Clebsch Potentials in the Variational Principle for a Perfect Fluid

Equations for a perfect fluid can be obtained by means of the variational principle both in the Lagrangian description and in the Eulerian one. It is known that we need additional fields somehow to describe a rotational isentropic flow in the latter description. We give a simple explanation for these fields; they are introduced to fix both ends of a pathline in the variational calculus. This restriction is imposed in the former description, and should be imposed in the latter description. It is also shown that we can derive a canonical Hamiltonian formulation for a perfect fluid by regarding the velocity field as the input in the framework of control theory.Comment: 15 page

A Variational Principle for Dissipative Fluid Dynamics

In the variational principle leading to the Euler equation for a perfect fluid, we can use the method of undetermined multiplier for holonomic constraints representing mass conservation and adiabatic condition. For a dissipative fluid, the latter condition is replaced by the constraint specifying how to dissipate. Noting that this constraint is nonholonomic, we can derive the balance equation of momentum for viscous and viscoelastic fluids by using a single variational principle. We can also derive the associated Hamiltonian formulation by regarding the velocity field as the input in the framework of control theory.Comment: 15 page

Ion engine thrust vector study, phase 2 Quarterly report

Performance prediction for expected thrust misalignment in electron bombardment ion thruste

Ion engine thrust vector study

Probability of thrust vector misalignment in ion thrustor arra

Dynamical systems with time-dependent coupling: Clustering and critical behaviour

We study the collective behaviour of an ensemble of coupled motile elements whose interactions depend on time and are alternatively attractive or repulsive. The evolution of interactions is driven by individual internal variables with autonomous dynamics. The system exhibits different dynamical regimes, with various forms of collective organization, controlled by the range of interactions and the dispersion of time scales in the evolution of the internal variables. In the limit of large interaction ranges, it reduces to an ensemble of coupled identical phase oscillators and, to some extent, admits to be treated analytically. We find and characterize a transition between ordered and disordered states, mediated by a regime of dynamical clustering.Comment: to appear in Physica

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

Dynamical model of sequential spatial memory: winnerless competition of patterns

We introduce a new biologically-motivated model of sequential spatial memory which is based on the principle of winnerless competition (WLC). We implement this mechanism in a two-layer neural network structure and present the learning dynamics which leads to the formation of a WLC network. After learning, the system is capable of associative retrieval of pre-recorded sequences of spatial patterns.Comment: 4 pages, submitted to PR

Plasticity and learning in a network of coupled phase oscillators

A generalized Kuramoto model of coupled phase oscillators with slowly varying coupling matrix is studied. The dynamics of the coupling coefficients is driven by the phase difference of pairs of oscillators in such a way that the coupling strengthens for synchronized oscillators and weakens for non-synchronized pairs. The system possesses a family of stable solutions corresponding to synchronized clusters of different sizes. A particular cluster can be formed by applying external driving at a given frequency to a group of oscillators. Once established, the synchronized state is robust against noise and small variations in natural frequencies. The phase differences between oscillators within the synchronized cluster can be used for information storage and retrieval.Comment: 10 page

Strongly Non-Equilibrium Bose-Einstein Condensation in a Trapped Gas

We present a qualitative (and quantitative, at the level of estimates) analysis of the ordering kinetics in a strongly non-equilibrium state of a weakly interacting Bose gas, trapped with an external potential. At certain conditions, the ordering process is predicted to be even more rich than in the homogeneous case. Like in the homogeneous case, the most characteristic feature of the full-scale non-equilibrium process is the formation of superfluid turbulence.Comment: 4 pages, revtex, no figures. Submitted to PR

On Relativistic Material Reference Systems

This work closes certain gaps in the literature on material reference systems in general relativity. It is shown that perfect fluids are a special case of DeWitt's relativistic elastic media and that the velocity--potential formalism for perfect fluids can be interpreted as describing a perfect fluid coupled to a fleet of clocks. A Hamiltonian analysis of the elastic media with clocks is carried out and the constraints that arise when the system is coupled to gravity are studied. When the Hamiltonian constraint is resolved with respect to the clock momentum, the resulting true Hamiltonian is found to be a functional only of the gravitational variables. The true Hamiltonian is explicitly displayed when the medium is dust, and is shown to depend on the detailed construction of the clocks.Comment: 18 pages, ReVTe