On the Asymptotics of the Schrödinger Equation Solutions and the Euler Model of an Ideal Fluid

In this paper we analyze the asymptotics of the Schrödinger equation solutions with respect to a small parameter ~. It is well known, that short- waveasymptoticstosolutionsofthisequationleadstothepairofequations— the Hamilton–Jacobi equation for the phase and the continuity equation. These equations coincide with the ones for the potential flows of an ideal fluid. The wave function is invariant with respect to the complex plane rotations group, and the asymptotics is constructed as a point-dependent action of this group on some function that is found by solving the transfer equation. It is shown in the paper, that if the Heisenberg group is used instead of the rotation group, then the limit of the equations solutions with ~ tending to zero leads to the equations for vortex flows of an ideal fluid in a potential field of forces. If the original Schrödinger equation is nonlinear, then equations for barotropic processes in an ideal fluid are obtained.

Killing Tensor and Carter Constant for Painlevé–Gullstrand Form of Lense–Thirring Spacetime

Recently, the authors have formulated and explored a novel Painlevé–Gullstrand variant of the Lense–Thirring spacetime, which has some particularly elegant features, including unit-lapse, intrinsically flat spatial 3-slices, and some particularly simple geodesics—the “rain” geodesics. At the linear level in the rotation parameter, this spacetime is indistinguishable from the usual slow-rotation expansion of Kerr. Herein, we shall show that this spacetime possesses a nontrivial Killing tensor, implying separability of the Hamilton–Jacobi equation. Furthermore, we shall show that the Klein–Gordon equation is also separable on this spacetime. However, while the Killing tensor has a 2-form square root, we shall see that this 2-form square root of the Killing tensor is not a Killing–Yano tensor. Finally, the Killing-tensor-induced Carter constant is easily extracted, and now, with a fourth constant of motion, the geodesics become (in principle) explicitly integrable.

The Lagrange–Charpit Theory of the Hamilton–Jacobi Problem

The Lagrange–Charpit theory is a geometric method of determining a complete integral by means of a constant of the motion of a vector field defined on a phase space associated to a nonlinear PDE of first order. In this article, we establish this theory on the symplectic structure of the cotangent bundle $$T^{*}Q$$ T ∗ Q of the configuration manifold Q. In particular, we use it to calculate explicitly isotropic submanifolds associated with a Hamilton–Jacobi equation.

Quasipotentials in the nonequilibrium stationary states or a method to get explicit solutions of Hamilton–Jacobi equations

We assume that a system at a mesoscopic scale is described by a field ϕ(x, t) that evolves by a Langevin equation with a white noise whose intensity is controlled by a parameter 1 / Ω . The system stationary state distribution in the small noise limit (Ω → ∞) is of the form P st [ϕ] ≃ exp(−ΩV 0[ϕ]), where V 0[ϕ] is called the quasipotential. V 0 is the unknown of a Hamilton–Jacobi equation. Therefore, V 0 can be written as an action computed along a path that is the solution from Hamilton’s equation that typically cannot be solved explicitly. This paper presents a theoretical scheme that builds a suitable canonical transformation that permits us to do such integration by deforming the original path into a straight line and including some weights along with it. We get the functional form of such weights through conditions on the existence and structure of the canonical transformation. We apply the scheme to get the quasipotential algebraically for several one-dimensional nonequilibrium models as the diffusive and reaction–diffusion systems.

Solution of the Hamilton – Jacobi Equation in a Central Potential Using the Separation of Variables Method with Staeckel Boundary Conditions

This manuscript aims at solving Hamilton-Jacobi equation in a central potential using the separation of variables technique with Staeckel boundary conditions. Our results show that the Hamilton – Jacobi variables can be completely separated, which agrees with other results employing different methods. Keywords: Lagrangian mechanics, Hamilton-Jacobi, Staeckel boundary conditions, Staeckel matrix, Staeckel vector, Hamilton's characteristic function, Hamilton's principal function.

The Hamilton–Jacobi Theory for Contact Hamiltonian Systems

The aim of this paper is to develop a Hamilton–Jacobi theory for contact Hamiltonian systems. We find several forms for a suitable Hamilton–Jacobi equation accordingly to the Hamiltonian and the evolution vector fields for a given Hamiltonian function. We also analyze the corresponding formulation on the symplectification of the contact Hamiltonian system, and establish the relations between these two approaches. In the last section, some examples are discussed.

Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry

If $U:[0,+\infty [\times M$ U : [ 0 , + ∞ [ × M is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation $$ \partial _{t}U+ H(x,\partial _{x}U)=0, $$ ∂ t U + H ( x , ∂ x U ) = 0 , where $M$ M is a not necessarily compact manifold, and $H$ H is a Tonelli Hamiltonian, we prove the set $\Sigma (U)$ Σ ( U ) , of points in $]0,+\infty [\times M$ ] 0 , + ∞ [ × M where $U$ U is not differentiable, is locally contractible. Moreover, we study the homotopy type of $\Sigma (U)$ Σ ( U ) . We also give an application to the singularities of the distance function to a closed subset of a complete Riemannian manifold.