Abstract: This paper presents a geometric-variational approach to continuous and discrete mechanics and field theories. Using multisymplectic geometry, we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle. In particular, we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function, and use this structure to prove covariant generalizations of conservation of symplecticity and Noether’s theorem. Natural discretization schemes for PDEs, which have these important preservation properties, then follow by choosing a discrete action functional. In the case of mechanics, we recover the variational symplectic integrators of Veselov type, while for PDEs we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries. We show that the usual notion of symplecticity along an infinite-dimensional space of fields can be naturally obtained by making a spacetime split. All of the aspects of our method are demonstrated with a nonlinear sine-Gordon equation, including computational results and a comparison with other discretization

We review the properties of the nonlinearly dispersive Navier–Stokes-alpha (NS-α) model of incompressible fluid turbulence — also called the viscous Camassa–Holm equations in the literature. We first re-derive the NS-α model by filtering the velocity of the fluid loop in Kelvin’s circulation theorem for the Navier–Stokes equations. Then we show that this filtering causes the wavenumber spectrum of the translational kinetic energy for the NS-α model to roll off as k −3 for kα> 1 in three dimensions, instead of continuing along the slower Kolmogorov scaling law, k −5/3, that it follows for kα < 1. This roll off at higher wavenumbers shortens the inertial range for the NS-α model and thereby makes it more computable. We also explain how the NS-α model is related to large eddy simulation (LES) turbulence modeling and to the stress tensor for second-grade fluids. We close by surveying recent results in the literature for the NS-α model and its inviscid limit (the Euler-α model).

Abstract. In this paper we discuss recent progress in using the Camassa-Holm equations to model turbulent flows. The Camassa-Holm equations, given their special geometric and physical properties, appear particularly well suited for studying turbulent flows. We identify the steady solution of the Camassa-Holm equation with the mean flow of the Reynolds equation and compare the results with empirical data for turbulent flows in channels and pipes. The data suggests that the constant α version of the Camassa-Holm equations, derived under the assumptions that the fluctuation statistics are isotropic and homogeneous, holds to order α distance from the boundaries. Near a boundary, these assumptions are no longer valid and the length scale α is seen to depend on the distance to the nearest wall. Thus, a turbulent flow is divided into two regions: the constant α region away from boundaries, and the near wall region. In the near wall region, Reynolds number scaling conditions imply that α decreases as Reynolds number increases. Away from boundaries, these scaling conditions imply α is independent of Reynolds number. Given the agreement with empirical and numerical data, our current work indicates that the Camassa-Holm equations provide a promising theoretical framework from which to understand some turbulent flows.

This paper surveys selected recent progress in geometric mechanics, focussing on Lagrangian reduction and gives some new applications to nonholonomic systems, that is, mechanical systems with constraints typified by rolling without slipping. Reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, Poincaré and others. The modern vision of mechanics includes, besides the traditional mechanics of particles and rigid bodies, field theories such as electromagnetism, fluid mechanics, plasma physics, solid mechanics as well as quantum mechanics, and relativistic theories, including gravity. Symmetries in mechanics ranges from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas, to subtle symmetries underlying integrable systems. Reduction theory concerns the removal of symmetries and utilizing their associated conservation laws. Reduction theory has been extremely useful in a wide variety of areas, from a deeper understanding of many

Reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, Poincaré and others. The modern vision of mechanics includes, besides the traditional mechanics of particles and rigid bodies, field theories such as electromagnetism, fluid mechanics, plasma physics, solid mechanics as well as quantum mechanics, and relativistic theories, including gravity. Symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas, to subtle symmetries underlying integrable systems. Reduction theory concerns the removal of symmetries and their associated conservation laws. Variational principles along with symplectic and Poisson geometry, provide fundamental tools for this endeavor. Reduction theory has been extremely useful in a wide variety of areas, from a deeper understanding of many physical theories, including new variational and Poisson structures, stability theory, integrable systems, as well as geometric phases.

Cotangent bundle reduction theory is a basic and well developed subject in which one performs symplectic reduction on cotangent bundles. One starts with a (free and proper) action of a Lie group G on a configuration manifold Q, considers its natural cotangent lift to T ∗ Q and then one seeks realizations of the corresponding symplectic or Poisson reduced space. We further develop this theory by explicitly identifying the symplectic leaves of the Poisson manifold T ∗ Q/G, decomposed as a Whitney sum bundle, T ∗ (Q/G) � �g ∗ over Q/G. The splitting arises naturally from a choice of connection on the G-principal bundle Q → Q/G. The symplectic leaves are computed and a formula for the reduced symplectic form is found.

This paper proves a symplectic reduction by stages theorem in the context of geometric mechanics on symplectic manifolds with symmetry groups that are group extensions. We relate the work to the semidirect product reduction theory developed in the 1980’s by Marsden, Ratiu, Weinstein, Guillemin and Sternberg as well as some more recent results and we recall how semidirect product reduction finds use in examples, such as the dynamics of an underwater vehicle. We shall start with the classical cases of commuting reduction (first appearing in Marsden and Weinstein [1974]) and present a new proof 1 and approach to semidirect product theory. We shall then give an idea of how the more general theory of group extensions proceeds (the details of which are given in Marsden, Misio̷lek, Perlmutter and Ratiu [1998]). The case of central extensions is illustrated in this paper with

We begin by placing the generalized Lagrangian mean (GLM) equations for a compressible adiabatic fluid into the Euler–Poincaré (EP) variational framework of fluid dynamics, for an averaged Lagrangian. We then state the EP Averag-ing Result—that GLM equations arise from GLM Hamilton’s principles in the EP framework. Next, we derive a new set of approximate small-amplitude GLM equations (gm equations) at second order in the fluctuating displacement of a Lagrangian trajectory from its mean position. These equations express the linear and nonlinear back-reaction effects on the Eulerian mean fluid quantities by the fluctuating displacements of the Lagrangian trajectories in terms of their Eulerian second moments. The derivation of the gm equations uses the linearized relations between Eulerian and Lagrangian fluctuations, in the tradition of Lagrangian stability analysis for fluids. The gm derivation also uses the method of averaged Lagrangians, in the tradition of wave, mean flow interaction (WMFI). The gm EP motion equations for compressible and incompressible ideal fluids are compared with the Euler-alpha turbulence closure equations. An alpha model is a GLM (or gm) fluid theory with a Taylor hypothesis closure (THC). Such closures are based on the linearized fluctuation relations that determine the dynamics of the Lagrangian statistical quantities in the Euler-alpha closure equations. We use the EP Averaging Result to bridge between the GLM equations and the Euler-alpha closure equations. Hence, combining the small-amplitude approximation with THC yields in new gm turbulence closure equations for compressible fluids in the EP variational framework.

This paper develops the geometry and analysis of the averaged Euler equations for ideal incompressible flow in domains in Euclidean space and on Riemannian manifolds, possibly with boundary. The averaged Euler equations involve a parameter α; one interpretation is that they are obtained by ensemble averaging the Euler equations in Lagrangian representation over rapid fluctuations whose amplitudes are of order α. The particle flows associated with these equations are shown to be geodesics on a suitable group of volume preserving diffeomorphisms, just as with the Euler equations themselves (according to Arnold’s theorem), but with respect to a right invariant H 1 metric instead of the L 2 metric. The equations are also equivalent to those for a certain second grade fluid. Additional properties of the Euler equations, such as smoothness of the geodesic spray (the Ebin-Marsden theorem) are also shown to hold. Using this nonlinear analysis framework, the limit of zero viscosity for the corresponding viscous equations is shown to be a regular limit, even in the

This paper was also one of the first to notice deep links between reduction and integrable systems, a subject continued by, for example, Bobenko, Reyman and Semenov-Tian-Shansky [1989].