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Journal of Geometric Mechanics

December 2019 , Volume 11 , Issue 4

Special issue dedicated to Darryl D. Holm on the occasion of his 70th birthday. II

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Networks of coadjoint orbits: From geometric to statistical mechanics
Alexis Arnaudon and So Takao
2019, 11(4): 447-485 doi: 10.3934/jgm.2019023 +[Abstract](39) +[HTML](19) +[PDF](5188.61KB)
Abstract:

A class of network models with symmetry group \begin{document}$ G $\end{document} that evolve as a Lie-Poisson system is derived from the framework of geometric mechanics, which generalises the classical Heisenberg model studied in statistical mechanics. We considered two ways of coupling the spins: one via the momentum and the other via the position and studied in details the equilibrium solutions and their corresponding nonlinear stability properties using the energy-Casimir method. We then took the example \begin{document}$ G = SO(3) $\end{document} and saw that the momentum-coupled system reduces to the classical Heisenberg model with massive spins and the position-coupled case reduces to a new system that has a broken symmetry group \begin{document}$ SO(3)/SO(2) $\end{document} similar to the heavy top. In the latter system, we numerically observed an interesting synchronisation-like phenomenon for a certain class of initial conditions. Adding a type of noise and dissipation that preserves the coadjoint orbit of the network model, we found that the invariant measure is given by the Gibbs measure, from which the notion of temperature is defined. We then observed a surprising 'triple-humped' phase transition in the heavy top-like lattice model, where the spins switched from one equilibrium position to another before losing magnetisation as we increased the temperature. This work is only a first step towards connecting geometric mechanics with statistical mechanics and several interesting problems are open for further investigation.

Morse families and Dirac systems
María Barbero Liñán, Hernán Cendra, Eduardo García Toraño and David Martín de Diego
2019, 11(4): 487-510 doi: 10.3934/jgm.2019024 +[Abstract](39) +[HTML](40) +[PDF](445.94KB)
Abstract:

Dirac structures and Morse families are used to obtain a geometric formalism that unifies most of the scenarios in mechanics (constrained calculus, nonholonomic systems, optimal control theory, higher-order mechanics, etc.), as the examples in the paper show. This approach generalizes the previous results on Dirac structures associated with Lagrangian submanifolds. An integrability algorithm in the sense of Mendella, Marmo and Tulczyjew is described for the generalized Dirac dynamical systems under study to determine the set where the implicit differential equations have solutions.

Variational integrators for anelastic and pseudo-incompressible flows
Werner Bauer and François Gay-Balmaz
2019, 11(4): 511-537 doi: 10.3934/jgm.2019025 +[Abstract](40) +[HTML](18) +[PDF](1479.71KB)
Abstract:

The anelastic and pseudo-incompressible equations are two well-known soundproof approximations of compressible flows useful for both theoretical and numerical analysis in meteorology, atmospheric science, and ocean studies. In this paper, we derive and test structure-preserving numerical schemes for these two systems. The derivations are based on a discrete version of the Euler-Poincaré variational method. This approach relies on a finite dimensional approximation of the (Lie) group of diffeomorphisms that preserve weighted-volume forms. These weights describe the background stratification of the fluid and correspond to the weighted velocity fields for anelastic and pseudo-incompressible approximations. In particular, we identify to these discrete Lie group configurations the associated Lie algebras such that elements of the latter correspond to weighted velocity fields that satisfy the divergence-free conditions for both systems. Defining discrete Lagrangians in terms of these Lie algebras, the discrete equations follow by means of variational principles. Descending from variational principles, the schemes exhibit further a discrete version of Kelvin circulation theorem, are applicable to irregular meshes, and show excellent long term energy behavior. We illustrate the properties of the schemes by performing preliminary test cases.

The problem of Lagrange on principal bundles under a subgroup of symmetries
Marco Castrillón López and Pedro Luis García Pérez
2019, 11(4): 539-552 doi: 10.3934/jgm.2019026 +[Abstract](43) +[HTML](21) +[PDF](345.31KB)
Abstract:

Given a Lagrangian density \begin{document}$ L{\bf{v}} $\end{document} defined on the \begin{document}$ 1 $\end{document}-jet extension \begin{document}$ J^1P $\end{document} of a principal \begin{document}$ G $\end{document}-bundle \begin{document}$ \pi \colon P\to M $\end{document} invariant under the action of a closed subgroup \begin{document}$ H\subset G $\end{document}, its Euler-Poincaré reduction in \begin{document}$ J^1P/H = C(P)\times_M P/H $\end{document} (\begin{document}$ C(P)\to M $\end{document} being the bundle of connections of \begin{document}$ P $\end{document} and \begin{document}$ P/H\to M $\end{document} being the bundle of \begin{document}$ H $\end{document}-structures) induces a Lagrange problem defined in \begin{document}$ J^1(C(P)\times_M P/H) $\end{document} by a reduced Lagrangian density \begin{document}$ l{\bf{v}} $\end{document} together with the constraints \begin{document}$ {\rm{Curv}}\sigma = 0, \nabla ^\sigma \bar{s} = 0 $\end{document}, for \begin{document}$ \sigma $\end{document} and \begin{document}$ \bar{s} $\end{document} sections of \begin{document}$ C(P) $\end{document} and \begin{document}$ P/H $\end{document} respectively. We prove that the critical section of this problem are solutions of the Euler-Poincaré equations of the reduced problem. We also study the Hamilton-Cartan formulation of this Lagrange problem, where we find some common points with Pontryagin's approach to optimal control problems for \begin{document}$ \sigma $\end{document} as control variables and \begin{document}$ \bar{s} $\end{document} as dynamical variables. Finally, the theory is illustrated with the case of affine principal fiber bundles and its application to the modelisation of the molecular strands on a Lorentzian plane.

Navier-Stokes and stochastic Navier-Stokes equations via Lagrange multipliers
Ana Bela Cruzeiro
2019, 11(4): 553-560 doi: 10.3934/jgm.2019027 +[Abstract](33) +[HTML](17) +[PDF](241.53KB)
Abstract:

We show that the Navier-Stokes as well as a random perturbation of this equation can be derived from a stochastic variational principle where the pressure is introduced as a Lagrange multiplier. Moreover we describe how to obtain corresponding constants of the motion.

Remarks on certain two-component systems with peakon solutions
Mike Hay, Andrew N. W. Hone, Vladimir S. Novikov and Jing Ping Wang
2019, 11(4): 561-573 doi: 10.3934/jgm.2019028 +[Abstract](42) +[HTML](25) +[PDF](319.59KB)
Abstract:

We consider a Lax pair found by Xia, Qiao and Zhou for a family of two-component analogues of the Camassa-Holm equation, including an arbitrary function \begin{document}$ H $\end{document}, and show that this apparent freedom can be removed via a combination of a reciprocal transformation and a gauge transformation, which reduces the system to triangular form. The resulting triangular system may or may not be integrable, depending on the choice of \begin{document}$ H $\end{document}. In addition, we apply the formal series approach of Dubrovin and Zhang to show that scalar equations of Camassa-Holm type with homogeneous nonlinear terms of degree greater than three are not integrable.

Non-Abelian momentum polytopes for products of $ \mathbb{CP}^2 $
James Montaldi and Amna Shaddad
2019, 11(4): 575-599 doi: 10.3934/jgm.2019029 +[Abstract](30) +[HTML](17) +[PDF](488.79KB)
Abstract:

This is the first of two companion papers. The joint aim is to study a generalization to higher dimension of the familiar point vortex systems in 2 dimensions. In this paper we classify the momentum polytopes for the action of the Lie group SU(3) on products of copies of complex projective 2-space (a real 4-dimensional manifold). For 2 copies, the momentum polytope is simply a line segment, which can sit in the positive Weyl chamber in a small number of ways. For a product of 3 copies there are 8 different types of generic momentum polytope, and numerous transition polytopes, all of which are classified here. The type of polytope depends on the weights of the symplectic form on each copy of projective space. In the second paper we use techniques of symplectic reduction to study the possible dynamics of interacting generalized point vortices.

The results of this paper can be applied to determine the inequalities satisfied by the eigenvalues of the sum of up to three 3x3 Hermitian matrices where each has a double eigenvalue.

Generalized point vortex dynamics on $ \mathbb{CP} ^2 $
James Montaldi and Amna Shaddad
2019, 11(4): 601-619 doi: 10.3934/jgm.2019030 +[Abstract](37) +[HTML](24) +[PDF](949.04KB)
Abstract:

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 \begin{document}$ \mathbb{CP} ^2 $\end{document} 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 \begin{document}$ \mathbb{CP} ^2 $\end{document}. 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.

Global well-posedness of a 3D MHD model in porous media
Edriss S. Titi and Saber Trabelsi
2019, 11(4): 621-637 doi: 10.3934/jgm.2019031 +[Abstract](39) +[HTML](20) +[PDF](338.88KB)
Abstract:

In this paper we show the global well-posedness of solutions to a three-dimensional magnetohydrodynamical (MHD) model in porous media. Compared to the classical MHD equations, our system involves a nonlinear damping term in the momentum equations due to the "Brinkman-Forcheimer-extended-Darcy" law of flow in porous media.

Momentum maps for mixed states in quantum and classical mechanics
Cesare Tronci
2019, 11(4): 639-656 doi: 10.3934/jgm.2019032 +[Abstract](36) +[HTML](49) +[PDF](401.35KB)
Abstract:

This paper presents the momentum map structures which emerge in the dynamics of mixed states. Both quantum and classical mechanics are shown to possess analogous momentum map pairs associated to left and right group actions. In the quantum setting, the right leg of the pair identifies the Berry curvature, while its left leg is shown to lead to different realizations of the density operator, which are of interest in quantum molecular dynamics. Finally, the paper shows how alternative representations of both the density matrix and the classical density are equivariant momentum maps generating new Clebsch representations for both quantum and classical dynamics. Uhlmann's density matrix [58] and Koopman wavefunctions [41] are shown to be special cases of this construction.

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