi, A. Soulière and T. Tokieda, Vortex dynamics on cylinders, SIAM J. on Appl. Dyn. Sys., 2 (2003), 417-430.  doi: 10.1137/S1111111102415569.

[23]

J. Montaldi and A. Shaddad, Non-Abelian momentum polytopes for products of $\mathbb{CP}^2$, J. Geom. Mechanics, (this volume).

[24]

J. Montaldi and T. Tokieda, Openness of momentum maps and persistence of extremal relative equilibria, Topology, 42 (2003), 833-844.  doi: 10.1016/S0040-9383(02)00047-2.

[25]

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.

[26]

J.-P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Progress in Mathematics, 222. Birkhäuser Boston, Inc., Boston, MA, 2004. doi: 10.1007/978-1-4757-3811-7.

[27]

S. Pekarsky and J. E. Marsden, Point vortices on a sphere: Stability of relative equilibria, J. Math. Phys., 39 (1998), 5894-5907.  doi: 10.1063/1.532602.

[28]

A. R. Rodrigues, C. Castilho and J. Koiller, Vortex dynamics on a triaxial ellipsoid and Kimura's conjecture, J. Geom. Mech., 10 (2018), 189-208.  doi: 10.3934/jgm.2018007.

[29]

T. Sakajo and Y. Shimizu, Point vortex interactions on a toroidal surface, k Proc. R. Soc. A, 472 (2016), 20160271, 24 pp, http://dx.doi.org/10.1098/rspa.2016.0271. doi: 10.1098/rspa.2016.0271.

[30]

A. Shaddad, The Classification and Dynamics of the Momentum Polytopes of the SU(3) Action on Points in the Complex Projective Plane with an Application to Point Vortices, Ph.D. thesis, University of Manchester, 2018.

[31]

R. Sjamaar, Convexity properties of the moment mapping re-examined, Advances in Math., 138 (1998), 46-91.  doi: 10.1006/aima.1998.1739.

Figure 3.  Possible non-trivial reduced spaces for the 3-vortex problem
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Figure 2.  This figure and associated table show the orbit type stratification of $ M=\mathbb{CP}^2\times\mathbb{CP}^2\times\mathbb{CP}^2 $; the table in (ⅱ) shows the geometry corresponding to the different stabilizers, while in (i) we see the adjacencies of the strata. The (a) and (b) refer in each case to two different geometry types for the same stabilizer, and hence different components of the corresponding fixed point space. (Note that strata marked with (a) contain the vertex $ a $ in their image, the strata marked with (b) contain the vertex $ b $ in their image, and the image of the $ \mathbb{T}^2_{(c)} $-strata consists of the points $ c_j $.)
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Figure 4.  The generic momentum polytopes from [23], showing the type of reduced space. The salmon coloured regions (including plain boundary points) are where the reduced space is a smooth 2-sphere, the black lines or dots are where it is a once-pointed 2-sphere, and the thick dashed lines represent where the reduced space is a point.
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Figure 1.  Three of the generic momentum polytopes from [23]. They illustrate in particular how the points $ a $ and $ b $ are always vertices, but the $ c_j $ may or may not be vertices of the polytope
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Figure 5.  The roots $\alpha_1, \alpha_2, \alpha_3$ and positive Weyl chamber of $SU(3)$, see 23
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Figure 6.  The green lines are contours of constant volume of reduced space; see Remark 3 and Fig. 4 for the key
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Table 1.  The allowed velocity spaces for relative equilibria for 2 vortices on $ \mathbb{CP}^2 $ (allowing for collisions).
$ \Gamma_1=\Gamma_2 $ equal: $ G_x=U(2)=G_\mu $ $R_0\simeq(\mathfrak{u}(2)/\mathfrak{u}(2))^{U(2)}=\{0\}$
orthogonal: $ G_x=\mathbb{T}^2 $, $ G_\mu= $U(2) $R_0\simeq(\mathfrak{u}(2)/\mathfrak{t}^2)^{\mathbb{T}^2}=\{0\}$
generic: $ G_x=U(1) $, $ G_\mu=\mathbb{T}^2 $ $ R_0\simeq(\mathfrak{t}^2/\mathfrak{u}(1))^{U(1)}=\mathbb{R} $
$ \Gamma_1=-\Gamma_2 $ equal: $ G_x=U(2) $, $ G_\mu=SU(3) $ $ R_0\simeq(\mathfrak{su}(3)/\mathfrak{u}(2))^{U(2)}=\left\{ 0 \right\} $
orthogonal: $ G_x=G_\mu=\mathbb{T}^2 $ $ R_0\simeq(\mathfrak{t}^2/\mathfrak{t}^2)^{\mathbb{T}^2}=\left\{ 0 \right\} $
generic: $ G_x=U(1) $, $ G_\mu=\mathbb{T}^2 $ $ R_0\simeq(\mathfrak{t}^2/\mathfrak{u}(1))^{U(1)}=\mathbb{R} $
Otherwise equal: $ G_x=G_\mu=U(2) $ $ R_0 = \left\{ 0 \right\} $
orthogonal: $ G_x=G_\mu=\mathbb{T}^2 $ $ R_0 = \left\{ 0 \right\} $
generic: $ G_x=U(1) $, $ G_\mu=\mathbb{T}^2 $ $ R_0\simeq(\mathfrak{t}^2/\mathfrak{u}(1))^{U(1)}=\mathbb{R} $
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$ \Gamma_1=\Gamma_2 $ equal: $ G_x=U(2)=G_\mu $ $R_0\simeq(\mathfrak{u}(2)/\mathfrak{u}(2))^{U(2)}=\{0\}$
orthogonal: $ G_x=\mathbb{T}^2 $, $ G_\mu= $U(2) $R_0\simeq(\mathfrak{u}(2)/\mathfrak{t}^2)^{\mathbb{T}^2}=\{0\}$
generic: $ G_x=U(1) $, $ G_\mu=\mathbb{T}^2 $ $ R_0\simeq(\mathfrak{t}^2/\mathfrak{u}(1))^{U(1)}=\mathbb{R} $
$ \Gamma_1=-\Gamma_2 $ equal: $ G_x=U(2) $, $ G_\mu=SU(3) $ $ R_0\simeq(\mathfrak{su}(3)/\mathfrak{u}(2))^{U(2)}=\left\{ 0 \right\} $
orthogonal: $ G_x=G_\mu=\mathbb{T}^2 $ $ R_0\simeq(\mathfrak{t}^2/\mathfrak{t}^2)^{\mathbb{T}^2}=\left\{ 0 \right\} $
generic: $ G_x=U(1) $, $ G_\mu=\mathbb{T}^2 $ $ R_0\simeq(\mathfrak{t}^2/\mathfrak{u}(1))^{U(1)}=\mathbb{R} $
Otherwise equal: $ G_x=G_\mu=U(2) $ $ R_0 = \left\{ 0 \right\} $
orthogonal: $ G_x=G_\mu=\mathbb{T}^2 $ $ R_0 = \left\{ 0 \right\} $
generic: $ G_x=U(1) $, $ G_\mu=\mathbb{T}^2 $ $ R_0\simeq(\mathfrak{t}^2/\mathfrak{u}(1))^{U(1)}=\mathbb{R} $
Table 2.  The allowed velocity spaces for relative equilibria for 3 vortices on $ \mathbb{CP}^2 $ (allowing for collisions), and for generic $ \Gamma_j $. See text for explanations.
$ \mu\in $ Wall: $ G_\mu=U(2) $
triple point $ G_x=U(2) $ $ R_0\simeq(\mathfrak{u}(2)/\mathfrak{u}(2))^{U(2)}=\left\{ 0 \right\} $
other vertices $ G_x=U(1) $ $ R_0\simeq(\mathfrak{u}(2)/\mathfrak{u}(1))^{U(1)} =\mathbb{R}^3 $
generic $ G_x=\bf{1} $ $ R_0\simeq\mathfrak{u}(2) =\mathbb{R}^4 $
$\mu\not\in$ Wall: $G_\mu=\mathbb{T}^2$
double point$G_x=U(1)$$R_0\simeq(\tt^2/\mathfrak{u}(1))^{U(1)}=\mathbb{R}$
double+orthogonal$G_x=\mathbb{T}^2$$R_0\simeq(\mathfrak{t}^2/\tt^2)^{\mathbb{T}^2}=\{0\}$
distinct coplanar$G_x=U(1)$$R_0\simeq(\mathfrak{t}^2/\mathfrak{u}(1))^{U(1)}=\mathbb{R}$
totally orthogonal$G_x=\mathbb{T}^2$$R_0\simeq(\mathfrak{t}^2/\tt^2)^{\mathbb{T}^2}=\{0\}$
semi-orthogonal$G_x=U(1)$$R_0\simeq(\mathfrak{t}^2/\mathfrak{u}(1))^{U(1)}=\mathbb{R}$
generic$G_x=\bf{1}$$R_0\simeq\mathfrak{t}^2=\mathbb{R}^2$
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$ \mu\in $ Wall: $ G_\mu=U(2) $
triple point $ G_x=U(2) $ $ R_0\simeq(\mathfrak{u}(2)/\mathfrak{u}(2))^{U(2)}=\left\{ 0 \right\} $
other vertices $ G_x=U(1) $ $ R_0\simeq(\mathfrak{u}(2)/\mathfrak{u}(1))^{U(1)} =\mathbb{R}^3 $
generic $ G_x=\bf{1} $ $ R_0\simeq\mathfrak{u}(2) =\mathbb{R}^4 $
$\mu\not\in$ Wall: $G_\mu=\mathbb{T}^2$
double point$G_x=U(1)$$R_0\simeq(\tt^2/\mathfrak{u}(1))^{U(1)}=\mathbb{R}$
double+orthogonal$G_x=\mathbb{T}^2$$R_0\simeq(\mathfrak{t}^2/\tt^2)^{\mathbb{T}^2}=\{0\}$
distinct coplanar$G_x=U(1)$$R_0\simeq(\mathfrak{t}^2/\mathfrak{u}(1))^{U(1)}=\mathbb{R}$
totally orthogonal$G_x=\mathbb{T}^2$$R_0\simeq(\mathfrak{t}^2/\tt^2)^{\mathbb{T}^2}=\{0\}$
semi-orthogonal$G_x=U(1)$$R_0\simeq(\mathfrak{t}^2/\mathfrak{u}(1))^{U(1)}=\mathbb{R}$
generic$G_x=\bf{1}$$R_0\simeq\mathfrak{t}^2=\mathbb{R}^2$
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