How do I transform a time-dependent acceleration between to rest frames? I was given a particular problem and while I found possibilities [1] [2] for acceleration-transformations when the acceleration $a$ is uniform, I did not find anything about transforming an $a(t)$.
I was thinking about using infinite Lorentz boosts after an infinitesimal time $\delta t$ to accomondate for the change in $\gamma=(1-v^2/c^2)^{-1}$, but didn't get anywhere.
Another idea was to consider the Lagrangian of a free particle where the potential results in a time-dependent acceleration.
In my special case, the acceleration is given in a train's proper time $\tau$ as $a(\tau) = (1-\alpha\tau)^{-1}$ m/s² with a constant $\alpha$. The question is to compute the train's speed as a function of the proper time in the train station's frame. The train left the station at $\tau = 0$.
My take using the Lagrangian was the following:
$ L = T - V$ -- where $V = 0 \Rightarrow L = -mc^2/\gamma$.
Now consider the acceleration to result from an external potential (the train's engines for that matter) with a force $\vec F = -\vec \nabla V = ma \vec e_x$. This leads to $V(\tau) = -\frac{mx(\tau)}{1-\alpha \tau} + \mathcal C(\tau)$. Using the Euler-Lagrange equation $\frac{d}{dt}\frac{\partial L}{\partial \dot x} - \frac{dL}{dx} = 0$ I unfortunately get nowhere as $\partial L / \partial \dot x = 0$ and $dL/dx = - m/(1-\alpha\tau)$ which therefore should equal zero.
So I think I didn't understand the concept on a fundamental level and hope you can enlighten me.
