The task of the relativistic oscillator in a non-inertial frame of reference

In this paper the analogues of the Lorentz transformations for non-inertial reference frames have been obtained. A common case when the movement speed of one coordinate frame in relation to another one can have time derivatives of higher orders. The obtained transformations conserve invariance of the space-time interval, and in the particular case of inertial frames become the well-known Lorentz transformations. It is shown that the transition from classical mechanics to the theory of relativity is analogous to the consideration of the vibrating system described by the equation of the sine-Gordon type. In this case, if the amplitude of the elliptic functions is $k\to 0$ the fluctuations can be considered small, and that leads to classical mechanics. With $k\to 1$ time depends on the vibration amplitude, which leads to the theory of relativity. In the case of inertial frames the amplitude is $k=\beta ={v}/{c}\;$.