Lagrangian to terms of second order LL2 section 65

Classical mechanics: instantaneous interactions. – In Lagrangian L depends on q and dq/dt for all particles at the same time t. – Field is just a mathematical convenience

Relativistic mechanics: finite velocity of propagation. – Field is independent system – Lagrangian for interaction of particles must include degrees of freedom for the field

If v<<c, we seek the Lagrangian to order (v/c) 2 – Electromagnetic radiation appears only in the next (3 rd ) approximation.

Zeroth approximation (no powers of v/c, classical) Exact Lagrangian for a particle e a in given fields At the position of potentials are determined from positions and motions of all other charges Retarded potentials

If v << c for all charges, then  changes little during time R/c Expandin powers of R/c All terms evaluated at R/c = 0, i.e. at the present time t, Total charge of system = constant Second term is zero

Scalar potential to second order in 1/c The expression for A already contains a factor 1/c. Then A gets multiplied again by 1/c in the Lagrangian. Keep only the 1 st term in the expansion. Evaluated at present time t Substitute these potentials in the Lagrangian to find approximate L to order (v/c) 2.

Single point charge R(t) = |r – r 0 (t)| is time dependent We can get rid of this by gauge transform

Gauge transform Let Now the extra term appears in A’ Lorentz condition is not satisfied. Neither is

With respect to field point coordinates What is this?

For multiple charges, sum their contributions, e.g.

Lagrangian for charge e a using and ignoring constant mc 2 Compare to zeroth order Lagrangian Exact Approximate to 2 nd order in v/c

Small corrections Mechanics, volume 1, equation (40.7): For small changes to L and H, the additions are equal in magnitude, but opposite in sign.