[talanum1 (OP)]

They say the lagrangian has a term: ∑_i qx'_i A_i.

Shouldn't it read: ∑_i,j qx'_i A_j with i ≠ j, since particle i is moving in the field created by particle j?

Do you just sum similarly if there are 3 particles?

[talanum1 (OP)]

I mean: ∑_i x_ikA_il

where i is components of the vectors and k and l are worldline numbers.