What examples are there of simple (special) relativistic systems in which the equations of motion are solvable? There are countless examples of these in non-relativistic mechanics, e.g. the simple pendulum, but I presume due to the non-linearity introduced by the relativistic gamma factor it is very challenging. The only example I know of is the solution to the equations of motion of a charged particle in a plane electromagnetic wave (in Landau & Lifshitz), but this is at a more advanced level and their solution is only given implicitly. I would also be interested in approximate solutions, e.g. those made possible by expanding the gamma factor as a function of momentum.
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Probably the simplest one is the case of an object which experiences an acceleration which is constant in its own frame i.e. a rocket in deep space, where the change in mass of the rocket due to fuel loss is negligible (or maybe it is powered by a light sail).
Let $\bf{A}$ be the four-acceleration, s.t. $$\textbf{A} \cdot \textbf{A} = a_0^2 = -\dot{\gamma}^2+\gamma^2[\gamma^2a^2+2\gamma\dot{\gamma}\beta a]$$ is the invariant acceleration $a_0$ in the object's instantaneous rest frame squared, and $a$ is the acceleration in some frame where the velocity is $\beta$. Let $\eta$ be the rapidity in that frame, s.t. $\gamma = \cosh{\eta}$. (I am taking $c = 1$). Then we have, after some simplifications: $$\textbf{A} \cdot \textbf{A} = \dot{\eta}^2 \cosh^2{\eta}$$ This gives us the result for 1d acceleration (i.e. without assuming constant acceleration) $$a_0 = \dot{\eta}\cosh{\eta} = \frac{d}{dt}\sinh{\eta}$$. For constant acceleration, if we take the rocket to start at rest in the lab frame, we get: $$\beta\gamma = \sinh{\eta} = a_0t$$ from which it is not too difficult to calculate the velocity and position as a function of time.
