Hamilton–Jacobi equation

In spherical coordinates the Hamiltonian of a free particle moving in a conservative potential U can be written

${\displaystyle H={\frac {1}{2m}}\left[p_{r}^{2}+{\frac {p_{\theta }^{2}}{r^{2}}}+{\frac {p_{\phi }^{2}}{r^{2}\sin ^{2}\theta }}\right]+U(r,\theta ,\phi ).}$

The Hamilton–Jacobi equation is completely separable in these coordinates provided that there exist functions:${\displaystyle U_{r}(r),U_{\theta }(\theta ),U_{\phi }(\phi )}$ such that ${\displaystyle U}$ can be written in the analogous form

${\displaystyle U(r,\theta ,\phi )=U_{r}(r)+{\frac {U_{\theta }(\theta )}{r^{2}}}+{\frac {U_{\phi }(\phi )}{r^{2}\sin ^{2}\theta }}.}$

Substitution of the completely separated solution

${\displaystyle S=S_{r}(r)+S_{\theta }(\theta )+S_{\phi }(\phi )-Et}$

into the HJE yields

${\displaystyle {\frac {1}{2m}}\left({\frac {dS_{r}}{dr}}\right)^{2}+U_{r}(r)+{\frac {1}{2mr^{2}}}\left[\left({\frac {dS_{\theta }}{d\theta }}\right)^{2}+2mU_{\theta }(\theta )\right]+{\frac {1}{2mr^{2}\sin ^{2}\theta }}\left[\left({\frac {dS_{\phi }}{d\phi }}\right)^{2}+2mU_{\phi }(\phi )\right]=E.}$

This equation may be solved by successive integrations of ordinary differential equations, beginning with the equation for ${\displaystyle \phi }$

${\displaystyle \left({\frac {dS_{\phi }}{d\phi }}\right)^{2}+2mU_{\phi }(\phi )=\Gamma _{\phi }}$

where ${\displaystyle \Gamma _{\phi }}$ is a constant of the motion that eliminates the ${\displaystyle \phi }$ dependence from the Hamilton–Jacobi equation

${\displaystyle {\frac {1}{2m}}\left({\frac {dS_{r}}{dr}}\right)^{2}+U_{r}(r)+{\frac {1}{2mr^{2}}}\left[\left({\frac {dS_{\theta }}{d\theta }}\right)^{2}+2mU_{\theta }(\theta )+{\frac {\Gamma _{\phi }}{\sin ^{2}\theta }}\right]=E.}$

The next ordinary differential equation involves the ${\displaystyle \theta }$ generalized coordinate

${\displaystyle \left({\frac {dS_{\theta }}{d\theta }}\right)^{2}+2mU_{\theta }(\theta )+{\frac {\Gamma _{\phi }}{\sin ^{2}\theta }}=\Gamma _{\theta }}$

where ${\displaystyle \Gamma _{\theta }}$ is again a constant of the motion that eliminates the ${\displaystyle \theta }$ dependence and reduces the HJE to the final ordinary differential equation

${\displaystyle {\frac {1}{2m}}\left({\frac {dS_{r}}{dr}}\right)^{2}+U_{r}(r)+{\frac {\Gamma _{\theta }}{2mr^{2}}}=E}$

whose integration completes the solution for ${\displaystyle S}$.

The Hamiltonian in elliptic cylindrical coordinates can be written

${\displaystyle H={\frac {p_{\mu }^{2}+p_{\nu }^{2}}{2ma^{2}\left(\sinh ^{2}\mu +\sin ^{2}\nu \right)}}+{\frac {p_{z}^{2}}{2m}}+U(\mu ,\nu ,z)}$

where the foci of the ellipses are located at ${\displaystyle \pm a}$ on the ${\displaystyle x}$-axis. The Hamilton–Jacobi equation is completely separable in these coordinates provided that ${\displaystyle U}$ has an analogous form

${\displaystyle U(\mu ,\nu ,z)={\frac {U_{\mu }(\mu )+U_{\nu }(\nu )}{\sinh ^{2}\mu +\sin ^{2}\nu }}+U_{z}(z)}$

where : ${\displaystyle U_{\mu }(\mu )}$, ${\displaystyle U_{\nu }(\nu )}$ and ${\displaystyle U_{z}(z)}$ are arbitrary functions. Substitution of the completely separated solution

${\displaystyle S=S_{\mu }(\mu )+S_{\nu }(\nu )+S_{z}(z)-Et}$ into the HJE yields

${\displaystyle {\frac {1}{2m}}\left({\frac {dS_{z}}{dz}}\right)^{2}+U_{z}(z)+{\frac {1}{2ma^{2}\left(\sinh ^{2}\mu +\sin ^{2}\nu \right)}}\left[\left({\frac {dS_{\mu }}{d\mu }}\right)^{2}+\left({\frac {dS_{\nu }}{d\nu }}\right)^{2}+2ma^{2}U_{\mu }(\mu )+2ma^{2}U_{\nu }(\nu )\right]=E.}$

Separating the first ordinary differential equation

${\displaystyle {\frac {1}{2m}}\left({\frac {dS_{z}}{dz}}\right)^{2}+U_{z}(z)=\Gamma _{z}}$

yields the reduced Hamilton–Jacobi equation (after re-arrangement and multiplication of both sides by the denominator)

${\displaystyle \left({\frac {dS_{\mu }}{d\mu }}\right)^{2}+\left({\frac {dS_{\nu }}{d\nu }}\right)^{2}+2ma^{2}U_{\mu }(\mu )+2ma^{2}U_{\nu }(\nu )=2ma^{2}\left(\sinh ^{2}\mu +\sin ^{2}\nu \right)\left(E-\Gamma _{z}\right)}$

which itself may be separated into two independent ordinary differential equations

${\displaystyle \left({\frac {dS_{\mu }}{d\mu }}\right)^{2}+2ma^{2}U_{\mu }(\mu )+2ma^{2}\left(\Gamma _{z}-E\right)\sinh ^{2}\mu =\Gamma _{\mu }}$

${\displaystyle \left({\frac {dS_{\nu }}{d\nu }}\right)^{2}+2ma^{2}U_{\nu }(\nu )+2ma^{2}\left(\Gamma _{z}-E\right)\sin ^{2}\nu =\Gamma _{\nu }}$

that, when solved, provide a complete solution for ${\displaystyle S}$.

The Hamiltonian in parabolic cylindrical coordinates can be written

${\displaystyle H={\frac {p_{\sigma }^{2}+p_{\tau }^{2}}{2m\left(\sigma ^{2}+\tau ^{2}\right)}}+{\frac {p_{z}^{2}}{2m}}+U(\sigma ,\tau ,z).}$

The Hamilton–Jacobi equation is completely separable in these coordinates provided that ${\displaystyle U}$ has an analogous form

${\displaystyle U(\sigma ,\tau ,z)={\frac {U_{\sigma }(\sigma )+U_{\tau }(\tau )}{\sigma ^{2}+\tau ^{2}}}+U_{z}(z)}$

where ${\displaystyle U_{\sigma }(\sigma )}$, ${\displaystyle U_{\tau }(\tau )}$, and ${\displaystyle U_{z}(z)}$ are arbitrary functions. Substitution of the completely separated solution

${\displaystyle S=S_{\sigma }(\sigma )+S_{\tau }(\tau )+S_{z}(z)-Et+{\text{constant}}}$

into the HJE yields

${\displaystyle {\frac {1}{2m}}\left({\frac {dS_{z}}{dz}}\right)^{2}+U_{z}(z)+{\frac {1}{2m\left(\sigma ^{2}+\tau ^{2}\right)}}\left[\left({\frac {dS_{\sigma }}{d\sigma }}\right)^{2}+\left({\frac {dS_{\tau }}{d\tau }}\right)^{2}+2mU_{\sigma }(\sigma )+2mU_{\tau }(\tau )\right]=E.}$

Separating the first ordinary differential equation

${\displaystyle {\frac {1}{2m}}\left({\frac {dS_{z}}{dz}}\right)^{2}+U_{z}(z)=\Gamma _{z}}$

yields the reduced Hamilton–Jacobi equation (after re-arrangement and multiplication of both sides by the denominator)

${\displaystyle \left({\frac {dS_{\sigma }}{d\sigma }}\right)^{2}+\left({\frac {dS_{\tau }}{d\tau }}\right)^{2}+2mU_{\sigma }(\sigma )+2mU_{\tau }(\tau )=2m\left(\sigma ^{2}+\tau ^{2}\right)\left(E-\Gamma _{z}\right)}$

which itself may be separated into two independent ordinary differential equations

${\displaystyle \left({\frac {dS_{\sigma }}{d\sigma }}\right)^{2}+2mU_{\sigma }(\sigma )+2m\sigma ^{2}\left(\Gamma _{z}-E\right)=\Gamma _{\sigma }}$

${\displaystyle \left({\frac {dS_{\tau }}{d\tau }}\right)^{2}+2mU_{\tau }(\tau )+2m\tau ^{2}\left(\Gamma _{z}-E\right)=\Gamma _{\tau }}$

that, when solved, provide a complete solution for ${\displaystyle S}$.

The HJE establishes a duality between trajectories and wave fronts.[6] For example, in geometrical optics, light can be considered either as “rays” or waves. The wave front can be defined as the surface ${\textstyle {\cal {C}}_{t}}$ that the light emitted at time ${\textstyle t=0}$ has reached at time ${\textstyle t}$. 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 ${\textstyle T}$ along a path,

$${\displaystyle T={\frac {1}{c}}\int _{A}^{B}nds}$$

where ${\textstyle n}$ is the medium's index of refraction and ${\textstyle ds}$ 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 ${\textstyle t}$, for a system initially at ${\textstyle {\textbf {q}}_{0}}$ at time ${\textstyle t_{0}}$, is defined as the collection of points ${\textstyle {\textbf {q}}}$ such that ${\textstyle S({\textbf {q}},t)={\text{const}}}$. If ${\textstyle S({\textbf {q}},t)}$ is known, the momentum is immediately deduced.

$${\displaystyle {\textbf {p}}={\frac {\partial S}{\partial {\textbf {q}}}}.}$$

Once ${\textstyle {\textbf {p}}}$ is known, tangents to the trajectories ${\textstyle {\dot {\textbf {q}}}}$ are computed by solving the equation

$${\displaystyle {\frac {\partial {\cal {L}}}{\partial {\dot {\textbf {q}}}}}={\boldsymbol {p}}}$$

for ${\textstyle {\dot {\textbf {q}}}}$, where ${\textstyle {\cal {L}}}$ is the Lagrangian. The trajectories are then recovered from the knowledge of ${\textstyle {\dot {\textbf {q}}}}$.

The isosurfaces of the function ${\displaystyle S({\textbf {q}},t)}$ can be determined at any time t. The motion of an ${\displaystyle S}$-isosurface as a function of time is defined by the motions of the particles beginning at the points ${\displaystyle {\textbf {q}}}$ on the isosurface. The motion of such an isosurface can be thought of as a wave moving through ${\displaystyle {\textbf {q}}}$-space, although it does not obey the wave equation exactly. To show this, let S represent the phase of a wave

${\displaystyle \psi =\psi _{0}e^{iS/\hbar }}$

where ${\displaystyle \hbar }$ 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 ${\displaystyle S}$ be a complex number. The Hamilton–Jacobi equation is then rewritten as

${\displaystyle {\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi -U\psi ={\frac {\hbar }{i}}{\frac {\partial \psi }{\partial t}}}$

which is the Schrödinger equation.

Conversely, starting with the Schrödinger equation and our ansatz for ${\displaystyle \psi }$, it can be deduced that[7]

${\displaystyle {\frac {1}{2m}}\left(\nabla S\right)^{2}+U+{\frac {\partial S}{\partial t}}={\frac {i\hbar }{2m}}\nabla ^{2}S.}$

The classical limit (${\displaystyle \hbar \rightarrow 0}$) of the Schrödinger equation above becomes identical to the following variant of the Hamilton–Jacobi equation,

${\displaystyle {\frac {1}{2m}}\left(\nabla S\right)^{2}+U+{\frac {\partial S}{\partial t}}=0.}$

Using the energy–momentum relation in the form[8]

${\displaystyle g^{\alpha \beta }P_{\alpha }P_{\beta }-(mc)^{2}=0}$

for a particle of rest mass ${\displaystyle m}$ travelling in curved space, where ${\displaystyle g^{\alpha \beta }}$are the contravariant coordinates of the metric tensor (i.e., the inverse metric) solved from the Einstein field equations, and ${\displaystyle c}$ is the speed of light. Setting the four-momentum ${\displaystyle P_{\alpha }}$ equal to the four-gradient of the action ${\displaystyle S}$,

${\displaystyle P_{\alpha }=-{\frac {\partial S}{\partial x^{\alpha }}}}$

gives the Hamilton–Jacobi equation in the geometry determined by the metric ${\displaystyle g}$:

${\displaystyle g^{\alpha \beta }{\frac {\partial S}{\partial x^{\alpha }}}{\frac {\partial S}{\partial x^{\beta }}}-(mc)^{2}=0,}$

in other words, in a gravitational field.

For a particle of rest mass ${\displaystyle m}$ and electric charge ${\displaystyle e}$ moving in electromagnetic field with four-potential ${\displaystyle A_{i}=(\phi ,\mathrm {A} )}$ in vacuum, the Hamilton–Jacobi equation in geometry determined by the metric tensor ${\displaystyle g^{ik}=g_{ik}}$ has a form

${\displaystyle g^{ik}\left({\frac {\partial S}{\partial x^{i}}}+{\frac {e}{c}}A_{i}\right)\left({\frac {\partial S}{\partial x^{k}}}+{\frac {e}{c}}A_{k}\right)=m^{2}c^{2}}$

and can be solved for the Hamilton principal action function ${\displaystyle S}$ to obtain further solution for the particle trajectory and momentum:[9]

${\displaystyle x=-{\frac {e}{c\gamma }}\int A_{z}\,d\xi ,}$

${\displaystyle y=-{\frac {e}{c\gamma }}\int A_{y}\,d\xi ,}$

${\displaystyle z=-{\frac {e^{2}}{2c^{2}\gamma ^{2}}}\int (\mathrm {A} ^{2}-{\overline {\mathrm {A} ^{2}}})\,d\xi ,}$

${\displaystyle \xi =ct-{\frac {e^{2}}{2\gamma ^{2}c^{2}}}\int (\mathrm {A} ^{2}-{\overline {\mathrm {A} ^{2}}})\,d\xi ,}$

${\displaystyle p_{x}=-{\frac {e}{c}}A_{x}}$, ${\displaystyle p_{y}=-{\frac {e}{c}}A_{y},}$

${\displaystyle p_{z}={\frac {e^{2}}{2\gamma c}}(\mathrm {A} ^{2}-{\overline {\mathrm {A} ^{2}}}),}$

${\displaystyle {\mathcal {E}}=c\gamma +{\frac {e^{2}}{2\gamma c}}(\mathrm {A} ^{2}-{\overline {\mathrm {A} ^{2}}}),}$

where ${\displaystyle \xi =ct-z}$ and ${\displaystyle \gamma ^{2}=m^{2}c^{2}+{\frac {e^{2}}{c^{2}}}{\overline {A}}^{2}}$ with ${\displaystyle {\overline {\mathbf {A} }}}$ the cycle average of the vector potential.

In the case of circular polarization,

${\displaystyle E_{x}=E_{0}\sin \omega \xi _{1}}$, ${\displaystyle E_{y}=E_{0}\cos \omega \xi _{1},}$

${\displaystyle A_{x}={\frac {cE_{0}}{\omega }}\cos \omega \xi _{1}}$, ${\displaystyle A_{y}=-{\frac {cE_{0}}{\omega }}\sin \omega \xi _{1}.}$

Hence

${\displaystyle x=-{\frac {ecE_{0}}{\omega }}\sin \omega \xi _{1},}$

${\displaystyle y=-{\frac {ecE_{0}}{\omega }}\cos \omega \xi _{1},}$

${\displaystyle p_{x}=-{\frac {eE_{0}}{\omega }}\cos \omega \xi _{1},}$

${\displaystyle p_{y}={\frac {eE_{0}}{\omega }}\sin \omega \xi _{1},}$

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

For the flat, monochromatic, linearly polarized wave with a field ${\displaystyle E}$ directed along the axis ${\displaystyle y}$

${\displaystyle E_{y}=E_{0}\cos \omega \xi _{1},}$

${\displaystyle A_{y}=-{\frac {cE_{0}}{\omega }}\sin \omega \xi _{1},}$

hence

${\displaystyle x={\text{const}},}$

${\displaystyle y_{0}=-{\frac {ecE_{0}}{\gamma \omega ^{2}}},}$

${\displaystyle y=y_{0}\cos \omega \xi _{1}}$, ${\displaystyle z=C_{z}y_{0}\sin 2\omega \xi _{1},}$

${\displaystyle C_{z}={\frac {eE_{0}}{8\gamma \omega }}}$, ${\displaystyle \gamma ^{2}=m^{2}c^{2}+{\frac {e^{2}E_{0}^{2}}{2\omega ^{2}}},}$

${\displaystyle p_{x}=0,}$

${\displaystyle p_{y,0}={\frac {eE_{0}}{\omega }},}$

${\displaystyle p_{y}=p_{y,0}\sin \omega \xi _{1},}$

${\displaystyle p_{z}=-2C_{z}p_{y,0}\cos 2\omega \xi _{1}}$

implying the particle figure-8 trajectory with a long its axis oriented along the electric field ${\displaystyle E}$ vector.

For the electromagnetic wave with axial (solenoidal) magnetic field:[10]

${\displaystyle E=E_{\phi }={\frac {\omega \rho _{0}}{c}}B_{0}\cos \omega \xi _{1},}$

${\displaystyle A_{\phi }=-\rho _{0}B_{0}\sin \omega \xi _{1}=-{\frac {L_{s}}{\pi \rho _{0}N_{s}}}I_{0}\sin \omega \xi _{1},}$

hence

${\displaystyle x={\text{constant}},}$

${\displaystyle y_{0}=-{\frac {e\rho _{0}B_{0}}{\gamma \omega }},}$

${\displaystyle y=y_{0}\cos \omega \xi _{1},}$

${\displaystyle z=C_{z}y_{0}\sin 2\omega \xi _{1},}$

${\displaystyle C_{z}={\frac {e\rho _{0}B_{0}}{8c\gamma }},}$

${\displaystyle \gamma ^{2}=m^{2}c^{2}+{\frac {e^{2}\rho _{0}^{2}B_{0}^{2}}{2c^{2}}},}$

${\displaystyle p_{x}=0,}$

${\displaystyle p_{y,0}={\frac {e\rho _{0}B_{0}}{c}},}$

${\displaystyle p_{y}=p_{y,0}\sin \omega \xi _{1},}$

${\displaystyle p_{z}=-2C_{z}p_{y,0}\cos 2\omega \xi _{1},}$

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