Massless Charged Particle

Massless charged particle in 2D

The Lagrangian is given by

\[L(x, \dot{x}) = A(x) \cdot \dot{x} - \phi (x) ,\]

with magnetic vector potential

\[A(x) = \frac{A_0}{2} \big( 1 + x_1^2 + x_2^2 \big) \begin{pmatrix} - x_2 \\ + x_1 \\ \end{pmatrix} ,\]

electrostatic potential

\[\phi(x) = E_0 \, \big( \cos (x_1) + \sin(x_2) \big) ,\]

and magnetic and electric fields

\[\begin{aligned} B(x) &= \nabla \times A(x) = A_0 \, (1 + 2 x_1^2 + 2 x_2^2) , \\ E(x) &= - \nabla \phi(x) = E_0 \, \big( \sin x_1, \, - \cos x_2 \big)^T . \end{aligned}\]

The Hamiltonian form of the equations of motion reads

\[\dot{x} = \frac{1}{B(x)} \begin{pmatrix} \hphantom{-} 0 & + 1 \\ - 1 & \hphantom{+} 0 \\ \end{pmatrix} \nabla \phi (x) .\]

Creates an implicit DAE object for the massless charged particle in 2D.

Creates an implicit DAE object for the massless charged particle in 2D.

Creates an implicit ODE object for the massless charged particle in 2D.

Creates an ODE object for the massless charged particle in 2D.