Search

Some mathematical facts about Beltrami fields, which are time-independent solutions of the three-dimensional incompressible Euler equations with nontrivial helicity, are assembled. The linearization about an arbitrary, fixed Beltrami field is studied in a Hamiltonian framework. A factorization of this linearization is introduced and used to characterize null spaces for Beltrami fields with ergodic streamlines. An interesting property of unstable modes is noticed and its consequences concerning the nature of potential instabilities are discussed. [ABSTRACT FROM AUTHOR]

Time evolution of relativistic particle systems can be accomplished by means of a Schrödinger wave equation, provided the Hamiltonian of the N particle system satisfies some commutator equations involving the other generators of a suitable representation of the Poincaré group on the initial data space of the Schrödinger equation. This set of operator equations is solved in the N free particle case. Further, the structure of N interacting particle Hamiltonians is worked out to include second-order relativistic corrections, i.e., (1/c2)-terms. It is shown that Breit and Barker–O’Connell Hamiltonians (nonzero spin particles) and the Bazˇanski Hamiltonian (zero-spin particles) are particular cases of this more general Hamiltonian. [ABSTRACT FROM AUTHOR]

In this paper the ADM (Arnowitt, Deser, and Misner) reduction of Einstein’s equations for three-dimensional ‘‘space-times’’ defined on manifolds of the form Σ×R, where Σ is a compact orientable surface, is discussed. When the genus g of Σ is greater than unity it is shown how the Einstein constraint equations can be solved and certain coordinate conditions imposed so as to reduce the dynamics to that of a (time-dependent) Hamiltonian system defined on the 12g-12-dimensional cotangent bundle, T*T(Σ), of the Teichmüller space, T(Σ), of Σ. The Hamiltonian is only implicitly defined (in terms of the solution of an associated Lichnerowicz equation), but its existence, uniqueness, and smoothness are established by standard analytical methods. Similar results are obtained for the case of genus g=1, where, in fact, the Hamiltonian can be computed explicitly and Hamilton’s equations integrated exactly (as was found previously by Martinec). The results are relevant to the problem of the reduction of the 3+1-dimensional Einstein equations (formulated on circle bundles over Σ×R and with a spacelike Killing field tangent to the fibers of the chosen bundle) and to the recent discussion by Witten of the possible exact solvability of the ‘‘topological dynamics’’ associated with Einstein’s equations in 2+1 dimensions. [ABSTRACT FROM AUTHOR]

The discussion is limited to finite-dimensional parametrized systems with one constraint—the so-called super-Hamiltonian. The topology of the corresponding Hamiltonian vector field on the constraint hypersurface, in particular the question of the existence of cross sections, is studied. This has some bearing on the applicability of the reduction method, as well as on unitarity within the Dirac method, of canonical quantization [see the previous paper, Phys. Rev. D 34, 1040 (1986)]. The main theorem of the present paper states that a cross section will exist if and only if the following conditions are both satisfied: (i) no dynamical trajectory is closed or almost closed; and (ii) the quotient set topology of the set of all dynamical trajectories is Hausdorff. Examples of systems that separately violate each of these conditions are given. [ABSTRACT FROM AUTHOR]

Conserved quantities, bi-Hamiltonian formulation, and recursive structure of the relativistic Toda lattice (RT) are obtained in an algorithmic way without making use of the Lax representation. Furthermore, for the multisoliton solutions the gradients of the angle variables are described in terms of mastersymmetries. A new hierarchy of completely integrable systems is discovered, which turns out to correspond to the ‘‘negative’’ of the hierarchy of RT. Thus it is shown that the full algebra of time-dependent symmetry group generators for each member of the RT hierarchy is isomorphic to the algebra of first order differential operators with Laurent polynomials as coefficients. The surprising phenomenon is revealed that the members of the RT hierarchy are connected to their negative counterparts by explicit Bäcklund transformation. [ABSTRACT FROM AUTHOR]

A simple algorithm for casting two-dimensional hyperbolic systems into the Hamiltonian form is given. First-order conserved quantities are found and expressed in terms of eigenvalues. It is shown that for some systems the dependence of the eigenvalues on the field variables can be used for classification. [ABSTRACT FROM AUTHOR]

In the haromonic oscillator basis, the effective Hamiltonians are constructed for the polynomial interactions V(r) = &Zigma;[sup t, sub i] = ¹g[sub i]r[sup 2i] and t ≤ 4. Their form (factorized in terms of the vectorial continued fractions) is simpler in a semirelativistic (c < ∞ ) reformulation. Its degeneracy in the c → ∞ limit (confluence of the physical and unphysical solutions) is removed by solving a coupled set of the nonlinear algebraic equations in REDUCE. [ABSTRACT FROM AUTHOR]

Some basic properties of the energy level sets for parameter-dependent systems are analyzed. A class of Hamiltonians depending linearly on two parameters is considered as a working example. It is shown that an approximate computation of the boundary of the level set can be useful to derive analytical representations of the eigenvalues of the system. The expressions deduced for the eigenenergies have the correct analytic behavior in the whole range of all physical parameters. In the simplest case, a nonvariational upper bound to the ground state energy is derived that satisfies both virial and Hellmann-Feynman theorems, as well as firstorder perturbation theory. Furthermore, some approximations are obtained, within the framework of the sealing variational method, that are numerical upper and lower bounds to the exact eigenvalues. Applications are illustrated with a family of anharmonic oscillators. [ABSTRACT FROM AUTHOR]

To every Hamiltonian system is associated a degenerate Lagrangian formulation. Following Dirac's theory of constraints one finds a family of lifts of the system to the cotangent bundle of the underlying manifold. The lifted system admits a reduction to the original equation such that the original Hamiltonian formulation is the pullback of the canonical symplectic form. Invariants of the original equation can be lifted to invariants of the extended system. This procedure is applied to integrable systems such as the Korteweg-de Vries equation. [ABSTRACT FROM AUTHOR]

Nonautonomous Hamiltonian systems of one degree of freedom close to integrable ones are considered. Let ε be a positive parameter measuring the strength of the perturbation and denote by εc the critical value at which a given KAM (Kolmogorov–Arnold–Moser) torus breaks down. A computer-assisted method that allows one to give rigorous lower bounds for εc is presented. This method has been applied in Celletti–Falcolini–Porzio (to be published in Ann. Inst. H. Poincaré) to the Escande and Doveil pendulum yielding a bound which is within a factor 40.2 of the value indicated by numerical experiments. [ABSTRACT FROM AUTHOR]

The equivalence between the Lagrangian and Hamiltonian formalism is studied for constraint systems. A procedure to construct the Lagrangian constraints from the Hamiltonian constraints is given. Those Hamiltonian constraints that are first class with respect to the Hamiltonian constraints produce Lagrangian constraints that are FL-projectable. [ABSTRACT FROM AUTHOR]

G. Thompson [J. Math. Phys. 25, 3474 (1984)], by way of example of a general method, examined the existence of integrals of the form I = (yp[sub x] - xp[sub y])³ + A (x,y)p[sub x] + B (x,y)p[sub y], for the Hamiltonian H = ½ (p²[sub x] + p²[sub y]) + V(x,y). The treatment was not complete. A complete solution is given here as well as indications for further development to integrals of higher order in the momenta. [ABSTRACT FROM AUTHOR]

A generating function, or Molien function, the coefficients of which give the number of independent polynomial invariants in G, has been useful in the Landau and renormalization group theories of phase transitions. Here a generalized Molien function for a field theoretical Hamiltonian (with short-range interactions) of the most general form invariant in a group G is derived. This form is useful for more general renormalization group calculations. Its Taylor series is calculated to low order for the FΓ[sup -, sub 2] representation of the space group R &3macr;c and also for the l = 1 (faithful) representation of SO(3). [ABSTRACT FROM AUTHOR]

The article presents the authors' comment on the article "Quantum mechanics of the 1/x2 potential," by Andrew M. Essin and David J. Griffiths. According to the authors, the quantum mechanical properties of the attractive -1/x/x2 potential needs further explanation of quantum properties such as the need for considering the restrictions imposed by the Hamiltonian H and the renormalization that is required to obtain physically proper Reference 1 solutions. The added that both H and H operators should be defined exactly the same set of functions in Hilbert space.

The problem of obtaining two Hamiltonian functions linked by a canonoid transformation is solved in the realm of generalized mechanics. By pairing the canonically conjugated pairs of variable a technique that also appears to be useful in solving many other problems in generalized mechanics is introduced. [ABSTRACT FROM AUTHOR]

The Schrödinger equation with a time-dependent quadratic Hamiltonian is investigated. The time-evolution operator is written as a product of exponential operators determined by the Heisenberg equations of motion. This product operator is shown to be global in the occupation number representation when the Hamiltonian is Hermitian. The success of some physical applications of the product-form representation is explained. [ABSTRACT FROM AUTHOR]

The problem of the evolution of SU(2) and SU (1,1) states is analyzed from a unified point of view. A generalized Rabi matrix for a time-dependent SU (2)-SU (1,1) coherence preserving Hamiltonian is also derived. [ABSTRACT FROM AUTHOR]

Dirac's theory of constraints is used to derive the Hamiltonian structure of the modified and fifth-order Korteweg-de Vries equations. A transformation to "physical" variables is performed on the canonical structure and is shown to be equivalent to the Dirac bracket. [ABSTRACT FROM AUTHOR]

Using Madelung’s hydrodynamical variables, it is shown that the bi-Hamiltonian structure of the nonlinear Schrödinger equation goes over to a bi-Hamiltonian structure of the shallow water wave equations in the classical limit. [ABSTRACT FROM AUTHOR]

For a certain choice of pressure the equations of gas dynamics can be diagonalized, making it easy to calculate some first-order conserved densities of the system. Using the Hamiltonian structure of these equations, third-order symmetries are seen to exist by the Noether correspondence. Such higher-order phenomena are usually associated with a linearization, but in this case, no linearization is obvious. [ABSTRACT FROM AUTHOR]

In this paper we construct a class of integrable Hamiltonian nonlinear evolution equations generated by a purely differential recursion operator. It turns out that this hierarchy is a complex version of the Burgers hierarchy and can be linearized through a generalization of the Cole–Hopf transformation. [ABSTRACT FROM AUTHOR]

A new approach to the equivalence problem (in phase space) is presented. Given a Hamiltonian describing a system of particles with two degrees of freedom (and the corresponding HamiltonJacobi equation), it is shown how to find the most general family of Hamiltonian functions that generates a new Hamilton-Jacobi equation with the following (and essential) characteristic, here defined as equivalence: Every new solution is also a solution of the original Hamilton-Jacobi equation and vice versa. [ABSTRACT FROM AUTHOR]

The most general form of the Hamiltonian is obtained under the restriction that initially coherent states of a multimode para-Bose system remain coherent at all times. The equation of motion for the square of the annihilation operator for the multimode system using this Hamiltonian is solved. [ABSTRACT FROM AUTHOR]

We derive a nearly optimal upper bound on the running time in the adiabatic theorem for a switching family of Hamiltonians. We assume the switching Hamiltonian is in the Gevrey class Gα as a function of time, and we show that the error in adiabatic approximation remains small for running times of order g-2 |ln g |6α. Here g denotes the minimal spectral gap between the eigenvalue(s) of interest and the rest of the spectrum of the instantaneous Hamiltonian. [ABSTRACT FROM AUTHOR]

Dynamical systems of the billiard type are of fundamental importance for the description of numerous phenomena observed in many different fields of research, including statistical mechanics, Hamiltonian dynamics, nonlinear physics, and many others. This Focus Issue presents the recent progress in this area with contributions from the mathematical as well as physical stand point. [ABSTRACT FROM AUTHOR]

We discuss spectral and resonance properties of exactly solvable Dirac Hamiltonians corresponding to a surface δ and δ[sup ′] interactions. First, we study spectral properties for δ-sphere and δ[sup ′]-sphere models. Next, we analyze the resonance phenomena for the same models with specific boundary conditions. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Continuous and discrete superselection rules induced by the interaction with the environment are investigated for a class of exactly soluble Hamiltonian models. The environment is given by a boson field. Stable superselection sectors can only emerge if the low frequences dominate and the ground state of the boson field disappears due to infrared divergence. The models allow uniform estimates of all transition matrix elements between different superselection sectors. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]

In this paper we study the behavior of the moments of a particle distribution as it is transported through a Hamiltonian system. Functions of moments that remain invariant for an arbitrary nonlinear Hamiltonian system are constructed perturbatively. The perturbation expansion is carried out according to the degree of nonlinearity of the symplectic mapping associated with the Hamiltonian. The existence of these moment invariants is entirely a consequence of the symplectic nature of the flow generated by Hamiltonian systems. These invariants may be of use in accelerator design, in the general area of charged particle beam transport, and in other areas of Hamiltonian dynamics. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]

It is shown that the criterium to decide whether two Hamiltonians are equivalent given by Espindola et al. [ O. Espindola, N. L. Teixeira, and M. L. Espindola, J. Math. Phys. 27, 151 (1986) ] is incorrect. [ABSTRACT FROM AUTHOR]