Some basic properties of infinite dimensional Hamiltonian systems

We consider some fundamental properties of infinite dimensional Hamiltonian systems, both linear and nonlinear. For exemple, in the case of linear systems, we prove a symplectic version of the teorem of M. Stone. In the general case we establish conservation of energy and the moment function for system with symmetry. (The moment function was introduced by B. Kostant and J .M. Souriau). For infinite dimensional systems these conservation laws are more delicate than those for finite dimensional systems because we are dealing with partial as opposed to ordinary differential equations

Kiss and Fly - a study of the impacts at a UK regional airport

In the light of the forecast growth in air transport the UK Government has placed a requirement on all airports with substantial air transport movements to implement surface access strategies. The emphasis of surface access policy has been to increase the proportion of people arriving at airports by public transport by a variety of means such as managing parking supply and pricing and improving public transport. The extent to which these policies will be effective will depend on a number of factors such as the quality and availability of the alternatives, the availability of competing off-site parking and the extent to which kiss and fly is feasible. This paper reports on two studies of passenger access to Leeds-Bradford International Airport in the summers of 2004 and 2005. The airport has an aspiration to increase public transport use to the airport from its current level of 3% to 10% by 2010. The principal means by which this is currently planned to be achieved is through the expansion of scheduled bus services to Leeds, Bradford and Harrogate. The 2004 study found that 49% of passengers were dropped off at the airport by friends and that the potential for larger quantities of people to reach the airport by conventional bus services was limited. The 2005 study investigated the extent to which these kiss and fly journeys generate extra travel on the road network. The results show that for an airport with around 2.5 million passengers the Kiss & Fly journeys are creating an extra 19.4 million kilometres, an increase of 36% over the distance that would have been travelled if people had driven and parked. The paper concludes that a charge levied on all vehicles accessing the airport, similar to a congestion charge, is likely to have the greatest impact on travel behaviour and will have a far greater impact on the environment than the current emphasis on public transport improvements and parking restrictions

On uniformly rotating fluid drops trapped between two parallel plates

This contribution is about the dynamics of a liquid bridge between two fixed parallel plates. We consider a mathematical model and present some results from the doctoral thesis [10] of the first author. He showed that there is a Poisson bracket and a corresponding Hamiltonian, so that the model equations are in Hamiltonian form. The result generalizes previous results of Lewis et al. on the dynamics of free boundary problems for "free" liquid drops to the case of a drop between two parallel plates, including, especially the effect of capillarity and the angle of contact between the plates and the free fluid surface. Also, we prove the existence of special solutions which represent uniformly rotating fluid ridges, and we present specific stability conditions for these solutions. These results extend work of Concus and Finn [2] and Vogel [18],[19] on static capillarity problems (see also Finn [5]). Using the Hamiltonian structure of the model equations and symmetries of the solutions, the stability conditions can be derived in a systematic way. The ideas that are described will be useful for other situations involving capillarity and free boundary problems as well

The Hamiltonian Structure of Nonlinear Elasticity: The Material and Convective Representations of Solids, Rods, and Plates

No Abstrac

Arnold diffusion in the dynamics of a 4-machine power system undergoing a large fault

We focus on the seemingly complicated dynamics of a four-machine power system which is undergoing a sudden fault. Adopting a Hamiltonian (energy) formulation, we consider the system as an interconnection of (one degree of freedom) subsystems. Under certain configuration (a star network) and parameter values we establish the presence of Arnold diffusion which entails periodic, almost periodic, and complicated nonperiodic dyanmics all simultaneously present; and erratic transfer of energies between the subsystems. In section 1 we introduce the transient stability problem in a mathematical setting and explain what our results mean in the power systems context. Section 2 provides insights into Arnold diffusion and summarizes its mathematical formulation as in [8], [1]. Section 3 gives conditions for which Arnold diffusion arises on certain energy levels of the swing equations. These conditions are verified analytically in the case when all but one subsystem (machine) undergo relatively small oscillations

Discrete Routh Reduction

This paper develops the theory of abelian Routh reduction for discrete mechanical systems and applies it to the variational integration of mechanical systems with abelian symmetry. The reduction of variational Runge-Kutta discretizations is considered, as well as the extent to which symmetry reduction and discretization commute. These reduced methods allow the direct simulation of dynamical features such as relative equilibria and relative periodic orbits that can be obscured or difficult to identify in the unreduced dynamics. The methods are demonstrated for the dynamics of an Earth orbiting satellite with a non-spherical $J_2$ correction, as well as the double spherical pendulum. The $J_2$ problem is interesting because in the unreduced picture, geometric phases inherent in the model and those due to numerical discretization can be hard to distinguish, but this issue does not appear in the reduced algorithm, where one can directly observe interesting dynamical structures in the reduced phase space (the cotangent bundle of shape space), in which the geometric phases have been removed. The main feature of the double spherical pendulum example is that it has a nontrivial magnetic term in its reduced symplectic form. Our method is still efficient as it can directly handle the essential non-canonical nature of the symplectic structure. In contrast, a traditional symplectic method for canonical systems could require repeated coordinate changes if one is evoking Darboux' theorem to transform the symplectic structure into canonical form, thereby incurring additional computational cost. Our method allows one to design reduced symplectic integrators in a natural way, despite the noncanonical nature of the symplectic structure.Comment: 24 pages, 7 figures, numerous minor improvements, references added, fixed typo

Sensorimotor processing for balance in spinocerebellar ataxia type 6.

We investigated whether balance impairments caused by cerebellar disease are associated with specific sensorimotor processing deficits that generalize across all sensory modalities. Experiments focused on the putative cerebellar functions of scaling and coordinate transformation of balance responses evoked by stimulation of single sensory channels. Vestibular, visual, and proprioceptive sensory channels were stimulated in isolation using galvanic vestibular stimulation, moving visual scenery, and muscle vibration, respectively, in 16 subjects with spinocerebellar ataxia type 6 (SCA6) and 16 matched healthy controls. Two polarities of each stimulus type evoked postural responses of similar form in the forward and backward directions. Disease severity was assessed using the Scale for Assessment and Rating of Ataxia. Impaired balance of SCA6 subjects during unperturbed stance was reflected in faster than normal body sway (P = 0.009), which correlated with disease severity (r = 0.705, P < 0.001). Sensory perturbations revealed a sensorimotor processing abnormality that was specific to response scaling for the visual channel. This manifested as visually evoked postural responses that were approximately three times larger than normal (backward, P < 0.001; forward P = 0.005) and correlated with disease severity (r = 0.543, P = 0.03). Response direction and habituation properties were no different from controls for all three sensory modalities. Cerebellar degeneration disturbs the scaling of postural responses evoked by visual motion, possibly through disinhibition of extracerebellar visuomotor centers. The excessively high gain of the visuomotor channel without compensatory decreases in gains of other sensorimotor channels provides a potential mechanism for instability of the balance control system in cerebellar disease. © 2015 International Parkinson and Movement Disorder Society

Altered muscarinic and nicotinic receptor densities in cortical and subcortical brain regions in Parkinson's disease

Muscarinic and nicotinic cholinergic receptors and choline acetyltransferase activity were studied in postmortem brain tissue from patients with histopathologically confirmed Parkinson's disease and matched control subjects. Using washed membrane homogenates from the frontal cortex, hippocampus, caudate nucleus, and putamen, saturation analysis of specific receptor binding was performed for the total number of muscarinic receptors with [3H]quinuclidinyl benzilate, for muscarinic M1 receptors with [3H]pirenzepine, for muscarinic M2 receptors with [3H]oxotremorine-M, and for nicotinic receptors with (-)-[3H]nicotine. In comparison with control tissues, choline acetyl-transferase activity was reduced in the frontal cortex and hippocampus and unchanged in the caudate nucleus and putamen of parkinsonian patients. In Parkinson's disease the maximal binding site density for [3H]quinuclidinyl benzilate was increased in the frontal cortex and unaltered in the hippocampus, caudate nucleus, and putamen. Specific [3H]pirenzepine binding was increased in the frontal cortex, unaltered in the hippocampus, and decreased in the caudate nucleus and putamen. In parkinsonian patients Bmax values for specific [3H]oxotremorine-M binding were reduced in the cortex and unchanged in the hippocampus and striatum compared with controls. Maximal (-)-[3H]nicotine binding was reduced in both the cortex and hippocampus and unaltered in both the caudate nucleus and putamen. Alterations of the equilibrium dissociation constant were not observed for any ligand in any of the brain areas examined. The present results suggest that both the innominatocortical and the septohippocampal cholinergic systems degenerate in Parkinson's disease.(ABSTRACT TRUNCATED AT 250 WORDS

Hamiltonian approach to hybrid plasma models

The Hamiltonian structures of several hybrid kinetic-fluid models are identified explicitly, upon considering collisionless Vlasov dynamics for the hot particles interacting with a bulk fluid. After presenting different pressure-coupling schemes for an ordinary fluid interacting with a hot gas, the paper extends the treatment to account for a fluid plasma interacting with an energetic ion species. Both current-coupling and pressure-coupling MHD schemes are treated extensively. In particular, pressure-coupling schemes are shown to require a transport-like term in the Vlasov kinetic equation, in order for the Hamiltonian structure to be preserved. The last part of the paper is devoted to studying the more general case of an energetic ion species interacting with a neutralizing electron background (hybrid Hall-MHD). Circulation laws and Casimir functionals are presented explicitly in each case.Comment: 27 pages, no figures. To appear in J. Phys.