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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A class of network models with symmetry group
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.
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.
Given a Lagrangian density
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.
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
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.
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
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
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.
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 [
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