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

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

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  • 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.

    Mathematics Subject Classification: 37J15, 53D20, 70H06.


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  • Figure 3.  Possible non-trivial reduced spaces for the 3-vortex problem

    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 $.)

    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.

    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

    Figure 5.  The roots $\alpha_1, \alpha_2, \alpha_3$ and positive Weyl chamber of $SU(3)$, see 23

    Figure 6.  The green lines are contours of constant volume of reduced space; see Remark 3 and Fig. 4 for the key

    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} $
     | Show Table
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    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)$$R_0\simeq(\tt^2/\mathfrak{u}(1))^{U(1)}=\mathbb{R}$
    distinct coplanar$G_x=U(1)$$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\}$
     | Show Table
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