Motion in an constant uniform Electric field.
We want to solve the problem
Unlike second year classical physics, we will use relativistic momentum, so for only a constant electric field, our Lorentz force equation to solve becomes
In components this is
Integrating the component we have
If we let , square and rearrange a bit we have
Now for the components, with , our equation to solve is
Squaring this one we have
Observe that our energy is
We can then write
Some messy substitution, using 1.8, yields
Solving for we have
Can solve with hyperbolic substitution or
Now we have something of the form
so our final solution for is
Now for we have
A final bit of substitution, including a sort of odd seeming parametrization of in terms of in terms of , we have
FIXME: check the checks.
An alternate way.
There’s also a tricky way (as in the text), with
We can solve this for
With the cross product zero, has only a component in the direction of , and we can invert to yield
and one can work from there as well.
Motion in an constant uniform Magnetic field.
Work by the magnetic field
Note that the magnetic field does no work
Because and are necessarily perpendicular we are reminded that the magnetic field does no work (even in this relativistic sense).
Initial energy of the particle
Because no work is done, the particle’s energy is only the initial time value
Simon asked if we’d calculated this (i.e. the Hamiltonian in class). We’d calculated the conservation for time invariance, the Hamiltonian (and called it ). We’d also calculated the Hamiltonian for the free particle
We had not done this calculation for the Lorentz force Lagrangian, so lets do it now. Recall that this Lagrangian was
with generalized momentum of
Our Hamiltonian is thus
which gives us
So we see that our “energy”, defined as a quantity that is conserved, as a result of the symmetry of time translation invariance, has a component due to the electric field (but not the vector potential field ), plus the free particle “energy”.
Is this right? With and being functions of space and time, perhaps we need to be more careful with this argument. Perhaps this actually only applies to a statics case where and are constant.
Since it was hinted to us that the energy component of the Lorentz force equation was proportional to , and we can peek ahead to find that , let’s compare to that
So if we have
I’d guess that we have
which is, using 2.40
Can the left hand side be integrated to yield ? Yes, but only in the statics case when , and for which we have
FIXME: My suspicion is that the result 2.43, is generally true, but that we have dropped terms from the Hamiltonian calculation that need to be retained when and are functions of time.
Expressing the field and the force equation.
We will align our field with the axis, and write
or, in components
Because the energy is only due to the initial value, we write
Evaluating the delta
Looks like circular motion, so it’s natural to use complex variables. With
Using this we have
which comes out nicely
Real and imaginary parts
Which is a helix.