This paper addresses the equivalence under state transformation of a discrete-time nonlinear control system to observer canonical form. Necessary and sufficient conditions for generic equivalence are given for the case when the state equations are not necessarily reversible. The proof is constructive and shows how to find the state transformation if the conditions are satisfied. The derived conditions are then compared with earlier conditions, obtained under more restrictive assumptions, to demonstrate that the earlier result follows directly from our theory. Two examples illustrate the new theory.
Using the concept of the world function, an exact deviation equation has been derived, which describes the relative motion of two accelerated point masses and is valid in the case of their arbitrary 4-velocities, 4-accelerations, parametrizations of their world lines, and arbitrary correspondence between the points of the world lines. The second approximation of the exact deviation equation, with respect to the components of deviation vector, has been elaborated in general coordinates and in the Fermi coordinates of an observer comoving with one of the point masses. The 3-acceleration of a freely falling test particle in the frame of reference of an accelerated observer in the Schwarzschild spacetime has been calculated in the second approximation. It turns out that the quadratic terms involve the observer's 3-velocity relative to the black hole and become significant compared to the linear terms if the observer moves at a relativistic 3-velocity. Moreover, in most cases, if the observer's 3-velocity approaches the velocity of light, the second-order deviation equation is no longer applicable as the relevant Taylor expansions do not converge. It is established that if a freely falling observer starts its motion from spacelike infinity at a nonrelativistic initial velocity, the problem of convergence does not appear. It is proved that free circular motion of an extended body is impossible near the 1.5 Schwarzschild radius as the body would be destroyed by the tidal forces.
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