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

Relating imperatives to action

The aim of this chapter is to provide an analysis of the use of logically complex imperatives, in particular, imperatives of the form Do A1 or A2 and Do A, if B. We argue for an analysis of imperatives in terms of classical logic which takes into account the influence of background information on imperatives. We show that by doing so one can avoid some counter-intuitive results which have been associated with analyses of imperatives in terms of classical logic. In particular, I address Hamblin's observations concerning rule-like imperatives and Ross' Paradox. The analysis is carried out within an agent-based logical framework. This analysis explicates what it means for an agent to have a successful policy for action with respect to satisfying his or her commitments, where some of these commitments have been introduced as a result of imperative language use

Nonlinear multi-state tunneling dynamics in a spinor Bose-Einstein condensate

We present an experimental realization of dynamic self-trapping and non-exponential tunneling in a multi-state system consisting of ultracold sodium spinor gases confined in moving optical lattices. Taking advantage of the fact that the tunneling process in the sodium spinor system is resolvable over a broader dynamic energy scale than previously observed in rubidium scalar gases, we demonstrate that the tunneling dynamics in the multi-state system strongly depends on an interaction induced nonlinearity and is influenced by the spin degree of freedom under certain conditions. We develop a rigorous multi-state tunneling model to describe the observed dynamics. Combined with our recent observation of spatially-manipulated spin dynamics, these results open up prospects for alternative multi-state ramps and state transfer protocols

Formulation and performance of variational integrators for rotating bodies

Variational integrators are obtained for two mechanical systems whose configuration spaces are, respectively, the rotation group and the unit sphere. In the first case, an integration algorithm is presented for Euler’s equations of the free rigid body, following the ideas of Marsden et al. (Nonlinearity 12:1647–1662, 1999). In the second example, a variational time integrator is formulated for the rigid dumbbell. Both methods are formulated directly on their nonlinear configuration spaces, without using Lagrange multipliers. They are one-step, second order methods which show exact conservation of a discrete angular momentum which is identified in each case. Numerical examples illustrate their properties and compare them with existing integrators of the literature

Illocutionary harm

A number of philosophers have become interested in the ways that individuals are subject to harm as the performers of illocutionary acts. This paper offers an account of the underlying structure of such harms: I argue that speakers are the subjects of illocutionary harm when there is interference in the entitlement structure of their linguistic activities. This interference comes in two forms: denial and incapacitation. In cases of denial, a speaker is prevented from achieving the outcomes to which they are entitled by their speech. In cases of incapacitation, a speaker’s standing to expect certain outcomes is itself undermined. I also discuss how individual speakers are subject to interference along two dimensions: as exercisers of certain non-linguistic capacities, and as producers of meaningful speech

Nonequilibrium Evolution of Correlation Functions: A Canonical Approach

We study nonequilibrium evolution in a self-interacting quantum field theory invariant under space translation only by using a canonical approach based on the recently developed Liouville-von Neumann formalism. The method is first used to obtain the correlation functions both in and beyond the Hartree approximation, for the quantum mechanical analog of the $\phi^{4}$ model. The technique involves representing the Hamiltonian in a Fock basis of annihilation and creation operators. By separating it into a solvable Gaussian part involving quadratic terms and a perturbation of quartic terms, it is possible to find the improved vacuum state to any desired order. The correlation functions for the field theory are then investigated in the Hartree approximation and those beyond the Hartree approximation are obtained by finding the improved vacuum state corrected up to ${\cal O}(\lambda^2)$. These correlation functions take into account next-to-leading and next-to-next-to-leading order effects in the coupling constant. We also use the Heisenberg formalism to obtain the time evolution equations for the equal-time, connected correlation functions beyond the leading order. These equations are derived by including the connected 4-point functions in the hierarchy. The resulting coupled set of equations form a part of infinite hierarchy of coupled equations relating the various connected n-point functions. The connection with other approaches based on the path integral formalism is established and the physical implications of the set of equations are discussed with particular emphasis on thermalization.Comment: Revtex, 32 pages; substantial new material dealing with non-equilibrium evolution beyond Hartree approx. based on the LvN formalism, has been adde