Hamilton–Jacobi equation
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In mathematics, the Hamilton–Jacobi equation (HJE) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the Hamilton–Jacobi–Bellman equation. It is named for William Rowan Hamilton and Carl Gustav Jacob Jacobi.
In physics, the HamiltonJacobi equation 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 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 longheld 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.^{[1]}^{[2]}
Notation
Boldface variables such as represent a list of generalized coordinates,
A dot over a variable or list signifies the time derivative (see Newton's notation), e.g.,
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, e.g.,
Mathematical formulation
Given the Hamiltonian of a mechanical system (where , are coordinates and momenta of the system and is time) the HamiltonJacobi equation is written as a firstorder, nonlinear partial differential equation for the Hamilton's principal function ^{[3]},
The Hamilton's principal function is defined as the function of the upper limit of the action integral taken along the minimal action trajectory of the system,
where is the Lagrangian of the system and where the trajectory satisfies the EulerLagrange equation of the system,
Calculating the variation of with respect to variation of the endpoint coordinate,
leads to
Using this result and calculating the variation of with respect to variation of the time of the endpoint leads directly to the HamiltonJacobi equation,
or
where is the change of the trajectory at the old endpoint due to the time shift and where
is the Hamiltonian of the system.
Alternatively, as described below, the HamiltonJacobi equation may be derived from Hamiltonian mechanics by treating S as the generating function for a canonical transformation of the classical Hamiltonian
The conjugate momenta correspond to the first derivatives of S with respect to the generalized coordinates
As a solution to the Hamilton–Jacobi equation, the principal function contains N + 1 undetermined constants, the first N of them denoted as α_{1}, α_{2} ... α_{N}, and the last one coming from the integration of .
The relationship between p and q then describes the orbit in phase space in terms of these constants of motion. Furthermore, the quantities
are also constants of motion, and these equations can be inverted to find q as a function of all the α and β constants and time.^{[4]}
Comparison with other formulations of mechanics
The HJE is a single, firstorder partial differential equation for the function S of the N generalized coordinates q_{1}...q_{N} and the time t. The generalized momenta do not appear, except as derivatives of S. Remarkably, the function 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 N, generally secondorder equations for the time evolution of the generalized coordinates. Similarly, Hamilton's equations of motion are another system of 2N firstorder equations for the time evolution of the generalized coordinates and their conjugate momenta p_{1}...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.
Derivation using canonical transformation
Any canonical transformation involving a type2 generating function G_{2}(q, P, t) leads to the relations
and Hamilton's equations in terms of the new variables P, Q and new Hamiltonian K have the same form:
To derive the HJE, we choose a generating function G_{2}(q, P, t) 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
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 α_{1}, α_{2} ... α_{N}, i.e. P_{m} = α_{m}, and the new generalized coordinates Q are typically denoted as β_{1}, β_{2} ... β_{N}, so Q_{m} = β_{m}.
Setting the generating function equal to Hamilton's principal function, plus an arbitrary constant A:
the HJE automatically arises:
Once we have solved for S(q, α, t), these also give us the useful equations
or written in components for clarity
Ideally, these N equations can be inverted to find the original generalized coordinates q as a function of the constants α, β and t, thus solving the original problem.
Action and Hamilton's functions
Hamilton's principal function S and classical function H are both closely related to action. The total differential of S is:
so the time derivative of S is
Therefore,
so S is actually the classical action plus an undetermined constant.
When H does not explicitly depend on time,
in this case W is the same as abbreviated action.
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 t can be separated if the Hamiltonian does not depend on time explicitly. In that case, the time derivative in the HJE must be a constant, usually denoted (–E), giving the separated solution
where the timeindependent function W(q) is sometimes called Hamilton's characteristic function. The reduced Hamilton–Jacobi equation can then be written
To illustrate separability for other variables, we assume that a certain generalized coordinate q_{k} and its derivative appear together as a single function
in the Hamiltonian
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
Substitution of these formulae into the Hamilton–Jacobi equation shows that the function ψ must be a constant (denoted here as Γ_{k}), yielding a firstorder ordinary differential equation for S_{k}(q_{k}).
In fortunate cases, the function S can be separated completely into N functions S_{m}(q_{m})
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 coordinatedependent 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
The Hamilton–Jacobi equation is completely separable in these coordinates provided that there exist functions U_{r}(r), U_{θ}(θ) and U_{ϕ}(ϕ) such that U can be written in the analogous form
Substitution of the completely separated solution
into the HJE yields
This equation may be solved by successive integrations of ordinary differential equations, beginning with the equation for ϕ
where Γ_{ϕ} is a constant of the motion that eliminates the ϕ dependence from the Hamilton–Jacobi equation
The next ordinary differential equation involves the θ generalized coordinate
where Γ_{θ} is again a constant of the motion that eliminates the θ dependence and reduces the HJE to the final ordinary differential equation
whose integration completes the solution for S.
Elliptic cylindrical coordinates
The Hamiltonian in elliptic cylindrical coordinates can be written
where the foci of the ellipses are located at on the axis. The Hamilton–Jacobi equation is completely separable in these coordinates provided that U has an analogous form
where : , and are arbitrary functions. Substitution of the completely separated solution
 into the HJE yields
Separating the first ordinary differential equation
yields the reduced Hamilton–Jacobi equation (after rearrangement and multiplication of both sides by the denominator)
which itself may be separated into two independent ordinary differential equations
that, when solved, provide a complete solution for S.
Parabolic cylindrical coordinates
The Hamiltonian in parabolic cylindrical coordinates can be written
The Hamilton–Jacobi equation is completely separable in these coordinates provided that U has an analogous form
where U_{σ}(σ), U_{τ}(τ) and U_{z}(z) are arbitrary functions. Substitution of the completely separated solution
into the HJE yields
Separating the first ordinary differential equation
yields the reduced Hamilton–Jacobi equation (after rearrangement and multiplication of both sides by the denominator)
which itself may be separated into two independent ordinary differential equations
that, when solved, provide a complete solution for S.
Eikonal approximation and relationship to the Schrödinger equation
The isosurfaces of the function S(q; t) can be determined at any time t. The motion of an Sisosurface 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
where ħ is a constant (Planck's 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. We may then rewrite the Hamilton–Jacobi equation as
which is a nonlinear variant of the Schrödinger equation.
Conversely, starting with the Schrödinger equation and our ansatz for ψ, we arrive at ^{[5]}
The classical limit (ħ → 0) of the Schrödinger equation above becomes identical to the following variant of the Hamilton–Jacobi equation,
HJE in a gravitational field
Using the energy–momentum relation in the form;^{[6]}
for a particle of rest mass m travelling in curved space, where g^{αβ} 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 fourmomentum P_{α} equal to the fourgradient of the action S;
gives the Hamilton–Jacobi equation in the geometry determined by the metric g:
in other words, in a gravitational field.
HJE in electromagnetic fields
For a particle of rest mass and electric charge moving in electromagnetic field with fourpotential in vacuum, the Hamilton–Jacobi equation in geometry determined by the metric tensor has a form
and can be solved for the Hamilton Principal Action function to obtain further solution for the particle trajectory and momentum:^{[7]}
 ,
where and with the cycle average of the vector potential. Therefore:
a) For a wave with the circular polarization:
 ,
 ,
Hence
where , implying the particle moving along a circular trajectory with a permanent radius and an invariable value of momentum directed along a magnetic field vector.
b) For the flat, monochromatic, linearly polarized wave with a field directed along the axis
hence
 ,
 ,
implying the particle figure8 trajectory with a long its axis oriented along the electric field vector.
c) For the electromagnetic wave with axial (solenoidal) magnetic field:^{[8]}
hence
where is the magnetic field magnitude in a solenoid with the effective radius , inductivity , number of windings , and an electric current magnitude through the solenoid windings. The particle motion occurs along the figure8 trajectory in plane set perpendicular to the solenoid axis with arbitrary azimuth angle due to axial symmetry of the solenoidal magnetic field.
See also
References
 ↑ Goldstein, Herbert (1980). Classical Mechanics (2nd ed.). Reading, MA: AddisonWesley. pp. 484–492. ISBN 0201029189. (particularly the discussion beginning in the last paragraph of page 491)
 ↑ Sakurai, pp. 103–107.
 ↑ Hand, L. N.; Finch, J. D. (2008). Analytical Mechanics. Cambridge University Press. ISBN 9780521575720.
 ↑ Goldstein, Herbert (1980). Classical Mechanics (2nd ed.). Reading, MA: AddisonWesley. p. 440. ISBN 0201029189.
 ↑ Goldstein, Herbert (1980). Classical Mechanics (2nd ed.). Reading, MA: AddisonWesley. pp. 490–491. ISBN 0201029189.
 ↑ J.A. Wheeler; C. Misner; K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. pp. 649, 1188. ISBN 0716703440.
 ↑ L. Landau and E. Lifshitz. THE CLASSICAL THEORY OF FIELDS. ADDISONWESLEY PUBLISHING COMPANY, INC., Reading, Massachusetts, USA 1959.
 ↑ E. V. Shun'ko; D. E. Stevenson; V. S. Belkin (2014). "Inductively Coupling Plasma Reactor With Plasma Electron Energy Controllable in the Range from ~6 to ~100 eV". IEEE Transactions on Plasma Science. 2014 IEEE Trans. On Plasma Science. 42, part II (3): 774–785. Bibcode:2014ITPS...42..774S. doi:10.1109/TPS.2014.2299954.
Further reading
 Hamilton, W. (1833). "On a General Method of Expressing the Paths of Light, and of the Planets, by the Coefficients of a Characteristic Function" (PDF). Dublin University Review: 795–826.
 Hamilton, W. (1834). "On the Application to Dynamics of a General Mathematical Method previously Applied to Optics" (PDF). British Association Report: 513–518.
 Goldstein, Herbert (2002). Classical Mechanics (3rd ed.). Addison Wesley. ISBN 0201657023.
 Fetter, A. & Walecka, J. (2003). Theoretical Mechanics of Particles and Continua. Dover Books. ISBN 0486432610.
 Landau, L. D.; Lifshitz, E. M. (1975). Mechanics. Amsterdam: Elsevier.
 Sakurai, J. J. (1985). Modern Quantum Mechanics. Benjamin/Cummings Publishing. ISBN 0805375015.
 Jacobi, C. G. J. (1884), Vorlesungen über Dynamik, C. G. J. Jacobi's Gesammelte Werke (in German), Berlin: G. Reimer
 Nakane, Michiyo; Fraser, Craig G. (2002). "The Early History of HamiltonJacobi Dynamics". Centaurus. Wiley. 44 (3–4): 161. doi:10.1111/j.16000498.2002.tb00613.x. PMID 17357243.