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

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

The problem is available in explicit (odeproblem), implicit (iodeproblem, idaeproblem, idaeproblem_spark) and Lagrangian (lodeproblem, ldaeproblem) form. The latter two are built from the Lagrangian $L(x, \dot{x}) = \vartheta (x) \cdot \dot{x} - H(x)$ with one-form $\vartheta = A$ and the symplectic two-form

\[\Omega_{ij} (x) = \frac{\partial \vartheta_i}{\partial x_j} - \frac{\partial \vartheta_j}{\partial x_i} = \begin{pmatrix} \hphantom{+} 0 & - B(x) \\ + B(x) & \hphantom{-} 0 \\ \end{pmatrix} ,\]

so that the Euler-Lagrange equations read $\Omega (x) \, \dot{x} = - \nabla \phi (x)$.

See GeometricProblems.MasslessChargedParticleSingular for a formulation with the same magnetic field but a "singular" (one-component) vector potential, suitable for degenerate variational integrators.

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 a variational DAE object for the massless charged particle in 2D.

Creates a variational (Lagrangian) ODE object for the massless charged particle in 2D.

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

Massless charged particle in 2D with "singular" vector potential

The Lagrangian is given by

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

with the "singular" (one-component) magnetic vector potential

\[A(x) = - A_0 \, x_2 \, \big( 1 + 2 x_1^2 + \tfrac{2}{3} x_2^2 \big) \begin{pmatrix} 1 \\ 0 \\ \end{pmatrix} ,\]

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

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

This is the same physical system as GeometricProblems.MasslessChargedParticle: the magnetic field $B$, electric field $E$ and hence the dynamics are identical. Only the vector potential differs by a gauge transformation, chosen here so that its second component vanishes ($A_2 = 0$). This "singular" one-form yields the same Euler-Lagrange equations but a different variational integrator, and is the form required by the degenerate variational integrator (DVI) method of GeometricIntegrators.

The problem is available in explicit (odeproblem), implicit (iodeproblem, idaeproblem, idaeproblem_spark) and Lagrangian (lodeproblem, ldaeproblem) form. The latter two are built from the Lagrangian $L(x, \dot{x}) = \vartheta (x) \cdot \dot{x} - H(x)$ with one-form $\vartheta = A$ and the symplectic two-form

\[\Omega_{ij} (x) = \frac{\partial \vartheta_i}{\partial x_j} - \frac{\partial \vartheta_j}{\partial x_i} = \begin{pmatrix} \hphantom{+} 0 & - B(x) \\ + B(x) & \hphantom{-} 0 \\ \end{pmatrix} ,\]

so that the Euler-Lagrange equations read $\Omega (x) \, \dot{x} = - \nabla \phi (x)$. Since the gauge transformation shifts $\vartheta$ only by an exact one-form, $\Omega$ is the same as for GeometricProblems.MasslessChargedParticle, while the Lagrangian differs.

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 a variational DAE object for the massless charged particle in 2D.

Creates a variational (Lagrangian) ODE object for the massless charged particle in 2D.

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