By J. W. Schutz
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Extra info for Foundations of Special Relativity. Kinematic Axioms for Minkowski Space-Time
7) D0 ≡ lim r→0 log( 1r ) While studying the geometry of the attractor, one is interested in the cubes in which the trajectory spends more time; to this end one can introduce the natural measure as the amount of time that the orbit spends in a given region of the phase space. 1) let z 0 be an initial condition in the basin of attraction of the attractor A and let z(t; z 0 ) be the trajectory at time t originating from z 0 . For a given cube C of the phase space, we deﬁne μ(C; z 0 , τ ) as the fraction of time that the orbit z(t; z 0 ) spends in the cube C during the time interval [0, τ ].
In general, if y0 = 2π pq with p, q positive integers (q = 0), one obtains a periodic orbit of period q. It is readily seen that the quantity p measures how many times the interval [0, 2π) is run before coming back to the starting position. The situation drastically changes when an irrational initial condition y0 is taken in place of a rational initial point. 5(a)). Such straight lines are quasi– periodic invariant curves, since on these curves a quasi–periodic motion takes place such that the dynamics comes indeﬁnitely close to the initial conditions at regular intervals of time, though never exactly retracing itself (as is the case for the periodic orbits).
5. (a) Variation of ω and (b) of the DFLI as a function of ε for 1000 values within the interval [0, 1] (reprinted from ). , ). In this chapter we concentrate on the mathematical description of the two–body problem. The starting point is the gravitational law and Newton’s three laws of dynamics. 67 · 10−11 m3 kg−1 s−2 , and e12 is the unit vector joining the two bodies. Newton’s laws of dynamics can be stated as follows: (i) First law (law of inertia): without external forces every body remains at rest or moves uniformly on a straight line.