Hamilton-Jacobi functional theory for the integration of classical field equations

Long time behavior of solutions of nonlinear classical field equations: the example of NLS defocusing

In the development of quantum mechanics in the 1920s, both matrix mechanics (developed by Born, Heisenberg and Jordon) and wave mechanics (developed by Schrödinger) prevailed. These early attempts corresponded to the quantum mechanics of particles. Matrix mechanics was found to lead directly to the Schrödinger equation, and the Schrödinger equation could be used to derive the alternative problem for matrix mechanics. Later emphasis lay on the development of the dynamics of fields, where the classical field equations were quantized (see, for example, Weinberg). Today, quantum field theory is one of the most successful physical theories ever developed. The symmetry between particle and wave mechanics is exploited herein. One of the important properties of quantum mechanics is that it is linear, leading to some confusion about how to treat the problem of nonlinear classical field equations. In the present paper we address the case of classical nonlinear soliton equations which are exactly integrable in terms of the periodic/quasiperiodic inverse scattering transform. This means that all physical spectral solutions of the soliton equations can be computed exactly for these specific boundary conditions. Unfortunately, such solutions are highly nonlinear, leading to difficulties in solving the associated quantum mechanical problems. Here we find a strategy for developing the quantum mechanical solutions for soliton dynamics. To address this difficulty, we apply a recently derived result for soliton equations, i.e., that all solutions can be written as quasiperiodic Fourier series. This means that soliton equations, in spite of their nonlinear solutions, are perfectly linearizable with quasiperiodic boundary conditions, the topic of finite gap theory, i.e., the inverse scattering transform with periodic/quasiperiodic boundary conditions. We then invoke the result that soliton equations are Hamiltonian, and we are able to show that the generalized coordinates and momenta also have quasiperiodic Fourier series, a generalized linear superposition law, which is valid in the case of nonlinear, integrable classical dynamics and is here extended to quantum mechanics. Hamiltonian dynamics with the quasiperiodicity of inverse scattering theory thus leads to matrix mechanics. This completes the main theme of our paper, i.e., that classical, nonlinear soliton field equations, linearizable with quasiperiodic Fourier series, can always be quantized in terms of matrix mechanics. Thus, the solitons and their nonlinear interactions are given an explicit description in quantum mechanics. Future work will be formulated in terms of the associated Schrödinger equation.

In 1967 Roger Penrose introduced what is now called twistor geometry (Penrose 1967). This geometry is an amplification of the geometry of lines in projective space studied extensively by Felix Klein and others in the late 19th century. Penrose made the fundamental observation that this classical geometry could be used in conjunction with a Radon transform style integral geometry to provide rather complete sets of solutions of numerous partial differential equations, both linear and nonlinear, which arise naturally in theoretical physics. The Radon transform, which involves integration over straight lines in R 3, was used by F. John to generate solutions of hyperbolic partial differential equations in 1942. Penrose integrates over complex projective lines, real 2-spheres, and the data of the integration is complex-analytic functions of 3 complex variables. This is analogous to (and indeed related to) Weierstrass' representation of solutions of the minimal surface equation in terms of triples of holomorphic functions of a single complex variable which satisfy an algebraic relation. In this paper we discuss the origins of the differential equations which are amiable to twistor-geometric study from the realms of theoretical physics. Twistor geometry is most suitable for physical theories which are relativistically invariant, including Maxwell's description of electromagnetic phenomena from the 19th century (Maxwell's equations), Einstein's theory of gravitation (the equations of general relativity), and the contemporary descriptions of elementary particles (quantum field theory). Twistor geometry interacts in a nontrivial manner with the mathematical models involved in all of the above physical theories. In particular, it has led to new solutions of some of the nonlinear partial differential equations involved (Einstein's equations, Yang-Mills equations, equations of a magnetic monopole). We will describe some recent progress on twistorgeometric representations of solutions of classical field equations. There are two aspects to this. First, there is the problem of translating a given system of differential equations into twistor-geometric language. This has been done successfully in many different cases, and will be described in Section 3. Once a given problem has been transferred to a twistor-geometric context, it becomes, in general, a specific problem involving algebraic geometry, several complex variables, and algebraic topology involving holomorphic data on complex manifolds. The only partial differential equations remaining in the twistor context are the CauchyRiemann equations characterizing holomorphic functions. In effect, the partial differential equation has been transformed into a geometric problem. This is analogous to the Fourier transform transforming a partial differential equation with constant coefficients into an algebraic problem, where differentiation is replaced by the (algebraic) symbol of the differential operator in question. The second aspect of the recent progress on the twistor-geometric representation of the solutions of the classical field equations concerns itself with solving the geometric problem in question and transforming back to space-time (or Euclidean space), generating (sometimes new) solutions of the classical field equations.

A survey of the results concerning the development of the theory of Hamilton-Jacobi equations for hereditary dynamical systems is presented. One feature of these systems is that the rate of change of their state depends not only on the current position, like in the classical case, but also on the full path travelled, that is, the history of the motion. Most of the paper is devoted to dynamical systems whose motion is described by functional differential equations of retarded type. In addition, more general systems described by functional differential equations of neutral type and closely related systems described by differential equations with fractional derivatives are considered. So-called path-dependent Hamilton-Jacobi equations are treated, which play for the above classes of systems a role similar to that of the classical Hamilton-Jacobi equations in dynamic optimization problems for ordinary differential systems. In the context of applications to control problems, the main attention is paid to the minimax approach to the concept of a generalized solution of the Hamilton-Jacobi equations under consideration and also to its relationship with the viscosity approach. Methods for designing optimal feedback control strategies with memory of motion history which are based on the constructions discussed are presented. Bibliography: 183 titles.

Relativistic solitons and superluminal signals

Mathematical aspects of classical nonlinear field equations

A space-time functional formalism for classical field equations

The idea of using material forces also termed configurational forces in a computational setting is presented. The theory of material forces is briefly recast in the terms of a non-linear elastic solid. It is shown, how in a computational setting with finite elements (FE) the discrete configurational forces are calculated once the classical field equations are solved. This post-process calculation is performed in a way, which is consistent with the approximation of the classical field equations. Possible physical meanings of this configurational forces are discussed. A purely computational aspect of material forces is pointed out, where material forces act as an indicator to obtain softer discretizations.

We discuss two aspects of the behavior of solitary waves in non-integrable classical field theories in one spatial dimension and contrast this behavior with that observed in closely related integrable theories. First, we examine the nature of kink/anti-kink interactions in the φ 4 theory, described by the classical field equation $$ {\phi _{tt}} - {\phi _{xx}} - \phi + {\phi ^3} = 0.$$

In the general relativity theory, using Einstein’s gravity field equation, we discover the spherical solution of the classical quantum gravity. The careful point is that this theory is different from the other quantum theory. This theory is made by the Einstein’s classical field equation.

In the general relativity theory, using Einstein’s gravity field equation, we discover the spherical solution of the classical quantum gravity. The careful point is that this theory is different from the other quantum theory. This theory is made by the Einstein’s classical field equation.

It is shown that the complex phase of the Feynman propagator is a solution of the quantum Hamilton–Jacobi equation, namely, it is the quantum Hamilton's principal function (or quantum action). Therefore, the Feynman propagator can be computed either by means of the path integration, or by the way of the Hamilton–Jacobi equation. This is analogous to what happens in classical mechanics, where the Hamilton's principal function can be computed either by integrating the Lagrangian along the extremal paths, or as a solution of partial differential equation, namely the classical Hamilton–Jacobi equation. If the path is decomposed in the classical one and quantum fluctuations, the contribution of these quantum fluctuations satisfies a non-linear partial differential equation, whose coefficients depend on the classical action. When the contribution of the quantum fluctuations depend only on the time, it can be computed by means of a simple integration. The final results for the propagators in this case are equal to the Van Vleck–Pauli–Morette expressions, even though the two derivations are quite different. Quanta 2023; 12: 22–26.

A detailed study of the Bianchi type IX cosmological model is done. Classical field equations are discussed with a massive scalar field. The Wheeler-DeWitt quantum equation is formed and is solved using a Born-Oppenheimer type of approximation.

Linear wave equations and Kacs-van Moebeke lattices

Exact solutions for a generalized Higgs equation