Question

Has the Helmholtz decomposition of the $\mathbf{E}$ field from the Liénard–Wiechert potentials been worked out?

Answer 1

Hey mate ^_^

Examined in light of Helmholtz decomposition, these equations could be read as: electric charge produces irrotational electric fields, and changing magnetic fields produce solenoidal electric fields....

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Answer 2

Hello mate here is your answer.

If you look at Maxwell's equations for E(x,t)E(x,t)they split neatly into two categories. They are:

Examined in light of Helmholtz decomposition, these equations could be read as: electric charge produces irrotational electric fields, and changing magnetic fields produce solenoidal electric fields.
The electric field from the Liénard–Wiechert potential is given by equation 14.14 from Jackson's "Classical Electrodynamics" (3rd Ed) 

Hope it helps you.