Question

2.2.6 The Lagrangian for a particle with charge q in an electromagnetic field described by scalar potential φ and vector potential A is Find the equation of motion of the charged particle. Hint. (d/dt) Aj = aA7/ar + Σ (aA7 /axi )xi. The dependence of the force fields E and B on the potentials ф and A is developed in Section 3.9; see in particular Eq. (3.108).
The page below is equation 3.108

Answer 1

L = 1/2 mv^2 - 9Q + 9 A^bar v^bar partial differentiate/partial differentiate t (partial differentiate L/partial differentiate x) = partial differentiate L/partial differentiate x: partial differentiate/partial differentiate t(mv) = partial differentiate L/partial differentiate x = mv + 9A partial differentiate L/partial differentiate x = -9 partial differentiate phi/partial differentiate x + 9 partial differentiate pi/partial differentiate x + 9 partial differentiate A^bar/partial differentiate x v^bar d/dt (mv + 9A^bar) = -9 partial differentiate phi/dx + 9 partial differentiate A^bar/partial differentiate x V mv + 9 partial differentiate A^bar/partial differentiate t = -9 partial differentiate phi/partial differentiate x + 9 partial differentiate A^bar/partial differentiate x v (1) Taking all three components eq^n (1) takes the form mx_i = 9[-(nabla phi + partial differentiate A^bar + partial differentiate A^bar/partial differentiate t) + V^bart times B^rightarrow] where B^bar = nabla^bar times A^bar: E^bar = -V^rightarrow = -v^rightarrow phi - partial differentiate A^rightarrow/partial differentiate t