the Hamilton–Jacobi equation in the geometry determined by the metric : 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 four-potential 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: , where and with the cycle average of the vector potential. A circularly polarized wave In the case of 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. A monochromatic linearly polarized plane wave For the flat, monochromatic, linearly polarized wave with a field directed along the axis hence , , implying the particle figure-8 trajectory with a long its axis oriented along the electric field vector. An electromagnetic wave with a solenoidal magnetic field For the electromagnetic wave with axial (solenoidal) magnetic field: 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 figure-8 trajectory in plane set perpendicular to the solenoid axis with arbitrary azimuth angle 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 References Further reading Hamiltonian mechanics Symplectic geometry Partial differential equations William Rowan Hamilton