Charged Particles in 3D

The motion of charged particles in an electromagnetic field $(E,B)$ is governed by the Lorentz force,

\[\ddot{x} (t) = \frac{e}{m} \big[ E(x(t)) + \dot{x} (t) \times B(x(t)) \big] ,\]

where $m$ and $e$ denote the particle's mass and charge, respectively.

Canonical Formulation

The canonical form of the equations can be obtained from the Hamiltonian

\[H (x,p) = \frac{1}{2m} (p-A(x))^2 + e \phi(x),\]

as

\[\begin{aligned} \dot{x} (t) &= \frac{\partial H}{\partial p} (x(t),p(t)) = \frac{1}{m} (p(t) - A(x(t))), \\ \dot{p} (t) &= - \frac{\partial H}{\partial x} (x(t),p(t)) = \frac{1}{m} \nabla A(x(t)) \cdot (p(t)-A(x(t))) - \nabla \phi(x(t)) , \end{aligned}\]

where the fields $(E,B)$ are related to the potentials $(\phi, A)$ by

\[\begin{aligned} E (x) &= - \nabla \phi (x) , & B (x) &= \nabla \times A (x) . \end{aligned}\]

Noncanonical Formulation

The noncanonical form of the equations can be obtained from the phasespace Lagrangian

\[\begin{aligned} L (x,\dot{x},v,\dot{v}) &= (e A(x) + mv) \cdot \dot{x} - H(x,v) , & H(x,v) &= \frac{m}{2} v^2 + e \phi(x), \end{aligned}\]

as

\[\begin{aligned} \dot{x} (t) &= v (t) , & \dot{v} (t) &= \frac{e}{m} \big[ \nabla A (x(t)) \cdot \dot{x}(t) - \dot{A} (x(t)) - \nabla \phi(x(t)) \big] . \end{aligned}\]

Computing the time derivative of $A$ and using the relation between the potentials $(\phi, A)$ and the fields $(E,B)$, this can be rewritten as

\[\begin{aligned} \dot{x} (t) &= v (t) , & \dot{v} (t) &= \frac{e}{m} \big[ E(x(t)) + v (t) \times B(x(t)) \big] . \end{aligned}\]

This constitutes a noncanonical Hamiltonian system of the form

\[\dot{z} (t) = \Omega^{-T} (z(t)) \nabla H(z(t)) ,\]

with $z = (x,v)$ and the symplectic matrix $\Omega$ given by

\[\Omega = \frac{1}{m} \begin{pmatrix} \mathbb{0} & \mathbb{1} \\ - \mathbb{1} & e \hat{B} \\ \end{pmatrix} ,\]

and

\[\hat{B} = \begin{pmatrix} 0 & -B_3 & B_2 \\ B_3 & 0 & - B_1 \\ - B_2 & B_1 & 0 \\ \end{pmatrix} .\]

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