The Hamiltonian structure of the Maxwell-Vlasov equations

Morrison [25] has observed that the Maxwell-Vlasov and Poisson-Vlasov equations for a collisionless plasma can be written in Hamiltonian form relative to a certain Poisson bracket. We derive another Poisson structure for these equations by using general methods of symplectic geometry. The main ingredients in our construction are the symplectic structure on the co-adjoint orbits for the group of canonical transformations, and the symplectic structure for the phase space of the electromagnetic field regarded as a gauge theory. Our Poisson bracket satisfies the Jacobi identity, whereas Morrison's does not [37]. Our construction also shows where canonical variables can be found and can be applied to the Yang-Mills-Vlasov equations and to electromagnetic fluid dynamics.     

A new Hamiltonian structure for the dynamics of general relativity

A new compact form of the dynamical equations of relativity is proposed. The new form clarifies the covariance of the equations under coordinate transformations of the space-time. On a deeper level, we obtain new insight into the infinite-dimensional symplectic geometry behind the dynamical equations, the decompositions of gravitational perturbations, and the space of gravitational degrees of freedom. Prospects for these results in studying fields coupled to gravity and the quantization of gravity are outlined.This essay was awarded the second prize for 1976 by the Gravity Research Foundation.     

Solution of the local mass problem in general relativity

The local mass problem is solved. That is, in suitable function spaces, it is shown that for any vacuum space-time near flat space, its massm is strictly positive. The relationship to other work in the field and some discussion of the global problem is given. Our proof is, in effect, a version of critical point analysis in infinite dimensions, but detailedL ^p and Sobolev-type estimates are needed for the precise proof, as well as careful attention to the coordinate invariance group. For the latter, we prove a suitable slice theorem based on the use of harmonic coordinates.Partially supported by the University of Toronto, Université de Paris VI and NSF Grant MPS-75-05576     

Structure of 72,74Se at high spin

Lifetimes of high spin states up to { }=22⁺ in the yrast positive parity bands have been measured to investigate the shape evolution with increasing spin in ^{72, 74}Se. The Q _t values derived from these measurements indicate that prolate shape stabilizes for ⁷²Se, while a triaxial shape develops for ⁷⁴Se at higher spins. Comparison of the observed trend in Q _t with spin for ^{72, 74}Se with that of the corresponding kryptones isotones emphasizes the stability provided by N=38 prolate shell gap even at high rotational frequency.     

Synthesis and bio-assay of RCM-derived Bowman-Birk inhibitor analogues

Bowman-Birk inhibitor analogues containing 2, 3 and 4-carbon analogues of the natural disulfide were synthesised via solid phase microwave-assisted RCM and found to have K(i) values against chymotrypsin in the low to sub-micromolar range, the best replacement for the disulfide arising from the linkage by RCM of two l-homoallylglycine residues.     

The Complex Geometry of Weak Piecewise Smooth Solutions of Integrable Nonlinear PDE's¶of Shallow Water and Dym Type

An extension of the algebraic-geometric method for nonlinear integrable PDE's is shown to lead to new piecewise smooth weak solutions of a class of N-component systems of nonlinear evolution equations. This class includes, among others, equations from the Dym and shallow water equation hierarchies. The main goal of the paper is to give explicit theta-functional expressions for piecewise smooth weak solutions of these nonlinear PDE's, which are associated to nonlinear subvarieties of hyperelliptic Jacobians. The main results of the present paper are twofold. First, we exhibit some of the special features of integrable PDE's that admit piecewise smooth weak solutions, which make them different from equations whose solutions are globally meromorphic, such as the KdV equation. Second, we blend the techniques of algebraic geometry and weak solutions of PDE's to gain further insight into, and explicit formulas for, piecewise-smooth finite-gap solutions. The basic technique used to achieve these aims is rather different from earlier papers dealing with peaked solutions. First, profiles of the finite-gap piecewise smooth solutions are linked to certain finite dimensional billiard dynamical systems and ellipsoidal billiards. Second, after reducing the solution of certain finite dimensional Hamiltonian systems on Riemann surfaces to the solution of a nonstandard Jacobi inversion problem, this is resolved by introducing new parametrizations. Amongst other natural consequences of the algebraic-geometric approach, we find finite dimensional integrable Hamiltonian dynamical systems describing the motion of peaks in the finite-gap as well as the limiting (soliton) cases, and solve them exactly. The dynamics of the peaks is also obtained by using Jacobi inversion problems. Finally, we relate our method to the shock wave approach for weak solutions of wave equations by determining jump conditions at the peak location. Received: 16 February 1999 / Accepted: 10 April 2001     

Dechanneling of MeV protons in tungsten

Abstract

A systematic investigation of the dechanneling of protons in tungsten has been completed for the <111> axial direction and also for the (110) plane. Measurements at room temperature have been made at energies of 0.5, 2.0, 3.5, 8.0 and 12.0 MeV; at 2.0 MeV, the effect of target temperature has been investigated over the range 77–873°K. In the axial case, the dechanneling rate increases strongly with the vibrational amplitude _ρ, but not as strongly as the ρ² dependence reported by the Catania group (Refs. 4–7) for silicon and germanium. Also, unlike silicon and germanium, the tungsten axial data exhibit practically no energy dependence and so cannot be fitted by the zρ²/E scaling parameter suggested by the Catania group. On the other hand, the planar dechanneling data in tungsten exhibit the predicted (E ^?1)dependence and relatively littie temperature dependence, in good agreement with previous work.     

A partial differential equation with infinitely many periodic orbits: Chaotic oscillations of a forced beam

This paper delineates a class of time-periodically perturbed evolution equations in a Banach space whose associated Poincaré map contains a Smale horseshoe. This implies that such systems possess periodic orbits with arbitrarily high period. The method uses techniques originally due to Melnikov and applies to systems of the form x=f _o(X)+f ₁(X,t), where f _o(X) is Hamiltonian and has a homoclinic orbit. We give an example from structural mechanics: sinusoidally forced vibrations of a buckled beam.     

Hamiltonian structures and stability for rigid bodies with flexible attachments

The dynamics of a rigid body with flexible attachments is studied. A general framework for problems of this type is established in the context of Poisson manifolds and reduction. A simple model for a rigid body with an attached linear extensible shear beam is worked out for illustration. Second, the Energy-Casimir method for proving nonlinear stability is recalled and specific stability criteria for our model example are worked out. The Poisson structure and stability results take into account vibrations of the string, rotations of the rigid body, their coupling at the point of attachment, and centrifugal and Coriolis forces.