On the equations of the inverse kinematics problem

TL;DR

This paper derives nonlinear differential equations to determine the motion parameters of a non-inertial reference frame from known acceleration and angular velocity, generalizing Lorentz transformations and accounting for Thomas precession.

Contribution

It introduces new differential equations for inverse kinematics of rigid frames, incorporating Thomas precession and enabling motion analysis relative to inertial frames.

Findings

Equations verified on a uniformly rotating disk example.

Provides a method to find non-inertial frame motion from acceleration and angular velocity.

Generalizes Lorentz transformation for non-inertial reference frames.

Abstract

The paper derived differential equations which solve the problem of restoration the motion parameters for a rigid reference frame from the known proper acceleration and angular velocity of its origin as functions of proper time. These equations are based on the well-known transformation to an arbitrary rigid non-inertial reference frame, which generalized the Lorentz transformation, takes into account the fact of her proper of Thomas precession and rigid fixation of the metric form reference frame. The role of this problem in physics is that all such reference frames found with the same characteristics will have the same properties. Another important advantage of these equations is that they allow you to find the motion of the non-inertial reference frame with respect to an arbitrary inertial frame. The resulting equations are nonlinear differential vector equation of the first order Riccati type \[\mathbf{{\dot{v}}'}=\frac{d\,\mathbf{{v}'}}{dt}=\mathbf{{W}'}-(\mathbf{{W}'{v}'})\mathbf{{v}'}-\mathbf{\Omega}'\times \mathbf{{v}'}\] and known system of 3 equations \[\frac {1}{2}\,e^{\alpha\lambda\nu }a^{ \lambda\beta } \frac{da^{\alpha \beta } }{dt}=\omega'^{\nu }\,,\] which allows to determine the orientation of the coordinate system for a given function on the right side of the equation \[ {\boldsymbol\omega }'=\mathbf{\Omega}'-\frac{1-\sqrt{1-{{{{v}'}}^{2}}}}{{{{{v}'}}^{2}}}\,\mathbf{{v}'}\times \mathbf{{W}'}\,,\] which are type angular velocity. The consequences of the obtained equations have been successfully verified on the example of a uniformly rotating disk.