Lorentz force from Lagrangian (non-covariant)
Posted by Peeter Joot on September 22, 2009
Jackson  gives the Lorentz force non-covariant Lagrangian
and leaves it as an exercise for the reader to verify that this produces the Lorentz force law. Felt like trying this anew since I recall having trouble the first time I tried it (the covariant derivation was easier).
Jackson gives a tip to use the convective derivative (yet another name for the chain rule), and using this in the Euler Lagrange equations we have
where is the spatial basis. The first order of business is calculating the gradient and conjugate momenta. For the latter we have
Applying the convective derivative we have
For the gradient we have
Rearranging 2 for this Lagrangian we have
The first two terms are the electric field
So it remains to be shown that the remaining two equal . Using the Hestenes notation using primes to denote what the gradient is operating on, we have
I’ve used the Geometric Algebra identities I’m familiar with to regroup things, but this last bit can likely be done with index manipulation too. The exercise is complete, and we have from the Lagrangian
 JD Jackson. Classical Electrodynamics Wiley. 2nd edition, 1975.