Hamiltonian Dynamics of Several Rigid Bodies Interacting with Point Vortices

We derive the dynamics of several rigid bodies of arbitrary shape in a two-dimensional inviscid and incompressible fluid, whose vorticity is given by point vortices. We adopt the idea of Vankerschaver et al. (J. Geom. Mech. 1(2): 223–226, 2009) to derive the Hamiltonian formulation via symplectic reduction from a canonical Hamiltonian system. The reduced system is described by a noncanonical symplectic form, which has previously been derived for a single circular disk using heavy differential-geometric machinery in an infinite-dimensional setting. In contrast, our derivation makes use of the fact that the dynamics of the fluid, and thus the point vortex dynamics, is determined from first principles. Using this knowledge we can directly determine the dynamics on the reduced, finite-dimensional phase space, using only classical mechanics. Furthermore, our approach easily handles several bodies of arbitrary shape. From the Hamiltonian description we derive a Lagrangian formulation, which enables the system for variational time integrators. We briefly describe how to implement such a numerical scheme and simulate different configurations for validation.

1. This is a special feature of the 2D case, where a single point vortex in an unbounded fluid domain will not move at all due to symmetry. In 3D, when considering vortex filaments, the self-energy has significant influence on the dynamics and cannot be ignored.

2. The physical meaning of these quantities is discussed in Sects. 3.1.3 and 3.1.4.

3. Any one-form \(\Theta \) on \(\mathrm {SE}(2)^n \times \mathbb {R}^{2 m}\) can be written as \(\Theta (\Xi , \dot{\gamma }) = \langle M_g(g, \gamma ), \Xi \rangle + \langle M_\gamma (g, \gamma ), \dot{\gamma } \rangle \).

4. As in the derivation of \(\sigma _{\text {can}}\) in Appendix 2, we use Eq. (39) and consider a two-parameter family \(g(s, t)\) with commuting partial derivatives. This makes the Lie bracket term in (39) vanish, but introduces the constraint (40) on \(\dot{\Gamma }\).

Ulrich Pinkall proposed the basic idea for deriving the symplectic form. It is my great pleasure to thank him for invaluable discussions and suggestions. Felix Knöppel and David Chubelaschwili helped to work out many of the details. Eva Kanso and the anonymous reviewers provided important feedback for improving the exposition. This work is supported by the DFG Research Center Matheon and the SFB/TR 109 “Discretization in Geometry and Dynamics.”

Authors and Affiliations

1. Institut für Mathematik, TU Berlin, Berlin, Germany

   Steffen Weißmann

Corresponding author

Correspondence to Steffen Weißmann.

Communicated by Eva Kanso.

Appendix 1: Cotangent Bundle Reduction of Fluid–Body Dynamics

In analogy to Arnold’s geometric description of fluid dynamics (Arnold 1966), the dynamics of rigid bodies interacting with a surrounding incompressible fluid can be viewed as a geodesic problem on a Riemannian manifold. The kinetic energy defines a Riemannian metric on the configuration space, and geodesics satisfy Hamilton’s equations on the cotangent bundle with kinetic energy as the Hamiltonian. This insight is due to Vankerschaver et al. (2009) (VKM). The authors use the framework of cotangent bundle reduction (Marsden et al. 2007) to obtain a reduced Hamiltonian system with magnetic symplectic form for the case of a single body in a fluid whose vorticity field is concentrated at point vortices.

The Hamiltonian formulation by VKM is an extension of Arnold’s original work (Arnold 1966), which describes the motion of an incompressible inviscid fluid in a fixed fluid domain \(\mathcal {F}\) as a geodesic on the group \(\mathop {\mathrm {Diff}}\nolimits _{\mathop {\mathrm {vol}}\nolimits }(\mathcal {F})\) of volume-preserving diffeomorphisms on \(\mathcal {F}\). However, when the fluid interacts with rigid bodies, the fluid domain is no longer fixed. The idea of VKM is to consider the space \(\mathop {\mathrm {Emb}}\nolimits _{\mathop {\mathrm {vol}}\nolimits }(\mathcal {F}^0, \mathbb {R}^2)\) of volume-preserving embeddings of an initial reference configuration \(\mathcal {F}^0\) into \(\mathbb {R}^2\) instead of \(\mathop {\mathrm {Diff}}\nolimits _{\mathop {\mathrm {vol}}\nolimits }(\mathcal {F})\). Any incompressible fluid motion is then described by a curve in the subset \(\mathcal {Q}^\mathcal {F}\subset \mathop {\mathrm {Emb}}\nolimits _{\mathop {\mathrm {vol}}\nolimits }(\mathcal {F}^0, \mathbb {R}^2)\) which is compatible with the body motion. The configuration space of the coupled system is \(\mathcal {Q} = \mathrm {SE}(2)^n \times \mathcal {Q}^\mathcal {F}\), and the dynamics is a canonical Hamiltonian system on \(T^* \mathcal {Q}\) with kinetic energy as the Hamiltonian.

The kinetic energy is invariant under volume-preserving diffeomorphisms of the initial fluid configuration \(\mathcal {F}^0\) (particle relabeling symmetry); i.e., the symmetry group \(\mathop {\mathrm {Diff}}\nolimits _{\mathop {\mathrm {vol}}\nolimits }(\mathcal {F}^0)\) acts from the right on \(\mathcal {Q}^\mathcal {F}\), and thus on \(\mathcal {Q}\). This action turns \(\mathcal {Q}\) into a principal fiber bundle over \(\mathrm {SE}(2)^n\). This structure allows one to follow the famous Kaluza–Klein approach to determine the Hamiltonian dynamics. In order to factor out the \(\mathop {\mathrm {Diff}}\nolimits _{\mathop {\mathrm {vol}}\nolimits }(\mathcal {F}^0)\)-symmetry one needs to fix a value of the associated momentum map, which corresponds to choosing an initial vorticity field of the fluid. This is where the assumption is used that vorticity is concentrated at \(m\) point vortices. The reduced phase space is \(\mathcal {M} = T^* SE(2)^n \times \mathbb {R}^{2 m}\) (see VKM, Sect. 4.2), and the dynamics is given by a reduced symplectic form \(\sigma \) on \(\mathcal {M}\) with kinetic energy as the Hamiltonian. The following theorem formulates the starting point for the derivations made in this paper:

Theorem 2

The dynamics of \(n\) rigid bodies interacting with \(m\) isolated point vortices is a Hamiltonian system. The Hamiltonian is the kinetic energy (21), and the phase space is

The cotangent bundle \(T^* \mathrm {SE}(2)^n\) corresponds to the rigid body configuration, and \(\mathbb {R}^{2 m}\) is the phase space for \(m\) point vortices. The symplectic form is

where \(\sigma _{\text {can}}\) is the canonical symplectic form on the cotangent bundle \(T^*\mathrm {SE}(2)^n\), \(\sigma _\gamma \) is the Kirillov–Kostant–Sariou form on the coadjoint orbit \(\mathbb {R}^{2 m}\), and \(\mathrm {d}\alpha \) is a magnetic term, i.e., a two-form on \(SE(2)^n \times \mathbb {R}^{2 m}\).

Proof

This is proven in VKS, Sect. 4. We emphasize here that the proofs do not rely on the fact that only a single rigid body was considered.

Appendix 2: The Cotangent Bundle of Euclidean Motions

In this appendix we consider the Lie group of Euclidean motions \(\mathrm {SE}(2)\) and denote the pairing between covectors and vectors by \(\left( \cdot , \cdot \right) \). For any covector \(\mu \in T_g^* \mathrm {SE}(2)\) we can find a body momentum \(M \in \mathbb {R}^3 \cong \mathfrak {se}^*(2)\) such that \(\left( \mu , \delta g\right) := \langle M, \Lambda \rangle \), for any \(\delta g = g \Lambda \). Note that \(\Theta (\delta \mu , \delta g) := \left( \mu , \delta g\right) \) is a one-form on the cotangent bundle \(T^*\mathrm {SE}(2)\), and \(\langle M, \Lambda \rangle \) is its push-forward to the left trivialization \(\mathbb {R}^3 \times \mathrm {SE}(2) \cong T^*\mathrm {SE}(2)\). It is the canonical one-form, and its exterior derivative gives the canonical symplectic form on \(T^*\mathrm {SE}(2)\). We now compute the symplectic form when pushed forward to the left trivialization \(\mathbb {R}^3 \times \mathrm {SE}(2)\), using the general formula for the exterior derivative of a one-form:

Here \(X\) and \(Y\) are vector fields and \([X, Y]\) is the Jacobi–Lie bracket of \(X\) and \(Y\). Consider a two-parameter family \((M(s,t), g(s,t))\) in \(\mathbb {R}^3 \times \mathrm {SE}(2)\), whose partial derivatives (denoted by \(\delta \) and \('\), respectively) commute. The vector fields will be \(X = (\delta M, \delta g)\) and \(Y = (M', g')\), where \(\delta g = g \Gamma \) and \(g' = g \Xi \) with \(\Xi = (\varOmega , V)\). One can check that the partial derivatives of \(g\) commute if and only if

The commuting partial derivatives ensure that the Jacobi–Lie bracket in (39) vanishes. The covariant derivatives are usual directional derivatives here, so we obtain the canonical symplectic two-form \(\sigma = \mathrm {d}\Theta \) in the left trivialization as

Here \(\mathop {\mathrm {ad}^*_{\Xi }}\nolimits \) is the matrix transpose of \(\mathop {\mathrm {ad}_{\Xi }}\nolimits \):

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