Wikipedia / Hamilton–Jacobi equation
Wikipedia / Hamilton–Jacobi equation
In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics.
The Hamilton–Jacobi equation is a formulation of mechanics in which the motion of a particle can be represented as a wave. In this sense, it fulfilled a long-held goal of theoretical physics (dating at least to Johann Bernoulli in the eighteenth century) of finding an analogy between the propagation of light and the motion of a particle. The wave equation followed by mechanical systems is similar to, but not identical with, the Schrödinger equation, as described below; for this reason, the Hamilton–Jacobi equation is considered the "closest approach" of classical mechanics to quantum mechanics. The qualitative form of this connection is called Hamilton's optico-mechanical analogy.
In mathematics, the Hamilton–Jacobi equation is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations. It can be understood as a special case of the Hamilton–Jacobi–Bellman equation from dynamic programming.
Overview
The Hamilton–Jacobi equation is a first-order, non-linear partial differential equation
for a system of particles at coordinates is the system's Hamiltonian giving the system's energy. The solution of this equation is the action, , called Hamilton's principal function. by an indefinite integral of the form used in the principle of least action: Geometrical surfaces of constant action are perpendicular to system trajectories, creating a wavefront-like view of the system dynamics. This property of the Hamilton–Jacobi equation connects classical mechanics to quantum mechanics.
Mathematical formulation
Notation
A dot over a variable or list signifies the time derivative (see Newton's notation
The dot product
The action functional (a.k.a. Hamilton's principal function)
Definition
Let the Hessian matrix shows that the Euler–Lagrange equations
the initial value problem into the action functional results in the Hamilton's principal function (HPF)
where
Formula for the momenta
The momenta disappears, once the HPF is known.
be the (unique) extremal from the definition of the Hamilton's principal function the velocity at . Then
Formula
Given the Hamiltonian of a mechanical system, the Hamilton–Jacobi equation is a first-order, non-linear partial differential equation,
Alternatively, as described below, the Hamilton–Jacobi equation may be derived from Hamiltonian mechanics as the generating function for a canonical transformation
.
then describes the orbit in phase space in terms of these constants of motion constants and time.
Comparison with other formulations of mechanics
The Hamilton–Jacobi equation is a single generalized coordinates, the classical action.
For comparison, in the equivalent Euler–Lagrange equations of motion of Lagrangian mechanics, the conjugate momenta also do not appear; however, those equations are a system, generally second-order equations for the time evolution of the generalized coordinates. Similarly, Hamilton's equations of motion are another system of 2N.
Since the HJE is an equivalent expression of an integral minimization problem such as Hamilton's principle, the HJE can be useful in other problems of the calculus of variations and, more generally, in other branches of mathematics and physics, such as dynamical systems, symplectic geometry and quantum chaos. For example, the Hamilton–Jacobi equations can be used to determine the geodesics on a Riemannian manifold, an important variational problem in Riemannian geometry. However as a computational tool, the partial differential equations are notoriously complicated to solve except when is it possible to separate the independent variables; in this case the HJE become computationally useful.
Derivation using a canonical transformation
Any canonical transformation involving a type-2 generating function
so the new generalized coordinates and momenta are constants of motion and the new generalized coordinates.
Ideally, these equations can be inverted to find the original generalized coordinates, thus solving the original problem.
Separation of variables
When the problem allows additive separation of variables, the HJE leads directly to constants of motion. For example, the time t is sometimes called the abbreviated action or Hamilton's characteristic function and sometimes (see action principle names
To illustrate separability for other variables, a certain generalized coordinate
In that case, the function S can be partitioned into two functions, one that depends only on qk and another that depends only on the remaining generalized coordinates
Substitution of these formulae into the Hamilton–Jacobi equation shows that the function ψ), yielding a first-order ordinary differential equation
ordinary differential equations.
The separability of S depends both on the Hamiltonian and on the choice of generalized coordinates. For orthogonal coordinates and Hamiltonians that have no time dependence and are quadratic will be completely separable if the potential energy is additively separable in each coordinate, where the potential energy term for each coordinate is multiplied by the coordinate-dependent factor in the corresponding momentum term of the Hamiltonian (the Staeckel conditions). For illustration, several examples in orthogonal coordinates are worked in the next sections.
Separation in spherical coordinates
In spherical coordinates the Hamiltonian of a free particle moving in a conservative potential U can be written
coordinate is cyclic resulting in two ordinary differential equations are constants of the motion. This reduces the HJE to the ordinary differential equations.
Waves and particles
Optical wave fronts and trajectories
The HJE establishes a duality between trajectories and wavefronts.. Light rays and wave fronts are dual: if one is known, the other can be deduced.
is the medium's index of refraction is an infinitesimal arc length. From the above formulation, one can compute the ray paths using the Euler–Lagrange formulation; alternatively, one can compute the wave fronts by solving the Hamilton–Jacobi equation. Knowing one leads to knowing the other.
The above duality is very general and applies to all systems that derive from a variational principle: either compute the trajectories using Euler–Lagrange equations or the wave fronts by using Hamilton–Jacobi equation.
.
Relationship to the Schrödinger equation
The isosurfaces can be determined at any time t on the isosurface. The motion of such an isosurface can be thought of as a wave-space, although it does not obey the wave equation exactly. To show this, let S represent the phase is a constant (the Planck constant is a scalar normalization constant; changes in the amplitude of the wave be a complex number.
Starting with the Schrödinger equation and our ansatz, it can be deduced that This is the Schrödinger equation in nonlinear Riccati form.
The following table summarizes the similarities between optics and quantum mechancis. The Hamilton-Jacobi equation is to quantum mechanics what the eikonal equation is to the wave equation of optics. Both the eikonal equation and the Hamilton-Jacobi equation are slowly varying approximations (WKB approximations) of the corresponding more fundamental wave equations.
Optics EM Wave equation Time dependant eikonal equation Eikonal equation
Quantum mechanics Schrödinger equation Hamilton–Jacobi equation
Applications
HJE in a gravitational field
Using the energy–momentum relation in the form for a particle of rest mass are the contravariant coordinates of the metric tensor (i.e., the inverse metric) solved from the Einstein field equations is the speed of light. Setting the four-momentum equal to the four-gradient in other words, in a gravitational field.
HJE in electromagnetic fields
For a particle of rest mass moving in electromagnetic field with four-potential to obtain further solution for the particle trajectory and momentum: the cycle average of the vector potential.
A circularly polarized wave
In the case of circular polarization
directed along a magnetic field vector.
A monochromatic linearly polarized plane wave
vector.
An electromagnetic wave with a solenoidal magnetic field
For the electromagnetic wave with axial (solenoidal) magnetic field: due to axial symmetry of the solenoidal magnetic field.
See also
References
Further reading