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The geometric nature of Euler fluids has been clearly identified and extensively studied over the years, culminating with Lagrangian and Hamiltonian descriptions of fluid dynamics where the configuration space is defined as the volume-preserving diffeomorphisms, and Kelvin's circulation theorem is viewed as a consequence of Noether's theorem associated with the particle relabeling symmetry of fluid mechanics. However computational approaches to fluid mechanics have been largely derived from a numerical-analytic point of view, and are rarely designed with structure preservation in mind, and often suffer from spurious numerical artifacts such as energy and circulation drift. In contrast, this paper geometrically derives discrete equations of motion for fluid dynamics from first principles in a purely Eulerian form. Our approach approximates the group of volume-preserving diffeomorphisms using a finite dimensional Lie group, and associated discrete Euler equations are derived from a variational principle with non-holonomic constraints. The resulting discrete equations of motion yield a structure-preserving time integrator with good long-term energy behavior and for which an exact discrete Kelvin's circulation theorem holds.

In this work, motivated by the problem of model-based predictive control of separated flows, we identify the key variables and the requirements on a model-based observer and construct a prototype low-dimensional model to be embedded in control applications. Namely, using a phenomenological physics-based approach and dynamical systems and singularity theories, we uncover the low-dimensional nature of the complex dynamics of actuated separated flows and capture the crucial bifurcation and hysteresis inherent in separation phenomena. This new look at the problem naturally leads to several important implications, such as, firstly, uncovering the physical mechanisms for hysteresis, secondly, predicting a finite amplitude instability of the bubble, and, thirdly, to new issues to be studied theoretically and tested experimentally.

In this work we discuss the instabilities in mechanical systems caused by two fundamentally different non-conservative forces, referred to as dissipative and positional forces, each of which may lead to energy dissipation. One of the objectives of this discussion is to recall and to put into the context of current research some of the important classical results by Thomson–Tait–Chetayev and Merkin, which are under-appreciated nowadays: many new examples, e.g. radiation in Hamiltonian systems, the Levitron, etc., appearing in recent literature can be interpreted with the help of these classical results. Next, in the spirit of the Lagrange–Dirichlet theory, we introduce the geometric picture of the phase space corresponding to the effects of destabilization in finite-dimensional systems. On the physical side, our objective is to demonstrate that both of these types of non-conservative forces appear quite commonly and often simultaneously in physical systems. As an illustration, we consider the Levitron, a system in which the dissipative effects have not heretofore been studied. Finally, using Nikolai’s elastic bar model as a paradigm, we discuss the notion of a secondary dissipation-induced instability, when the above two fundamental non-conservative forces interact. This unified way of thinking should help us to understand the intricate links among various mechanical systems through the most fundamental mechanisms in their behavior

This paper develops the theory and computation of Lagrangian Coherent Structures (LCS), which are defined as ridges of Finite-Time Lyapunov Exponent (FTLE) fields. These ridges can be seen as finite-time mixing templates. Such a framework is common in dynamical systems theory for autonomous and time-periodic systems, in which examples of LCS are stable and unstable manifolds of fixed points and periodic orbits. The concepts defined in this paper remain applicable to flows with arbitrary time dependence and, in particular, to flows that are only defined (computed or measured) over a finite interval of time. Previous work has demonstrated the usefulness of FTLE fields and the associated LCSs for revealing the Lagrangian behavior of systems with general time dependence. However, ridges of the FTLE field need not be exactly advected with the flow. The main result of this paper is an estimate for the flux across an LCS, which shows that the flux is small, and in most cases negligible, for well-defined LCSs or those that rotate at a speed comparable to the local Eulerian velocity field, and are computed from FTLE fields with a sufficiently long integration time. Under these hypotheses, the structures represent nearly invariant manifolds even in systems with arbitrary time dependence. The results are illustrated on three examples. The first is a simplified dynamical model of a double-gyre flow. The second is surface current data collected by high-frequency radar stations along the coast of Florida and the third is unsteady separation over an airfoil. In all cases, the existence of LCSs governs the transport and it is verified numerically that the flux of particles through these distinguished lines is indeed negligible.

This paper focuses on methods of constructing of reduced-order models of mechanical systems which preserve the Lagrangian structure of the original system. These methods may be used in combination with standard spatial decomposition methods, such as the Karhunen–Loeve expansion, balancing, and wavelet decompositions. The model reduction procedure is implemented for three-dimensional finite-element models of elasticity, and we show that using the standard Newmark implicit integrator, significant savings are obtained in the computational costs of simulation. In particular simulation of the reduced model scales linearly in the number of degrees of freedom, and parallelizes well.

We present a method for applying the Karhunen-Lo`eve decomposition to systems with continuous symmetry. The techniques in this paper contribute to the general procedure of removing variables associated with the symmetry of a problem, and related ideas have been used in previous works both to identify coherent structures in solutions of PDEs, and to derive low-order models via Galerkin projection. The main result of this paper is to derive a simple and easily implementable set of reconstruction equations which close the system of ODEs produced by Galerkin projection. The geometric interpretation of the method closely parallels techniques used in geometric phases and reconstruction techniques in geometric mechanics. We apply the method to the Kuramoto- Sivashinsky equation and are able to derive accurate models of considerably lower dimension than are possible with the traditional Karhunen-Loève expansion.

Hamiltonian Lie-Poisson structures of the three-wave equations associated with the Lie algebras su(3) and su(2; 1) are derived and shown to be compatible. Poisson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated. These results can be applied to applications of nonlinear-waves in, for instance, nonlinear optics. Some of the general structures presented in the latter part of this paper are implicit in the literature; our purpose is to put the three-wave interaction in the modern setting of geometric mechanics and to explore some new things, such as explicit geometric phase formulas, as well as some old things, such as integrability, in this context.

This paper develops the stability theory of relative equilibria for mechanical systems with symmetry. It is especially concerned with systems that have a noncompact symmetry group, such as the group of Euclidean motions, and with relative equilibria for such symmetry groups. For these systems with rigid motion symmetry, one gets stability but possibly with drift in certain rotational as well as translational directions. Motivated by questions on stability of underwater vehicle dynamics, it is of particular interest that, in some cases, we can allow the relative equilibria to have nongeneric values of their momentum. The results are proved by combining theorems of Patrick with the technique of reduction by stages. This theory is then applied to underwater vehicle dynamics. The stability of specific relative equilibria for the underwater vehicle is studied. For example, we find conditions for Liapunov stability of the steadily rising and possibly spinning, bottom-heavy vehicle, which corresponds to a relative equilibrium with nongeneric momentum. The results of this paper should prove useful for the control of underwater vehicles.

We derive and analyze several low dimensional Hamiltonian normal forms describing system symmetry breaking in ideal hydrodynamics. The equations depend on two parameters (^є, λ), where ^є is the strength of a system symmetry breaking perturbation and λ is a detuning parameter. In many cases the resulting equations are completely integrable and have an interesting Hamiltonian structure. Our work is motivated by three-dimensional instabilities of rotating columnar fluid flows with circular streamlines (such as the Burger vortex) subjected to precession, elliptical distortion or off-center displacement.

The Poisson bracket formulation of fluid, plasma and rigid body type systems has undergone considerable recent development using techniques of symmetry group reduction. The relationship between this approach and that using Lin constraints and Clebsch potentials is established. The connection is made in the setting of abstract Clebsch variables as well as that of variational principles on reduced spaces. Variational principles for both the Clebsch and reduced form (such as fluids in spatial representation) are derived from the standard variational principle of Hamilton in material (Lagrangian) representation using reduction theory.

We study generic bifurcations of equilibria in one-parameter Hamiltonian systems with symmetry group Γ on the generalized eigenvalues of the linearized system go through zero. Theorem 3.3 classifies expected actions of Γ on the generalized eigenspace of this zero eigenvalue. Generic one degree of freedom symmetric systems is section 4; remarks concerning systems with more degrees of freedom are given in section 5.

This paper is a study of incompressible fluids, especially their Clebsch variables and vortices, using symplectic geometry and the Lie-Poisson structure on the dual of a Lie algebra. Following ideas of Arnold and others it is shown that Euler's equations are Lie-Poisson equations associated to the group of volume-preserving diffeomorphisms. The dual of the Lie algebra is seen to be the space of vortices, and Kelvin's circulation theorem is interpreted as preservation of coadjoint orbits. In this context, Clebsch variables can be understood as momentum maps. The motion of N point vortices is shown to be identifiable with the dynamics on a special coadjoint orbit, and the standard canonical variables for them are a special kind of Clebsch variables. Point vortices with cores, vortex patches, and vortex filaments can be understood in a similar way. This leads to an explanation of the geometry behind the Hald-Beale-Majda convergence theorems for vorticity algorithms. Symplectic structures on the coadjoint orbits of a vortex patch and filament are computed and shown to be closely related to those commonly used for the KdV and the Schrödinger equations respectively.