classical electrodynamics - What is the Lagrangian for a relativistic charge that includes the self-force?

The usual Lagrangian for a relativistically moving charge, as found in most text books, doesn't take into account the self force from it radiating EM energy. So what is the Lagrangian for a relativistic charge that includes the self-force?

Answer

classical electrodynamics mainly deals with two kinds of proplems: a) The action of a field on a charged particle and b) the fields arising from the motion of such a field. Of course, this can only be approximative but it turns out that a lot of phenomena can be described in this way.

However, you are right, an entire treatment would include a) and b) simultaniously - including the whole dynamics of such a system with radiative reaction (or, the Abraham-Lorentz-Force and its relativistic counterpart).

But as Qmechanic pointed out in a comment, there may be no fully consistent way to do so within the framework of classical electrodynamics. Jackson (Chapter 16) states:

The difficulties presented by this problem touch one of the most fundamental aspects of physics, the nature of an elementary particle. Although partial solutions, workable within limiting areas, can be given, the basic problem remains unsolved.

Thus, you will have to search for a really satisfactory answer within the description of Quantum Electrodynamics. Otherwise, you may include the effect phenomenologically. This was done e.g. by Barone and Mendes in Lagrangian description of the radiation damping (PRA, 2007) and they give a Lagrangian of the form

$$\mathcal{L}=\frac{m}{2}\dot{\mathbf{r}}_1\cdot\dot{\mathbf{r}}_2 - \frac{\gamma}{2}\epsilon\left(\dot{\mathbf{r}}_1,\ddot{\mathbf{r}}_2\right) -V\left(\mathbf{r}_1,\mathbf{r}_2\right),$$

with $\gamma := 2e^2/3c^3$, with $\epsilon = \epsilon_{ij}dx^idx^j$ beeing the Levi-Cevita tensor, and the $\mathbf{r}_i$ arise from the special treatment of the problem used in the paper employing some kind of image phase-space representation of the system. Furthermore, $V\,$ is the potential related to the Abraham–Lorentz–Dirac-Force which is given explicitely in the paper.