The Homogeneous Hamilton–Jacobi and Bernoulli Equations Revisited, II

It is shown that the admissible solutions of the continuity and Bernoulli or Burgers' equations of a perfect one-dimensional liquid are conditioned by a relation established in 1949–1950 by Pauli, Morette, and Van Hove, apparently, overlooked so far, which, in our case, stipulates that the mass density is proportional to the second derivative of the velocity potential. Positivity of the density implies convexity of the potential, i.e., smooth solutions, no shock. Non-elementary and symmetric solutions of the above equations are given in analytical and numerical form. Analytically, these solutions are derived from the original Ansatz proposed in Ref. 1 and from the ensuing operations which show that they represent a particular case of the general implicit solutions of Burgers' equation. Numerically and with the help of an ad hoc computer program, these solutions are simulated for a variety of initial conditions called compatible if they satisfy the Morette–Van Hove formula and anti-compatible if the sign of the initial velocity field is reversed. In the latter case, singular behaviour is observed. Part of the theoretical development presented here is rephrased in the context of the Hopf–Lax formula whose domain of applicability for the solution of the Cauchy problem of the homogeneous Hamilton–Jacobi equation has recently been enlarged.     

Central Limit Theorem¶for Stochastic Hamilton–Jacobi Equations

We study the asymptotic behavior of , where u solves the Hamilton–Jacobi equation u _t +H(x,u _x ) ≡ 0 with H a stationary ergodic process in the x-variable. It was shown in Rezakhanlou–Tarver [RT] that u ^ɛ converges to a deterministic function provided H(x,p) is convex in p and the convex conjugate of H in the p-variable satisfies certain growth conditions. In this article we establish a central limit theorem for the convergence by showing that for a class of examples, u ^ɛ(x,t) can be (stochastically) represented as , where Z(x,t) is a suitable random field. In particular we establish a central limit theorem when the dimension is one and , where ω is a random function that enjoys some mild regularity. Received: 15 February 1999 / Accepted: 14 December 1999     

Hamilton–Jacobi diffieties

Diffieties formalize geometrically the concept of differential equations. We introduce and study Hamilton–Jacobi diffieties. They are finite dimensional subdiffieties of a given diffiety and appear to play a special role in the field theoretic version of the geometric Hamilton–Jacobi theory.     

Semi-Lagrangian Schemes for Hamilton–Jacobi Equations,Discrete Representation Formulae and Godunov Methods

We study a class of semi-Lagrangian schemes which can be interpreted as a discrete version of the Hopf–Lax–Oleinik representation formula for the exact viscosity solution of first order evolutive Hamilton–Jacobi equations. That interpretation shows that the scheme is potentially accurate to any prescribed order. We discuss how the method can be implemented for convex and coercive Hamiltonians with a particular structure and how this method can be coupled with a discrete Legendre trasform. We also show that in one dimension, the first-order semi-Lagrangian scheme coincides with the integration of the Godunov scheme for the corresponding conservation laws. Several test illustrate the main features of semi-Lagrangian schemes for evolutive Hamilton–Jacobi equations.     

Integration of the Hamilton–Jacobi and Maxwell Equations in Diagonal Metrics of Stäckel Spaces

Russian Physics Journal - A classification of metrics and electromagnetic potentials is carried out for the case when the Hamilton–Jacobi equation admits a complete separation of variables in...     

Dynamical Yang–Baxter Equations,Quasi-Poisson Homogeneous Spaces,and Quantization

The purpose of this Letter is twofold. First, we generalize the correspondence between dynamical r-matrices and Poisson homogeneous spaces provided in Karolinsky, E. and Stolin, A [Lett. Math. Phys. 60 (2002), 257–274]. Secondly, we propose a quantization of this quasi-classical result. In particular, we explain the relationship between irreducible highest weight modules and equivariant quantization of coadjoint orbits.*Supported in part by the Royal Swedish Academy of Sciences**Supported in part by RFFI grant 02-01-00085a and CRDF grant RM1-2334-MO-02Mathematical Subject Classification (2000). 17B10, 17B20, 17B35, 17B62, 53C30.     

Nonholonomic Hamilton–Jacobi theory via Chaplygin Hamiltonization

We develop Hamilton–Jacobi theory for Chaplygin systems, a certain class of nonholonomic mechanical systems with symmetries, using a technique called Hamiltonization, which transforms nonholonomic systems into Hamiltonian systems. We give a geometric account of the Hamiltonization, identify necessary and sufficient conditions for Hamiltonization, and apply the conventional Hamilton–Jacobi theory to the Hamiltonized systems. We show, under a certain sufficient condition for Hamiltonization, that the solutions to the Hamilton–Jacobi equation associated with the Hamiltonized system also solve the nonholonomic Hamilton–Jacobi equation associated with the original Chaplygin system. The results are illustrated through several examples.     

The Lagnese–Stellmacher Potentials Revisited

We give a new proof of a classical result of Lagnese and Stellmacher, characterizing all Huygens’ operators of the form , where q(x) depends on only one variable. The proof amounts to characterize the Schrödinger operators with a finite heat kernel expansion.     

The Weierstrass–Mandelbrot Process Revisited

We derive a functional central limit theorem for quasi-Gaussian processes. In particular, we prove that the limit of the Mandelbrot–Weierstrass process is a complex fractional Brownian motion.     

Generalized Euler–Poincaré Equations on Lie Groups and Homogeneous Spaces,Orbit Invariants and Applications

We develop the necessary tools, including a notion of logarithmic derivative for curves in homogeneous spaces, for deriving a general class of equations including Euler–Poincaré equations on Lie groups and homogeneous spaces. Orbit invariants play an important role in this context and we use these invariants to prove global existence and uniqueness results for a class of PDE. This class includes Euler–Poincaré equations that have not yet been considered in the literature as well as integrable equations like Camassa–Holm, Degasperis–Procesi, μCH and μDP equations, and the geodesic equations with respect to right-invariant Sobolev metrics on the group of diffeomorphisms of the circle.     

Quantization from Hamilton–Jacobi theory with a random constraint

We propose a method of quantization based on Hamilton–Jacobi theory in the presence of a random constraint due to the fluctuations of a set of hidden random variables. Given a Lagrangian, it reproduces the results of canonical quantization yet with a unique ordering of operators if the Lagrange multiplier that arises in the dynamical system with constraint can only take binary values ±?/2$\pm ?/2$ with equal probability.     

Hamilton–Jacobi Quantization of Singular Lagrangians with Linear Velocities

In this paper, constrained Hamiltonian systems with linear velocities are investigated by using the Hamilton–Jacobi method. The integrablity conditions are considered on the equations of motion and the action function as well in order to obtain the path integral quantization of singular Lagrangians with linear velocities.     

On Jacobi quasi-Nijenhuis algebroids and Courant–Jacobi algebroid morphisms

We propose a definition of a Jacobi quasi-Nijenhuis algebroid and show that any such Jacobi algebroid has an associated quasi-Jacobi bialgebroid. Therefore, an associated Courant–Jacobi algebroid is also obtained. We introduce the notions of quasi-Jacobi bialgebroid morphism and Courant–Jacobi algebroid morphism, also providing some examples of Courant–Jacobi algebroid morphisms.     

The Einstein–Maxwell Equations and Conformally Kähler Geometry

Page’s Einstein metric on \({{\mathbb{CP}}_2\#\overline{\mathbb{CP}}_2}\) is conformally related to an extremal Kähler metric. Here we construct a family of conformally Kähler solutions of the Einstein–Maxwell equations that deforms the Page metric, while sweeping out the entire Kähler cone of \({{\mathbb{CP}}_2\#\overline{\mathbb{CP}}_2}\). The same method also yields analogous solutions on every Hirzebruch surface. This allows us to display infinitely many geometrically distinct families of solutions of the Einstein–Maxwell equations on the smooth 4-manifolds \({S^2 \times S^2}\) and \({{\mathbb{CP}}_2\#\overline{\mathbb{CP}}_2}\).     

Massless Spin–Zero Particle and the Classical Action via Hamilton–Jacobi Equation in G?del Universe

In this letter we investigate the separability of the Klein–Gordon and Hamilton–Jacobi equation in G?del universe. We show that the Klein–Gordon eigen modes are quantized and the complete spectrum of the particle’s energy is a mixture of an azimuthal quantum number, m and a principal quantum number, n and a continuous wave number k. We also show that the Hamilton–Jacobi equation gives a closed function for classical action. These results may be used to calculate the Casimir vacuum energy in G?del universe.     

A Hamilton–Jacobi type equation for computing minimum potential energy paths

A new method for computing minimum-energy reaction paths is presented. Unlike existing approaches (e.g. intrinsic reaction coordinate methods), our approach works for any reactant configuration: the structure of the transition state, reactive intermediates and product will be determined by the algorithm, and so need not be known beforehand. The method we have developed is based on solving a Hamilton–Jacobi type equation. Specifically, we introduce a speed function so that the ‘first arrival times’ from the Hamilton–Jacobi equation correspond to least-potentials. Then, adopting a back-tracing method, we can use the first arrival times to determine the minimum-energy path between any classically allowed molecular conformation and the initial (reactant) conformation. The method is illustrated by applying it to six different systems: (1) a model system with four different minima in the potential energy surface, (2) a model Muller–Brown potential, (3) the isomerization reaction of malonaldehyde using a fitting potential energy surface, (4) a model Minyaev–Quapp potential representative of con- and dis-rotations of two BH₂ groups in the BH₂–CH₂–BH₂ molecule, (5) the F?+?H₂→FH?+?H reaction and (6) the H?+?FH?→?HF?+?H reaction. Our results demonstrate that the proposed method represents a robust alternative to existing techniques for finding chemical reaction paths.     

A local discontinuous Galerkin method for directly solving Hamilton–Jacobi equations

In this paper we propose a new local discontinuous Galerkin method to directly solve Hamilton–Jacobi equations. The scheme is a natural extension of the monotone scheme. For the linear case with constant coefficients, the method is equivalent to the discontinuous Galerkin method for conservation laws. Thus, stability and error analysis are obtained under the framework of conservation laws. For both convex and noneconvex Hamiltonian, optimal (k + 1)th order of accuracy for smooth solutions are obtained with piecewise kth order polynomial approximations. The scheme is numerically tested on a variety of one and two dimensional problems. The method works well to capture sharp corners (discontinuous derivatives) and have the solution converges to the viscosity solution.     

Hamilton—Jacobi quantization of constrained systems

The path integral quantization of contrained systems is analysed using Hamilton-Jacobi formalism. The integrability conditions are investigated and the results are in agreement with those obtained by Dirac’s method. Presented at the 10th Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June, 2001.     

The Linearized Vlasov and Vlasov–Fokker–Planck Equations in a Uniform Magnetic Field

Journal of Statistical Physics - We study the linearized Vlasov equations and the linearized Vlasov–Fokker–Planck equations in the weakly collisional limit in a uniform magnetic field....