The Charged Particles Gallery is a sub-gallery of the ODE gallery. The exhibits in this gallery are examples of second order ordinary differential equations (ODEs) in three dimensions. You can look at the documentation for the ODE(3D) 2nd Order Gallery for more information on such equations. You can find more general information on ODE exhibits in the ODE documentation.
A Charged Particle exhibit shows the motion of an electrically charged particle in a magnetic field in three dimensional space. The differential equation that determines the motion is Newton's third law of motion, F = ma, where F is the force that is exerted on the particle. In the equation, a stands for acceleration and m is a constant that represents mass (and can be taken to be 1). The force depends on both the magnetic field and on the velocity of the particle. It also depends on the particle's electric charge, but that can be taken to be a constant, e. Let p(t) be the function that gives the particle's position at time t, and let B represent the magnetic field. B is assumed to be a function of position (that is, we do not consider time-varying magnetic fields). Both p(t) and B(p) are vector-valued functions. Then the force acting on the particle is given by e p(t) × B(p(t)), where × represents the cross product of vectors. The ODE can therefore be written as
m p''(t) = e p(t) × B(p(t))
This is a second order autonomous ODE. Initial conditions for such an equation consist of the position and velocity at a given time. Recall that is is possible to input the initial condition by clicking-and-dragging with the middle mouse button (or by holding down the ALT/Option button while dragging). This allows you to "launch" a particle in the magnetic field and observe its motion.