Vector potential YouTube

Summary of formulations

We want to consider the electromagnetic field again. We are assuming that given fields are functions of the coordinates and momenta associated with the system, and not, explicitly, time-dependent.

The non-Newtonian formulations are somewhat non-intuitive, since they require that the magnetic portion of the field appears in the Lagrangian in terms of a vector potential from which the magnetic force is derived. Also, each of the conjugate momenta contains a term extra to the mechanical momentum.

All of the important equations of motion can be derived from the principle of least action even though, in principle, we could find examples that can't (at least in a mathematical sense). The motion due to gravitational fields, atomic force fields and electromagnetic fields can all be derived from the principle of least action.

Nb. The equations of motion of the fields themselves can also be derived from the principle of least action - something we shall not cover in this course.

The principle of least action, not of much practical use itself, leads us, via the calculus of variations, to the Lagrangian and Hamiltonian forms.

The Hamiltonian formulation is possibly/probably the least intuitive, but it happens to be the only one (other than Poisson brackets) that includes quantum mechanics. It also, has the very elegant description of trajectories in phase space, which allows us to see how many systems behave simultaneously, by looking at contour maps of the Hamiltonian itself.

We could apply any of the formulations to the problem of the motion of a particle in an electromagnetic field.

The simplest case

Figure 8.1 - magnetic force into the page

Figure 8.1 - magnetic force into the page

For simplicity, we take the magnetic force to be constant and pointing into the page, thus $$\vec{B} = -B \hat{z}$$

That way, if the electric field acts only in the $x,y$-plane and the velocity of a charged particle starts in the $x,y$-plane, then it will remain in the $x,y$-plane.

Of course, we could point the magnetic and electric components in any direction, and allow motion in all three directions, but the physics will remain the same.

Newton's form

Newton's form for the electromagnetic force is $$m \vec{a} = q\vec{E} + q\left(\vec{v} \times \vec{B} \right)$$

Figure 8.2 - The magnetic force

Figure 8.2 - The magnetic force

Taking the simplest case above, where $$\begin{align*}\vec{B} &= -B\hat{z}\\\vec{E} &= E_x\hat{x} + E_y\hat{y}\\\vec{v} &= v_x\hat{x} + v_y\hat{y}\end{align*}$$

This combination means the magnetic component of the Newtonian force $$q\left(\vec{v} \times \vec{B}\right)$$ is also in the $x,y$-plane, as shown in figure 8.2.

Also, the magnetic force does no work on the charge, since $$\left(\vec{v} \times \vec{B} \right) \perp \vec{v}$$ hence doesn't change the energy of the charge as it moves on its trajectory.

Divergence

To make the jump from using Newton's laws to any of the other versions, and in particular, the principle of least action, then we need the concept of the vector potential associated with the magnetic field.

If $\vec{A}$ is the vector potential of $\vec{B}$ then $\vec{B}$ is the curl of $\vec{A}$. That is, $$\vec{B} = \nabla \times \vec{A}$$

One consequence of this is that the divergence of $\vec{B}$ is zero, $$\nabla \cdot \vec{B} = \nabla \cdot \left(\nabla \times \vec{A} \right) = 0$$

The divergence of the magnetic field being zero is equivalent to saying there are no magnetic monopoles - which, if they exist, are the sources of the magnetic field and which would either carry one pole or the other (North or South), but not both.

We can contrast this with the divergence of the electric field, called the charge density of the field, which is not zero,$$\nabla \cdot \vec{E} \;\neq 0$$

In fact, Professor Susskind suggests that magnetic monopoles probably do exist, but as far as this course is concerned, it's assumed there aren't any magnetic sources of fields.