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In physics, the Hamilton–Jacobi equation (HJE) is a reformulation of classical mechanics and, thus, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics. The Hamilton–Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely.
The HJE is also the only formulation of mechanics in which the motion of a particle can be represented as a wave. In this sense, the HJE fulfilled a long-held goal of theoretical physics (dating at least to Johann Bernoulli in the 18th 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, Schrödinger's equation, as described below; for this reason, the HJE is considered the "closest approach" of classical mechanics to quantum mechanics.
Mathematical formulation
The Hamilton–Jacobi equation is a first-order, non-linear partial differential equation for a function Failed to parse (Missing texvc executable; please see math/README to configure.): S(q_{1},\dots,q_{N}; t)
called Hamilton's principal function
- Failed to parse (Missing texvc executable; please see math/README to configure.): H\left(q_{1},\dots,q_{N};\frac{\partial S}{\partial q_{1}},\dots,\frac{\partial S}{\partial q_{N}};t\right) + \frac{\partial S}{\partial t}=0.
As described below, this equation may be derived from Hamiltonian mechanics by treating Failed to parse (Missing texvc executable; please see math/README to configure.): S
as the generating function for a canonical transformation of the classical Hamiltonian Failed to parse (Missing texvc executable; please see math/README to configure.): H(q_{1},\dots,q_{N};p_{1},\dots,p_{N};t)
. The conjugate momenta correspond to the first derivatives of Failed to parse (Missing texvc executable; please see math/README to configure.): S
with respect to the generalized coordinates
- Failed to parse (Missing texvc executable; please see math/README to configure.): p_{k} \ \stackrel{\mathrm{def}}{=}\ \frac{\partial S}{\partial q_{k}}.
Similarly, the generalized coordinates can be obtained as derivatives with respect to the transformed momenta, as described below. By inverting these equations, one can determine the evolution of the mechanical system, i.e., determine the generalized coordinates as a function of time. The initial positions and velocities appear in the constants of integration for the solution Failed to parse (Missing texvc executable; please see math/README to configure.): S , which correspond to conserved quantities of the evolution such as the total energy, the angular momentum, or the Laplace-Runge-Lenz vector.
Comparison with other formulations of mechanics
The HJE is a single, first-order partial differential equation for the function Failed to parse (Missing texvc executable; please see math/README to configure.): S
of the Failed to parse (Missing texvc executable; please see math/README to configure.): N
generalized coordinates Failed to parse (Missing texvc executable; please see math/README to configure.): q_{1},\dots,q_{N}
and the time Failed to parse (Missing texvc executable; please see math/README to configure.): t
. The generalized momenta do not appear, except as derivatives of Failed to parse (Missing texvc executable; please see math/README to configure.): S . Remarkably, the function Failed to parse (Missing texvc executable; please see math/README to configure.): S
is equal to 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 of Failed to parse (Missing texvc executable; please see math/README to configure.): N , generally second-order equations for the time evolution of the generalized coordinates. As another comparison, Hamilton's equations of motion are likewise a system of Failed to parse (Missing texvc executable; please see math/README to configure.): 2N
first-order equations for the time evolution of the generalized coordinates and their conjugate momenta Failed to parse (Missing texvc executable; please see math/README to configure.): p_{1},\dots,p_{N}
.
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.
Notation
For brevity, we use boldface variables such as Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{q}
to represent the list of Failed to parse (Missing texvc executable; please see math/README to configure.): N
generalized coordinates
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{q} \ \stackrel{\mathrm{def}}{=}\ (q_{1}, q_{2}, \ldots, q_{N-1}, q_{N})
that need not transform like a vector under rotation. The dot product is defined here as the sum of the products of corresponding components, e.g.,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{p} \cdot \mathbf{q} \ \stackrel{\mathrm{def}}{=}\ \sum_{k=1}^{N} p_{k} q_{k}.
Derivation
Any canonical transformation involving a type-2 generating function Failed to parse (Missing texvc executable; please see math/README to configure.): G_{2}(\mathbf{q},\mathbf{P},t)
leads to the relations
- Failed to parse (Missing texvc executable; please see math/README to configure.): \qquad {\partial G_{2} \over \partial \mathbf{q}} = \mathbf{p}, \qquad {\partial G_{2} \over \partial \mathbf{P}} = \mathbf{Q}, \qquad K = H + {\partial G_{2} \over \partial t}
(See the canonical transformation article for more details.)
To derive the HJE, we choose a generating function Failed to parse (Missing texvc executable; please see math/README to configure.): S(\mathbf{q}, \mathbf{P}, t)
that makes the new Hamiltonian Failed to parse (Missing texvc executable; please see math/README to configure.): K
identically zero. Hence, all its derivatives are also zero, and Hamilton's equations become trivial
- Failed to parse (Missing texvc executable; please see math/README to configure.): {d\mathbf{P} \over dt} = {d\mathbf{Q} \over dt} = 0
i.e., the new generalized coordinates and momenta are constants of motion. The new generalized momenta Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{P}
are usually denoted Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha_{1}, \alpha_{2}, \ldots, \alpha_{N-1}, \alpha_{N}
, i.e., Failed to parse (Missing texvc executable; please see math/README to configure.): P_{m} = \alpha_{m} .
The HJE results from the equation for the transformed Hamiltonian Failed to parse (Missing texvc executable; please see math/README to configure.): K
- Failed to parse (Missing texvc executable; please see math/README to configure.): K(\mathbf{Q},\mathbf{P},t) = H(\mathbf{q},\mathbf{p},t) + {\partial S \over \partial t} = 0.
which is equivalent to the HJE
- Failed to parse (Missing texvc executable; please see math/README to configure.): H\left(\mathbf{q},{\partial S \over \partial \mathbf{q}},t\right) + {\partial S \over \partial t} = 0,
since Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{p}=\partial S/\partial \mathbf{q} .
The new generalized coordinates Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{Q}
are also constants, typically denoted as Failed to parse (Missing texvc executable; please see math/README to configure.): \beta_{1}, \beta_{2}, \ldots, \beta_{N-1}, \beta_{N}
. Once we have solved for Failed to parse (Missing texvc executable; please see math/README to configure.): S(\mathbf{q},\boldsymbol\alpha, t) , these also give useful equations
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{Q} = \boldsymbol\beta = {\partial S \over \partial \boldsymbol\alpha}
or written in components for clarity
- Failed to parse (Missing texvc executable; please see math/README to configure.): Q_{m} = \beta_{m} = \frac{\partial S(\mathbf{q},\boldsymbol\alpha, t)}{\partial \alpha_{m}}
Ideally, these Failed to parse (Missing texvc executable; please see math/README to configure.): N
equations can be inverted to find the original generalized coordinates Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{q}
as a function of the constants Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol\alpha
and Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol\beta
, thus solving the original problem.
Separation of variables
The HJE is most useful when it can be solved via additive separation of variables, which directly identifies constants of motion. For example, the time Failed to parse (Missing texvc executable; please see math/README to configure.): t
can be separated if the Hamiltonian does not depend on time explicitly. In that case, the time derivative Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial S}{\partial t}
in the HJE must be a constant (usually denoted Failed to parse (Missing texvc executable; please see math/README to configure.): -E
), giving the separated solution
- Failed to parse (Missing texvc executable; please see math/README to configure.): S = W(q_{1},\dots,q_{N}) - Et
where the time-independent function Failed to parse (Missing texvc executable; please see math/README to configure.): W(\mathbf{q})
is sometimes called Hamilton's characteristic function. The reduced Hamilton–Jacobi equation can then be written
- Failed to parse (Missing texvc executable; please see math/README to configure.): H\left(\mathbf{q},\frac{\partial S}{\partial \mathbf{q}} \right) = E
To illustrate separability for other variables, we assume that a certain generalized coordinate Failed to parse (Missing texvc executable; please see math/README to configure.): q_{k}
and its derivative Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial S}{\partial q_{k}}
appear together in the Hamiltonian as a single function Failed to parse (Missing texvc executable; please see math/README to configure.): \psi \left(q_{k}, \frac{\partial S}{\partial q_{k}} \right)
- Failed to parse (Missing texvc executable; please see math/README to configure.): H = H(q_{1},\dots,q_{k-1}, q_{k+1}, \ldots, q_{N};p_{1}, \dots, p_{k-1}, p_{k+1}, \ldots, p_{N}; \psi; t)
In that case, the function Failed to parse (Missing texvc executable; please see math/README to configure.): S
can be partitioned into two functions, one that depends only on Failed to parse (Missing texvc executable; please see math/README to configure.): q_{k}
and another that depends only on the remaining generalized coordinates
- Failed to parse (Missing texvc executable; please see math/README to configure.): S = S_{k}(q_{k}) + S_{rem}(q_{1}, \dots, q_{k-1}, q_{k+1}, \ldots, q_{N}; t)
Substitution of these formulae into the Hamilton–Jacobi equation shows that the function Failed to parse (Missing texvc executable; please see math/README to configure.): \psi
must be a constant (denoted here as Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma_{k}
), yielding a first-order ordinary differential equation for Failed to parse (Missing texvc executable; please see math/README to configure.): S_{k}(q_{k})
- Failed to parse (Missing texvc executable; please see math/README to configure.): \psi \left(q_{k}, \frac{d S_{k}}{d q_{k}} \right) = \Gamma_{k}
In fortunate cases, the function Failed to parse (Missing texvc executable; please see math/README to configure.): S
can be separated completely into Failed to parse (Missing texvc executable; please see math/README to configure.): N
functions Failed to parse (Missing texvc executable; please see math/README to configure.): S_{m}(q_{m})
- Failed to parse (Missing texvc executable; please see math/README to configure.): S=S_{1}(q_{1})+S_{2}(q_{2})+\cdots+S_{N}(q_{N})-Et
In such a case, the problem devolves to Failed to parse (Missing texvc executable; please see math/README to configure.): N
ordinary differential equations.
The separability of Failed to parse (Missing texvc executable; please see math/README to configure.): 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 in the generalized momenta, Failed to parse (Missing texvc executable; please see math/README to configure.): S
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.
Example of spherical coordinates
The Hamiltonian in spherical coordinates can be written
- Failed to parse (Missing texvc executable; please see math/README to configure.): H = \frac{1}{2m} \left[ p_{r}^{2} + \frac{p_{\theta}^{2}}{r^{2}} + \frac{p_{\phi}^{2}}{r^{2} \sin^{2} \theta} \right] + U(r, \theta, \phi)
The Hamilton–Jacobi equation is completely separable in these coordinates provided that Failed to parse (Missing texvc executable; please see math/README to configure.): U
has an analogous form
- Failed to parse (Missing texvc executable; please see math/README to configure.): U(r, \theta, \phi) = U_{r}(r) + \frac{U_{\theta}(\theta)}{r^{2}} + \frac{U_{\phi}(\phi)}{r^{2}\sin^{2}\theta}
where Failed to parse (Missing texvc executable; please see math/README to configure.): U_{r}(r) , Failed to parse (Missing texvc executable; please see math/README to configure.): U_{\theta}(\theta)
and Failed to parse (Missing texvc executable; please see math/README to configure.): U_{\phi}(\phi)
are arbitrary functions. Substitution of the completely separated solution Failed to parse (Missing texvc executable; please see math/README to configure.): S = S_{r}(r) + S_{\theta}(\theta) + S_{\phi}(\phi) - Et
into the HJE yields
- Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{2m} \left( \frac{dS_{r}}{dr} \right)^{2} + U_{r}(r) + \frac{1}{2m r^{2}} \left[ \left( \frac{dS_{\theta}}{d\theta} \right)^{2} + 2m U_{\theta}(\theta) \right] + \frac{1}{2m r^{2}\sin^{2}\theta} \left[ \left( \frac{dS_{\phi}}{d\phi} \right)^{2} + 2m U_{\phi}(\phi) \right] = E
This equation may be solved by successive integrations of ordinary differential equations, beginning with the Failed to parse (Missing texvc executable; please see math/README to configure.): \phi
equation
- Failed to parse (Missing texvc executable; please see math/README to configure.): \left( \frac{dS_{\phi}}{d\phi} \right)^{2} + 2m U_{\phi}(\phi) = \Gamma_{\phi}
where Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma_{\phi}
is a constant of the motion that eliminates the Failed to parse (Missing texvc executable; please see math/README to configure.): \phi
dependence from the Hamilton–Jacobi equation
- Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{2m} \left( \frac{dS_{r}}{dr} \right)^{2} + U_{r}(r) + \frac{1}{2m r^{2}} \left[ \left( \frac{dS_{\theta}}{d\theta} \right)^{2} + 2m U_{\theta}(\theta) + \frac{\Gamma_{\phi}}{\sin^{2}\theta} \right] = E
The next ordinary differential equation involves the Failed to parse (Missing texvc executable; please see math/README to configure.): \theta
generalized coordinate
- Failed to parse (Missing texvc executable; please see math/README to configure.): \left( \frac{dS_{\theta}}{d\theta} \right)^{2} + 2m U_{\theta}(\theta) + \frac{\Gamma_{\phi}}{\sin^{2}\theta} = \Gamma_{\theta}
where Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma_{\theta}
is again a constant of the motion that eliminates the Failed to parse (Missing texvc executable; please see math/README to configure.): \theta
dependence and reduces the HJE to the final ordinary differential equation
- Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{2m} \left( \frac{dS_{r}}{dr} \right)^{2} + U_{r}(r) + \frac{\Gamma_{\theta}}{2m r^{2}} = E
whose integration completes the solution for Failed to parse (Missing texvc executable; please see math/README to configure.): S .
Example of elliptic cylindrical coordinates
The Hamiltonian in elliptic cylindrical coordinates can be written
- Failed to parse (Missing texvc executable; please see math/README to configure.): H = \frac{p_{\mu}^{2} + p_{\nu}^{2}}{2ma^{2} \left( \sinh^{2} \mu + \sin^{2} \nu\right)} + \frac{p_{z}^{2}}{2m} + U(\mu, \nu, z)
where the foci of the ellipses are located at Failed to parse (Missing texvc executable; please see math/README to configure.): \pm a
on the Failed to parse (Missing texvc executable; please see math/README to configure.): x
-axis. The Hamilton–Jacobi equation is completely separable in these coordinates provided that Failed to parse (Missing texvc executable; please see math/README to configure.): U
has an analogous form
- Failed to parse (Missing texvc executable; please see math/README to configure.): U(\mu, \nu, z) = \frac{U_{\mu}(\mu) + U_{\nu}(\nu)}{\sinh^{2} \mu + \sin^{2} \nu} + U_{z}(z)
where Failed to parse (Missing texvc executable; please see math/README to configure.): U_{\mu}(\mu) , Failed to parse (Missing texvc executable; please see math/README to configure.): U_{\nu}(\nu)
and Failed to parse (Missing texvc executable; please see math/README to configure.): U_{z}(z)
are arbitrary functions. Substitution of the completely separated solution Failed to parse (Missing texvc executable; please see math/README to configure.): S = S_{\mu}(\mu) + S_{\nu}(\nu) + S_{z}(z) - Et
into the HJE yields
- Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{2m} \left( \frac{dS_{z}}{dz} \right)^{2} + U_{z}(z) + \frac{1}{2ma^{2} \left( \sinh^{2} \mu + \sin^{2} \nu\right)} \left[ \left( \frac{dS_{\mu}}{d\mu} \right)^{2} + \left( \frac{dS_{\nu}}{d\nu} \right)^{2} + 2m a^{2} U_{\mu}(\mu) + 2m a^{2} U_{\nu}(\nu)\right] = E
Separating the first ordinary differential equation
- Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{2m} \left( \frac{dS_{z}}{dz} \right)^{2} + U_{z}(z) = \Gamma_{z}
yields the reduced Hamilton–Jacobi equation (after re-arrangement and multiplication of both sides by the denominator)
- Failed to parse (Missing texvc executable; please see math/README to configure.): \left( \frac{dS_{\mu}}{d\mu} \right)^{2} + \left( \frac{dS_{\nu}}{d\nu} \right)^{2} + 2m a^{2} U_{\mu}(\mu) + 2m a^{2} U_{\nu}(\nu) = 2ma^{2} \left( \sinh^{2} \mu + \sin^{2} \nu\right) \left( E - \Gamma_{z} \right)
which itself may be separated into two independent ordinary differential equations
- Failed to parse (Missing texvc executable; please see math/README to configure.): \left( \frac{dS_{\mu}}{d\mu} \right)^{2} + 2m a^{2} U_{\mu}(\mu) + 2ma^{2} \left(\Gamma_{z} - E \right) \sinh^{2} \mu = \Gamma_{\mu}
- Failed to parse (Missing texvc executable; please see math/README to configure.): \left( \frac{dS_{\nu}}{d\nu} \right)^{2} + 2m a^{2} U_{\nu}(\nu) + 2ma^{2} \left(\Gamma_{z} - E \right) \sin^{2} \nu = \Gamma_{\nu}
that, when solved, provide a complete solution for Failed to parse (Missing texvc executable; please see math/README to configure.): S .
Example of parabolic cylindrical coordinates
The Hamiltonian in parabolic cylindrical coordinates can be written
- Failed to parse (Missing texvc executable; please see math/README to configure.): H = \frac{p_{\sigma}^{2} + p_{\tau}^{2}}{2m \left( \sigma^{2} + \tau^{2}\right)} + \frac{p_{z}^{2}}{2m} + U(\sigma, \tau, z)
The Hamilton–Jacobi equation is completely separable in these coordinates provided that Failed to parse (Missing texvc executable; please see math/README to configure.): U
has an analogous form
- Failed to parse (Missing texvc executable; please see math/README to configure.): U(\sigma, \tau, z) = \frac{U_{\sigma}(\sigma) + U_{\tau}(\tau)}{\sigma^{2} + \tau^{2}} + U_{z}(z)
where Failed to parse (Missing texvc executable; please see math/README to configure.): U_{\sigma}(\sigma) , Failed to parse (Missing texvc executable; please see math/README to configure.): U_{\tau}(\tau)
and Failed to parse (Missing texvc executable; please see math/README to configure.): U_{z}(z)
are arbitrary functions. Substitution of the completely separated solution Failed to parse (Missing texvc executable; please see math/README to configure.): S = S_{\sigma}(\sigma) + S_{\tau}(\tau) + S_{z}(z) - Et
into the HJE yields
- Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{2m} \left( \frac{dS_{z}}{dz} \right)^{2} + U_{z}(z) + \frac{1}{2m \left( \sigma^{2} + \tau^{2} \right)} \left[ \left( \frac{dS_{\sigma}}{d\sigma} \right)^{2} + \left( \frac{dS_{\tau}}{d\tau} \right)^{2} + 2m U_{\sigma}(\sigma) + 2m U_{\tau}(\tau)\right] = E
Separating the first ordinary differential equation
- Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{2m} \left( \frac{dS_{z}}{dz} \right)^{2} + U_{z}(z) = \Gamma_{z}
yields the reduced Hamilton–Jacobi equation (after re-arrangement and multiplication of both sides by the denominator)
- Failed to parse (Missing texvc executable; please see math/README to configure.): \left( \frac{dS_{\sigma}}{d\sigma} \right)^{2} + \left( \frac{dS_{\tau}}{d\tau} \right)^{2} + 2m U_{\sigma}(\sigma) + 2m U_{\tau}(\tau) = 2m \left( \sigma^{2} + \tau^{2} \right) \left( E - \Gamma_{z} \right)
which itself may be separated into two independent ordinary differential equations
- Failed to parse (Missing texvc executable; please see math/README to configure.): \left( \frac{dS_{\sigma}}{d\sigma} \right)^{2} + 2m U_{\sigma}(\sigma) + 2m\sigma^{2} \left(\Gamma_{z} - E \right) = \Gamma_{\sigma}
- Failed to parse (Missing texvc executable; please see math/README to configure.): \left( \frac{dS_{\tau}}{d\tau} \right)^{2} + 2m a^{2} U_{\tau}(\tau) + 2m \tau^{2} \left(\Gamma_{z} - E \right) = \Gamma_{\tau}
that, when solved, provide a complete solution for Failed to parse (Missing texvc executable; please see math/README to configure.): S .
Eikonal approximation and relationship to the Schrödinger equation
The isosurfaces of the function Failed to parse (Missing texvc executable; please see math/README to configure.): S(\mathbf{q}; t)
can be determined at any time Failed to parse (Missing texvc executable; please see math/README to configure.): t
. The motion of an Failed to parse (Missing texvc executable; please see math/README to configure.): S -isosurface as a function of time is defined by the motions of the particles beginning at the points Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{q}
on the isosurface. The motion of such an isosurface can be thought of as a wave moving through Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{q}
space, although it does not obey the wave equation exactly. To show this, let Failed to parse (Missing texvc executable; please see math/README to configure.): S
represent the phase of a wave
- Failed to parse (Missing texvc executable; please see math/README to configure.): \psi = \psi_{0} e^{iS/\hbar}
where Failed to parse (Missing texvc executable; please see math/README to configure.): \hbar
is a constant introduced to make the exponential argument unitless; changes in the amplitude of the wave can be represented by having Failed to parse (Missing texvc executable; please see math/README to configure.): S
be a complex number. We may then re-write the Hamilton–Jacobi equation as
- Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\hbar^{2}}{2m\psi} \left( \boldsymbol\nabla \psi \right)^{2} - U\psi = \frac{\hbar}{i} \frac{\partial \psi}{\partial t}
which is a nonlinear variant of the Schrödinger equation.
Conversely, starting with the Schrödinger equation and our Ansatz for Failed to parse (Missing texvc executable; please see math/README to configure.): \psi , we arrive at,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{2m} \left( \boldsymbol\nabla S \right)^{2} + U + \frac{\partial S}{\partial t} = \frac{i\hbar}{2m} \nabla^{2} S
The classical limit (Failed to parse (Missing texvc executable; please see math/README to configure.): \hbar \rightarrow 0 ) of the Schrödinger equation above becomes identical to the following variant of the Hamilton-Jacobi equation,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{2m} \left( \boldsymbol\nabla S \right)^{2} + U + \frac{\partial S}{\partial t} = 0
The Hamilton-Jacobi equation in the gravitational field
- Failed to parse (Missing texvc executable; please see math/README to configure.): g^{ik}\frac{\partial{S}}{\partial{x^{i}}}\frac{\partial{S}}{\partial{x^{k}}} - m^{2}c^{2} = 0
where Failed to parse (Missing texvc executable; please see math/README to configure.): g^{ik}
are the contravariant components of the metric tensor, m is the rest mass of the particle and c is the speed of light.
See also
References
- Hamilton W. (1833) "On a General Method of Expressing the Paths of Light, and of the Planets, by the Coefficients of a Characteristic Function", Dublin University Review, pp. 795-826.
- Hamilton W. (1834) "On the Application to Dynamics of a General Mathematical Method previously Applied to Optics", British Association Report, pp.513-518.
- A. Fetter and J. Walecka (2003). Theoretical Mechanics of Particles and Continua. Dover Books. ISBN 0-486-43261-0.
es:Ecuación de Hamilton-Jacobi it:teoria di Hamilton-Jacobi ru:Уравнения Гамильтона — Якоби uk:Рівняння Гамільтона-Якобі
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