Hamilton–Jacobi equation explained

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, Schrödinger's equation, as described below; for this reason, the Hamilton–Jacobi equation is considered the "closest approach" of classical mechanics to quantum mechanics.^{[1] [2]} 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.^[3]

Overview

The Hamilton–Jacobi equation is a first-order, non-linear partial differential equation

for a system of particles at coordinates . The function

H is the system's Hamiltonian giving the system's energy. The solution of the equation is the action functional,,^[4] called Hamilton's principal function in older textbooks. The solution can be related to the system Lagrangian l{L} by an indefinite integral of the form used in the principle of least action:$$\backslash \; S\; =\; \backslash int\; \backslash mathcal\backslash \; \backslash operatornamet\; +\; ~\backslash mathsf~$$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.^[5]

Mathematical formulation

Notation

Boldface variables such as

q represent a list of N generalized coordinates,$$\backslash mathbf\; =\; (q\_1,\; q\_2,\; \backslash ldots,\; q\_,\; q\_N)$$

A dot over a variable or list signifies the time derivative (see Newton's notation). For example,$$\backslash dot\; =\; \backslash frac.$$

The dot product notation between two lists of the same number of coordinates is a shorthand for the sum of the products of corresponding components, such as$$\backslash mathbf\; \backslash cdot\; \backslash mathbf\; =\; \backslash sum\_^N\; p\_k\; q\_k.$$

The action functional (a.k.a. Hamilton's principal function)

Definition

Let the Hessian matrix $H\_(\backslash mathbf,\backslash mathbf,t)\; =\; \backslash left\backslash \_$ be invertible. The relation$$\backslash frac\backslash frac\; =\backslash sum^n\_\backslash left(\backslash frac\; ^j+\; \backslash frac^j\; \backslash right)+\backslash frac,\backslash qquad\; i=1,\backslash ldots,n,$$shows that the Euler–Lagrange equations form a

n x n system of second-order ordinary differential equations. Inverting the matrix H _{\cal} transforms this system into$$\backslash ddot\; q^i\; =\; F\_i(\backslash mathbf,\backslash mathbf,t),\backslash \; i=1,\backslash ldots,\; n.$$

Let a time instant

t _{0} and a point q _{0} \in M in the configuration space be fixed. The existence and uniqueness theorems guarantee that, for every v _{0,} the initial value problem with the conditions \gamma| _{\tau = t _{0}} = q _{0} and • { \gamma}| ^{\tau = t _{0}} = v _{0} has a locally unique solution \gamma = \gamma(\tau; t _{0,q} _{0,v} _{0).} Additionally, let there be a sufficiently small time interval (t _{0,t} _{1)} such that extremals with different initial velocities v _{0} would not intersect in M x (t _{0,t} _{1).} The latter means that, for any q \in M and any t \in (t _{0,t} _{1),} there can be at most one extremal \gamma = \gamma(\tau;t,t _{0,q,q} _{0)} for which \gamma| _{\tau = t _{0}} = q _{0} and \gamma| _{\tau=t} = q . Substituting \gamma = \gamma(\tau;t,t _{0,q,q} _{0)} into the action functional results in the Hamilton's principal function (HPF)

where

\gamma = \gamma(\tau;t,t _{0,q,q} _{0),} \gamma| _{\tau = t _{0}} = q _{0,} \gamma| _{\tau=t} = q .

Formula for the momenta

The momenta are defined as the quantities $p\_i(\backslash mathbf,\backslash mathbf,t)\; =\; \backslash partial\; /\backslash partial\; \backslash dot\; q^i.$ This section shows that the dependency of

p _{i} on • q disappears, once the HPF is known.

Indeed, let a time instant

t _{0} and a point q _{0} in the configuration space be fixed. For every time instant t and a point q , let \gamma = \gamma(\tau;t,t _{0,q,q} _{0)} be the (unique) extremal from the definition of the Hamilton's principal function . Call v \stackrel{ def }\, \dot \gamma(\tau;t,t_0,\mathbf,\mathbf_0)|_ the velocity at . Then

Formula

H( q , p ,t) of a mechanical system, the Hamilton–Jacobi equation is a first-order, non-linear partial differential equation for the Hamilton's principal function S ,^[6]

Alternatively, as described below, the Hamilton–Jacobi equation may be derived from Hamiltonian mechanics by treating

S as the generating function for a canonical transformation of the classical Hamiltonian$$H\; =\; H(q\_1,q\_2,\backslash ldots,\; q\_N;p\_1,p\_2,\backslash ldots,\; p\_N;t).$$

The conjugate momenta correspond to the first derivatives of

S with respect to the generalized coordinates$$p\_k\; =\; \backslash frac.$$

As a solution to the Hamilton–Jacobi equation, the principal function contains

N +1 undetermined constants, the first N of them denoted as \alpha _{1,} \alpha _{2,} ... , \alpha _{N} , and the last one coming from the integration of (\partial S)/(\partial t) .

The relationship between

p and q then describes the orbit in phase space in terms of these constants of motion. Furthermore, the quantities$$\backslash beta\_k=\backslash frac,\backslash quad\; k=1,2,\; \backslash ldots,\; N$$are also constants of motion, and these equations can be inverted to find q as a function of all the \alpha and \beta constants and time.^[7]

Comparison with other formulations of mechanics

The Hamilton–Jacobi equation is a single, first-order partial differential equation for the function of the

N generalized coordinates q _{1,} q _{2,} ... , q _{N} and the time t . The generalized momenta do not appear, except as derivatives of S , 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

N , generally second-order equations for the time evolution of the generalized coordinates. Similarly, Hamilton's equations of motion are another system of 2N first-order equations for the time evolution of the generalized coordinates and their conjugate momenta p _{1,} p _{2,} ... , 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. 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.^[8]

Derivation using a canonical transformation

G _{2} ( q , P , t) leads to the relations$$\backslash mathbf\; =,\; \backslash quad\backslash mathbf\; =,\; \backslash quadK(\backslash mathbf,\backslash mathbf,t)\; =\; H(\backslash mathbf,\backslash mathbf,t)\; +$$and Hamilton's equations in terms of the new variables P , Q and new Hamiltonian K have the same form:$$\backslash dot\; =\; -,\backslash quad\; \backslash dot\; =\; +.$$

To derive the HJE, a generating function

G _{2} ( q , P , t) is chosen in such a way that, it will make the new Hamiltonian K =0 . Hence, all its derivatives are also zero, and the transformed Hamilton's equations become trivial$$\backslash dot\; =\; \backslash dot\; =\; 0$$so the new generalized coordinates and momenta are constants of motion. As they are constants, in this context the new generalized momenta P are usually denoted \alpha _{1,} \alpha _{2,} ... , \alpha _{N} , i.e. P _{m} = \alpha _{m} and the new generalized coordinates Q are typically denoted as \beta _{1,} \beta _{2,} ... , \beta _{N} , so Q _{m} = \beta _{m} .

Setting the generating function equal to Hamilton's principal function, plus an arbitrary constant

A :$$G\_2(\backslash mathbf,\backslash boldsymbol,t)=S(\backslash mathbf,t)+A,$$the HJE automatically arises$$\backslash mathbf=\backslash frac=\backslash frac\; \backslash ,\; \backslash rightarrow\; \backslash ,H(\backslash mathbf,\backslash mathbf,t)\; +\; =0\; \backslash ,\; \backslash rightarrow\; \backslash ,H\backslash left(\backslash mathbf,\backslash frac,t\backslash right)\; +\; =0.$$

When solved for

S( q ,\boldsymbol\alpha, t) , these also give us the useful equations$$\backslash mathbf\; =\; \backslash boldsymbol\backslash beta\; =,$$or written in components for clarity$$Q\_\; =\; \backslash beta\_\; =\; \backslash frac.$$q as a function of the constants \boldsymbol\alpha, \boldsymbol\beta, and t , 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 can be separated if the Hamiltonian does not depend on time explicitly. In that case, the time derivative

(\partial S)/(\partial t) in the HJE must be a constant, usually denoted (- E ), giving the separated solution$$S\; =\; W(q\_1,q\_2,\; \backslash ldots,\; q\_N)\; -\; Et$$where the time-independent function W( q ) is sometimes called the abbreviated action or Hamilton's characteristic function ^[8] and sometimes^[9] written S _{0} (see action principle names). The reduced Hamilton–Jacobi equation can then be written$$H\backslash left(\backslash mathbf,\backslash frac\; \backslash right)\; =\; E.$$q _{k} and its derivative (\partial S)/(\partial q _{k}) are assumed to appear together as a single function$$\backslash psi\; \backslash left(q\_k,\; \backslash frac\; \backslash right)$$in the Hamiltonian$$H\; =\; H(q\_1,q\_2,\backslash ldots,\; q\_,\; q\_,\backslash ldots,\; q\_N;\; p\_1,p\_2,\backslash ldots,\; p\_,\; p\_,\backslash ldots,\; p\_N;\; \backslash psi;\; t).$$

In that case, the function S can be partitioned into two functions, one that depends only on q_k and another that depends only on the remaining generalized coordinates$$S\; =\; S\_k(q\_k)\; +\; S\_\backslash text(q\_1,\backslash ldots,\; q\_,\; q\_,\; \backslash ldots,\; q\_N,\; t).$$

Substitution of these formulae into the Hamilton–Jacobi equation shows that the function ψ must be a constant (denoted here as

\Gamma _{k} ), yielding a first-order ordinary differential equation for S _{k} (q _{k),}

$$\backslash psi\; \backslash left(q\_k,\; \backslash frac\; \backslash right)\; =\; \backslash Gamma\_k.$$

In fortunate cases, the function

S can be separated completely into N functions S _{m} (q _{m),} $$S=S\_1(q\_1)+S\_2(q\_2)+\backslash cdots+S\_N(q\_N)-Et.$$

In such a case, the problem devolves to

N 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 in the generalized momenta,

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.

Examples in various coordinate systems

Spherical coordinates

In spherical coordinates the Hamiltonian of a free particle moving in a conservative potential U can be written$$H\; =\; \backslash frac\; \backslash left[p\_\{r\}^\{2\}\; +\; \backslash frac\{p\_\{\backslash theta\}^\{2\}\}\{r^\{2\}\}\; +\; \backslash frac\{p\_\{\backslash phi\}^\{2\}\}\{r^\{2\}\; \backslash sin^\{2\}\; \backslash theta\}\; \backslash right]\; +\; U(r,\; \backslash theta,\; \backslash phi).$$

The Hamilton–Jacobi equation is completely separable in these coordinates provided that there exist functions

U _{r} (r), U _{\theta} (\theta), U _{\phi} (\phi) such that U can be written in the analogous form$$U(r,\; \backslash theta,\; \backslash phi)\; =\; U\_(r)\; +\; \backslash frac\; +\; \backslash frac\; .$$

Substitution of the completely separated solution$$S\; =\; S\_(r)\; +\; S\_(\backslash theta)\; +\; S\_(\backslash phi)\; -\; Et$$into the HJE yields$$\backslash frac\; \backslash left(\backslash frac\; \backslash right)^\; +\; U\_(r)\; +\backslash frac\; \backslash left[\backslash left(\backslash frac\{\; dS\_\{\backslash theta\}\}\{\; d\backslash theta\}\; \backslash right)^\{2\}\; +\; 2m\; U\_\{\backslash theta\}(\backslash theta)\; \backslash right]\; +\backslash frac\; \backslash left[\backslash left(\backslash frac\{\; dS\_\{\backslash phi\}\}\{\; d\backslash phi\}\; \backslash right)^\{2\}\; +\; 2m\; U\_\{\backslash phi\}(\backslash phi)\; \backslash right]\; =\; E.$$

This equation may be solved by successive integrations of ordinary differential equations, beginning with the equation for

\phi $$\backslash left(\backslash frac\; \backslash right)^\; +\; 2m\; U\_(\backslash phi)\; =\; \backslash Gamma\_$$where \Gamma _{\phi} is a constant of the motion that eliminates the \phi dependence from the Hamilton–Jacobi equation$$\backslash frac\; \backslash left(\backslash frac\; \backslash right)^\; +\; U\_(r)\; +\; \backslash frac\; \backslash left[\backslash left(\backslash frac\{\; dS\_\{\backslash theta\}\}\{\; d\backslash theta\}\; \backslash right)^\{2\}\; +\; 2m\; U\_\{\backslash theta\}(\backslash theta)\; +\; \backslash frac\{\backslash Gamma\_\{\backslash phi\}\}\{\backslash sin^\{2\}\backslash theta\}\; \backslash right]\; =\; E.$$

The next ordinary differential equation involves the

\theta generalized coordinate$$\backslash left(\backslash frac\; \backslash right)^\; +\; 2m\; U\_(\backslash theta)\; +\; \backslash frac\; =\; \backslash Gamma\_$$where \Gamma _{\theta} is again a constant of the motion that eliminates the \theta dependence and reduces the HJE to the final ordinary differential equation$$\backslash frac\; \backslash left(\backslash frac\; \backslash right)^\; +\; U\_(r)\; +\; \backslash frac\; =\; E$$whose integration completes the solution for S .

Elliptic cylindrical coordinates

The Hamiltonian in elliptic cylindrical coordinates can be written$$H\; =\; \backslash frac\; +\; \backslash frac\; +\; U(\backslash mu,\; \backslash nu,\; z)$$where the foci of the ellipses are located at

\pm a on the x -axis. The Hamilton–Jacobi equation is completely separable in these coordinates provided that U has an analogous form$$U(\backslash mu,\; \backslash nu,\; z)\; =\; \backslash frac\; +\; U\_(z)$$where U _{\mu(\mu)} , U _{\nu(\nu)} and U _{z(z)} are arbitrary functions. Substitution of the completely separated solution$$S\; =\; S\_(\backslash mu)\; +\; S\_(\backslash nu)\; +\; S\_(z)\; -\; Et$$ into the HJE yields$$\backslash frac\; \backslash left(\backslash frac\; \backslash right)^\; +\; U\_(z)\; +\backslash frac\; \backslash left[\backslash left(\backslash frac\{\; dS\_\{\backslash mu\}\}\{\; d\backslash mu\}\; \backslash right)^\{2\}\; +\; \backslash left(\backslash frac\{\; dS\_\{\backslash nu\}\}\{\; d\backslash nu\}\; \backslash right)^\{2\}\; +\; 2m\; a^\{2\}\; U\_\{\backslash mu\}(\backslash mu)\; +\; 2m\; a^\{2\}\; U\_\{\backslash nu\}(\backslash nu)\backslash right]\; =\; E.$$

Separating the first ordinary differential equation$$\backslash frac\; \backslash left(\backslash frac\; \backslash right)^\; +\; U\_(z)\; =\; \backslash Gamma\_$$yields the reduced Hamilton–Jacobi equation (after re-arrangement and multiplication of both sides by the denominator)$$\backslash left(\backslash frac\; \backslash right)^\; +\; \backslash left(\backslash frac\; \backslash right)^\; +\; 2m\; a^\; U\_(\backslash mu)\; +\; 2m\; a^\; U\_(\backslash nu)\; =\; 2ma^\; \backslash left(\backslash sinh^\; \backslash mu\; +\; \backslash sin^\; \backslash nu\backslash right)\; \backslash left(E\; -\; \backslash Gamma\_\; \backslash right)$$which itself may be separated into two independent ordinary differential equations$$\backslash left(\backslash frac\; \backslash right)^\; +\; 2m\; a^\; U\_(\backslash mu)\; +\; 2ma^\; \backslash left(\backslash Gamma\_\; -\; E\; \backslash right)\; \backslash sinh^\; \backslash mu\; =\; \backslash Gamma\_$$$$\backslash left(\backslash frac\; \backslash right)^\; +\; 2m\; a^\; U\_(\backslash nu)\; +\; 2ma^\; \backslash left(\backslash Gamma\_\; -\; E\; \backslash right)\; \backslash sin^\; \backslash nu\; =\; \backslash Gamma\_$$that, when solved, provide a complete solution for

S .

Parabolic cylindrical coordinates

The Hamiltonian in parabolic cylindrical coordinates can be written$$H\; =\; \backslash frac\; +\; \backslash frac\; +\; U(\backslash sigma,\; \backslash tau,\; z).$$

The Hamilton–Jacobi equation is completely separable in these coordinates provided that

U has an analogous form$$U(\backslash sigma,\; \backslash tau,\; z)\; =\; \backslash frac\; +\; U\_(z)$$where U _{\sigma} (\sigma) , U _{\tau} (\tau) , and U _{z(z)} are arbitrary functions. Substitution of the completely separated solution$$S\; =\; S\_(\backslash sigma)\; +\; S\_(\backslash tau)\; +\; S\_(z)\; -\; Et\; +\; \backslash text$$into the HJE yields$$\backslash frac\; \backslash left(\backslash frac\; \backslash right)^\; +\; U\_(z)\; +\backslash frac\; \backslash left[\backslash left(\backslash frac\{\; dS\_\{\backslash sigma\}\}\{\; d\backslash sigma\}\; \backslash right)^\{2\}\; +\; \backslash left(\backslash frac\{\; dS\_\{\backslash tau\}\}\{\; d\backslash tau\}\; \backslash right)^\{2\}\; +\; 2m\; U\_\{\backslash sigma\}(\backslash sigma)\; +\; 2m\; U\_\{\backslash tau\}(\backslash tau)\backslash right]\; =\; E.$$

Separating the first ordinary differential equation$$\backslash frac\; \backslash left(\backslash frac\; \backslash right)^\; +\; U\_(z)\; =\; \backslash Gamma\_$$yields the reduced Hamilton–Jacobi equation (after re-arrangement and multiplication of both sides by the denominator)$$\backslash left(\backslash frac\; \backslash right)^\; +\; \backslash left(\backslash frac\; \backslash right)^\; +\; 2m\; U\_(\backslash sigma)\; +\; 2m\; U\_(\backslash tau)\; =\; 2m\; \backslash left(\backslash sigma^\; +\; \backslash tau^\; \backslash right)\; \backslash left(E\; -\; \backslash Gamma\_\; \backslash right)$$which itself may be separated into two independent ordinary differential equations$$\backslash left(\backslash frac\; \backslash right)^\; +\; 2m\; U\_(\backslash sigma)\; +\; 2m\backslash sigma^\; \backslash left(\backslash Gamma\_\; -\; E\; \backslash right)\; =\; \backslash Gamma\_$$$$\backslash left(\backslash frac\; \backslash right)^\; +\; 2m\; U\_(\backslash tau)\; +\; 2m\; \backslash tau^\; \backslash left(\backslash Gamma\_\; -\; E\; \backslash right)\; =\; \backslash Gamma\_$$that, when solved, provide a complete solution for

S .

Waves and particles

Optical wave fronts and trajectories

See main article: Hamilton's optico-mechanical analogy. The HJE establishes a duality between trajectories and wavefronts.^[10] For example, in geometrical optics, light can be considered either as “rays” or waves. The wave front can be defined as the surface $\_$ that the light emitted at time $t=0$ has reached at time $t$. Light rays and wave fronts are dual: if one is known, the other can be deduced.

More precisely, geometrical optics is a variational problem where the “action” is the travel time $T$ along a path,$$T\; =\; \backslash frac\backslash int\_^\; n\; \backslash ,\; ds$$ where $n$ is the medium's index of refraction and $ds$ 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.

The wave front at time $t$, for a system initially at $\backslash mathbf\_$ at time $t\_$, is defined as the collection of points $\backslash mathbf$ such that $S(\backslash mathbf,t)=\backslash text$. If $S(\backslash mathbf,t)$ is known, the momentum is immediately deduced.$$\backslash mathbf=\backslash frac.$$

Once $\backslash mathbf$ is known, tangents to the trajectories $\backslash dot$ are computed by solving the equation$$\backslash frac=\backslash boldsymbol$$for $\backslash dot$, where $$ is the Lagrangian. The trajectories are then recovered from the knowledge of $\backslash dot$.

Relationship to the Schrödinger equation

The isosurfaces of the function

S( q , t) can be determined at any time t. The motion of an S -isosurface as a function of time is defined by the motions of the particles beginning at the points q on the isosurface. The motion of such an isosurface can be thought of as a wave moving through q -space, although it does not obey the wave equation exactly. To show this, let S represent the phase of a wave$$\backslash psi\; =\; \backslash psi\_\; e^$$where \hbar is a constant (the Planck constant) introduced to make the exponential argument dimensionless; changes in the amplitude of the wave can be represented by having S be a complex number. The Hamilton–Jacobi equation is then rewritten as$$\backslash frac\; \backslash nabla^2\; \backslash psi\; -\; U\backslash psi\; =\; \backslash frac\; \backslash frac$$which is the Schrödinger equation.

Conversely, starting with the Schrödinger equation and our ansatz for

\psi , it can be deduced that^[11] $$\backslash frac\; \backslash left(\backslash nabla\; S\; \backslash right)^\; +\; U\; +\; \backslash frac\; =\; \backslash frac\; \backslash nabla^\; S.$$

The classical limit (

\hbar → 0 ) of the Schrödinger equation above becomes identical to the following variant of the Hamilton–Jacobi equation,$$\backslash frac\; \backslash left(\backslash nabla\; S\; \backslash right)^\; +\; U\; +\; \backslash frac\; =\; 0.$$

Applications

HJE in a gravitational field

m travelling in curved space, where g ^{\alpha} are the contravariant coordinates of the metric tensor (i.e., the inverse metric) solved from the Einstein field equations, and c is the speed of light. Setting the four-momentum P _{\alpha} equal to the four-gradient of the action S ,$$P\_\backslash alpha\; =-\backslash frac$$gives the Hamilton–Jacobi equation in the geometry determined by the metric g :$$g^\backslash frac\backslash frac\; -(mc)^2\; =\; 0,$$in other words, in a gravitational field.

HJE in electromagnetic fields

m and electric charge e moving in electromagnetic field with four-potential A _{i} = (\phi,\Alpha) in vacuum, the Hamilton–Jacobi equation in geometry determined by the metric tensor g ^{ik} = g _{ik} has a form$$g^\backslash left\; (\backslash frac\; +\; \backslash frac\; A\_i\; \backslash right)\; \backslash left\; (\backslash frac\; +\; \backslash frac\; A\_k\; \backslash right)\; =\; m^2\; c^2$$and can be solved for the Hamilton principal action function S to obtain further solution for the particle trajectory and momentum:^[13] $$x\; =\; -\; \backslash frac\; \backslash int\; A\_z\; \backslash ,d\backslash xi,$$$$y\; =\; -\; \backslash frac\; \backslash int\; A\_y\; \backslash ,d\backslash xi,$$$$z\; =\; -\; \backslash frac\; \backslash int\; (\backslash Alpha^2\; -\; \backslash overline)\; \backslash ,\; d\; \backslash xi,$$$$\backslash xi\; =\; ct\; -\; \backslash frac\backslash int\; (\backslash Alpha^2\; -\; \backslash overline)\; \backslash ,\; d\; \backslash xi,$$$$p\_x\; =\; -\; \backslash fracA\_x,\; \backslash quad\; p\_y\; =\; -\; \backslash fracA\_y,$$$$p\_z\; =\; \backslash frac(\backslash Alpha^2\; -\; \backslash overline),$$$$\backslash mathcal\; =\; c\backslash gamma\; +\; \backslash frac(\backslash Alpha^2\; -\; \backslash overline),$$where \xi = ct - z and \gamma ^{2} = m ^{2} c ^{2} + (e ^{2})/(c ^{2}) \overline{A} ^{2} with \overline{ A } the cycle average of the vector potential.

A circularly polarized wave

In the case of circular polarization,$$E\_x\; =\; E\_0\; \backslash sin\; \backslash omega\; \backslash xi\_1,\; \backslash quad\; E\_y\; =\; E\_0\; \backslash cos\; \backslash omega\; \backslash xi\_1,$$$$A\_x\; =\; \backslash frac\; \backslash cos\; \backslash omega\; \backslash xi\_1,\; \backslash quad\; A\_y\; =\; -\; \backslash frac\; \backslash sin\; \backslash omega\; \backslash xi\_1.$$

Hence$$x\; =\; -\; \backslash frac\; \backslash omega\; \backslash sin\; \backslash omega\; \backslash xi\_1,$$$$y\; =\; -\; \backslash frac\; \backslash omega\; \backslash cos\; \backslash omega\; \backslash xi\_1,$$$$p\_x\; =\; -\; \backslash frac\; \backslash omega\; \backslash cos\; \backslash omega\; \backslash xi\_1,$$$$p\_y\; =\; \backslash frac\; \backslash sin\; \backslash omega\; \backslash xi\_1,$$where

\xi _{1} = \xi /c , implying the particle moving along a circular trajectory with a permanent radius e cE _{0} / \gamma \omega ^{2} and an invariable value of momentum e E _{0} / \omega ^{2} directed along a magnetic field vector.

A monochromatic linearly polarized plane wave

For the flat, monochromatic, linearly polarized wave with a field

E directed along the axis y $$E\_y\; =\; E\_0\; \backslash cos\; \backslash omega\; \backslash xi\_1,$$$$A\_y\; =\; -\; \backslash frac\; \backslash sin\; \backslash omega\; \backslash xi\_1,$$hence$$x\; =\; \backslash text,$$$$y\_0\; =\; -\backslash frac,$$$$y\; =\; y\_0\; \backslash cos\; \backslash omega\; \backslash xi\_1,\; \backslash quad\; z\; =\; C\_z\; y\_0\; \backslash sin\; 2\backslash omega\; \backslash xi\_1,$$$$C\_z\; =\; \backslash frac,\; \backslash quad\; \backslash gamma^2\; =\; m^2\; c^2\; +\; \backslash frac,$$$$p\_x\; =\; 0,$$$$p\_\; =\; \backslash frac,$$$$p\_y\; =\; p\_\; \backslash sin\; \backslash omega\; \backslash xi\_1,$$$$p\_z\; =\; -\; 2C\_z\; p\_\; \backslash cos\; 2\backslash omega\; \backslash xi\_1$$implying the particle figure-8 trajectory with a long its axis oriented along the electric field E vector.

An electromagnetic wave with a solenoidal magnetic field

For the electromagnetic wave with axial (solenoidal) magnetic field:^[14] $$E\; =\; E\_\backslash phi\; =\; \backslash frac\; B\_0\; \backslash cos\; \backslash omega\; \backslash xi\_1,$$$$A\_\backslash phi\; =\; -\; \backslash rho\_0\; B\_0\; \backslash sin\; \backslash omega\; \backslash xi\_1\; =\; -\; \backslash frac\; I\_0\; \backslash sin\; \backslash omega\; \backslash xi\_1,$$hence$$x\; =\; \backslash text,$$$$y\_0\; =\; -\backslash frac,$$$$y\; =\; y\_0\; \backslash cos\; \backslash omega\; \backslash xi\_1,$$$$z\; =\; C\_z\; y\_0\; \backslash sin\; 2\backslash omega\; \backslash xi\_1,$$$$C\_z\; =\; \backslash frac,$$$$\backslash gamma^2\; =\; m^2\; c^2\; +\; \backslash frac,$$$$p\_x\; =\; 0,$$$$p\_\; =\; \backslash frac,$$$$p\_y\; =\; p\_\; \backslash sin\; \backslash omega\; \backslash xi\_1,$$$$p\_z\; =\; -\; 2C\_z\; p\_\; \backslash cos\; 2\; \backslash omega\; \backslash xi\_1,$$where

B _{0} is the magnetic field magnitude in a solenoid with the effective radius \rho _{0} , inductivity L _{s} , number of windings N _{s} , and an electric current magnitude I _{0} through the solenoid windings. The particle motion occurs along the figure-8 trajectory in yz plane set perpendicular to the solenoid axis with arbitrary azimuth angle \varphi due to axial symmetry of the solenoidal magnetic field.

See also

• Canonical transformation
• Constant of motion
• Hamiltonian vector field
• Hamilton–Jacobi–Einstein equation
• WKB approximation
• Action-angle coordinates

Further reading

• Book: Arnold. V.I.. Vladimir Arnold . Mathematical Methods of Classical Mechanics . 2. Springer . New York . 1989. 0-387-96890-3 .
• 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 . 795–826.
• Hamilton . W. . 1834 . On the Application to Dynamics of a General Mathematical Method previously Applied to Optics . British Association Report . 513–518.
• Book: A. . Fetter . amp . J. . Walecka . Theoretical Mechanics of Particles and Continua . Dover Books . 2003 . 978-0-486-43261-8 .
• Book: Landau . Lev Landau . L. D. . Lifshitz . E. M. . Mechanics . Elsevier . Amsterdam . 1975 .
• Book: Sakurai, J. J. . . Benjamin/Cummings Publishing . 1985 . 978-0-8053-7501-5 .
  • Michiyo . Nakane. Craig G. . Fraser . The Early History of Hamilton-Jacobi Dynamics . Centaurus. 44. 3–4. 161–227. 2002. 10.1111/j.1600-0498.2002.tb00613.x . 17357243.

Notes and References

1. Book: Goldstein, Herbert . Herbert Goldstein . 1980 . Classical Mechanics . 2nd . Addison-Wesley . Reading, MA . 978-0-201-02918-5 . 484–492. Classical Mechanics (textbook) . (particularly the discussion beginning in the last paragraph of page 491)
2. Book: Sakurai, J. J. . J. J. Sakurai . 1994 . Modern Quantum Mechanics . rev. . Addison-Wesley . Reading, MA . 0-201-53929-2 . 103–107. Modern Quantum Mechanics .
3. Book: Kálmán, Rudolf E. . The Theory of Optimal Control and the Calculus of Variations . Mathematical Optimization Techniques . Richard . Bellman . Berkeley . University of California Press . 1963 . 309–331 . 1033974 .
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