The free rigid body dynamics: generalized versus classic

In this paper we analyze the normal forms of a general quadratic Hamiltonian system defined on the dual of the Lie algebra $\mathfrak{o}(K)$ of real $K$ - skew - symmetric matrices, where $K$ is an arbitrary $3\times 3$ real symmetric matrix. A consequence of the main results is that any first-order autonomous three-dimensional differential equation possessing two independent quadratic constants of motion which admits a positive/negative definite linear combination, is affinely equivalent to the classical "relaxed" free rigid body dynamics with linear controls.Comment: 12 page

New variational and multisymplectic formulations of the Euler-Poincar\'e equation on the Virasoro-Bott group using the inverse map

We derive a new variational principle, leading to a new momentum map and a new multisymplectic formulation for a family of Euler--Poincar\'e equations defined on the Virasoro-Bott group, by using the inverse map (also called `back-to-labels' map). This family contains as special cases the well-known Korteweg-de Vries, Camassa-Holm, and Hunter-Saxton soliton equations. In the conclusion section, we sketch opportunities for future work that would apply the new Clebsch momentum map with $2$-cocycles derived here to investigate a new type of interplay among nonlinearity, dispersion and noise.Comment: 19 page

Two-component {CH} system: Inverse Scattering, Peakons and Geometry

An inverse scattering transform method corresponding to a Riemann-Hilbert problem is formulated for CH2, the two-component generalization of the Camassa-Holm (CH) equation. As an illustration of the method, the multi - soliton solutions corresponding to the reflectionless potentials are constructed in terms of the scattering data for CH2.Comment: 22 pages, 3 figures, draft, please send comment

Continuous and discrete Clebsch variational principles

The Clebsch method provides a unifying approach for deriving variational principles for continuous and discrete dynamical systems where elements of a vector space are used to control dynamics on the cotangent bundle of a Lie group \emph{via} a velocity map. This paper proves a reduction theorem which states that the canonical variables on the Lie group can be eliminated, if and only if the velocity map is a Lie algebra action, thereby producing the Euler-Poincar\'e (EP) equation for the vector space variables. In this case, the map from the canonical variables on the Lie group to the vector space is the standard momentum map defined using the diamond operator. We apply the Clebsch method in examples of the rotating rigid body and the incompressible Euler equations. Along the way, we explain how singular solutions of the EP equation for the diffeomorphism group (EPDiff) arise as momentum maps in the Clebsch approach. In the case of finite dimensional Lie groups, the Clebsch variational principle is discretised to produce a variational integrator for the dynamical system. We obtain a discrete map from which the variables on the cotangent bundle of a Lie group may be eliminated to produce a discrete EP equation for elements of the vector space. We give an integrator for the rotating rigid body as an example. We also briefly discuss how to discretise infinite-dimensional Clebsch systems, so as to produce conservative numerical methods for fluid dynamics

Complete integrability versus symmetry

The purpose of this article is to show that on an open and dense set, complete integrability implies the existence of symmetry

The Hamiltonian structure and Euler-Poincar\'{e} formulation of the Vlasov-Maxwell and gyrokinetic systems

We present a new variational principle for the gyrokinetic system, similar to the Maxwell-Vlasov action presented in Ref. 1. The variational principle is in the Eulerian frame and based on constrained variations of the phase space fluid velocity and particle distribution function. Using a Legendre transform, we explicitly derive the field theoretic Hamiltonian structure of the system. This is carried out with a modified Dirac theory of constraints, which is used to construct meaningful brackets from those obtained directly from Euler-Poincar\'{e} theory. Possible applications of these formulations include continuum geometric integration techniques, large-eddy simulation models and Casimir type stability methods. [1] H. Cendra et. al., Journal of Mathematical Physics 39, 3138 (1998)Comment: 36 pages, 1 figur

An Optimal Control Formulation for Inviscid Incompressible Ideal Fluid Flow

In this paper we consider the Hamiltonian formulation of the equations of incompressible ideal fluid flow from the point of view of optimal control theory. The equations are compared to the finite symmetric rigid body equations analyzed earlier by the authors. We discuss various aspects of the Hamiltonian structure of the Euler equations and show in particular that the optimal control approach leads to a standard formulation of the Euler equations -- the so-called impulse equations in their Lagrangian form. We discuss various other aspects of the Euler equations from a pedagogical point of view. We show that the Hamiltonian in the maximum principle is given by the pairing of the Eulerian impulse density with the velocity. We provide a comparative discussion of the flow equations in their Eulerian and Lagrangian form and describe how these forms occur naturally in the context of optimal control. We demonstrate that the extremal equations corresponding to the optimal control problem for the flow have a natural canonical symplectic structure.Comment: 6 pages, no figures. To appear in Proceedings of the 39th IEEEE Conference on Decision and Contro

Fluctuations of the vortex line density in turbulent flows of quantum fluids

We present an analytical study of fluctuations of the Vortex Line Density (VLD) $$ in turbulent flows of quantum fluids. Two cases are considered. The first one is the counterflowing (Vinen) turbulence, where the vortex lines are disordered, and the evolution of quantity $\mathcal{L}(t)$ obeys the Vinen equation. The second case is the quasi-classic turbulence, where vortex lines are believed to form the so called vortex bundles, and their dynamics is described by the HVBK equations. The latter case, is of a special interest, since a number of recent experiments demonstrate the $\omega ^{-5/3}$ dependence for spectrum VLD, instead of $\omega ^{1/3}$ law, typical for spectrum of vorticity. In nonstationary situation, in particular, in the fluctuating turbulent flow there is a retardation between the instantaneous value of the normal velocity and the quantity $\mathcal{L}$. This retardation tends to decrease in the accordance with the inner dynamics, which has a relaxation character. In both cases the relaxation dynamics of VLD is related to fluctuations of the relative velocity, however if for the Vinen case the rate of temporal change for $\mathcal{L}(t)$ is directly depends on $\delta \mathbf{v}_{ns}$, for the HVBK dynamics it depends on $\nabla \times \delta \mathbf{v}_{ns}$. As a result, for the disordered case the spectrum $<\delta \mathcal{L}(\omega) \delta \mathcal{L}(-\omega)>$ coincides with the spectrum $\omega ^{-5/3}$. In the case of the bundle arrangement, the spectrum of the VLD varies (at different temperatures) from $\omega ^{1/3}$ to $\omega ^{-5/3}$ dependencies. This conclusion may serve as a basis for the experimental determination of what kind of the turbulence is implemented in different types of generation.Comment: 8 pages, 29 reference

Finite Size Polyelectrolyte Bundles at Thermodynamic Equilibrium

We present the results of extensive computer simulations performed on solutions of monodisperse charged rod-like polyelectrolytes in the presence of trivalent counterions. To overcome energy barriers we used a combination of parallel tempering and hybrid Monte Carlo techniques. Our results show that for small values of the electrostatic interaction the solution mostly consists of dispersed single rods. The potential of mean force between the polyelectrolyte monomers yields an attractive interaction at short distances. For a range of larger values of the Bjerrum length, we find finite size polyelectrolyte bundles at thermodynamic equilibrium. Further increase of the Bjerrum length eventually leads to phase separation and precipitation. We discuss the origin of the observed thermodynamic stability of the finite size aggregates