10.1 Existence of an asymptotic velocity

That it is possible to define an asymptotic velocity in all generality was demonstrated in [79Jump To The Next Citation Point] (see also [12Jump To The Next Citation Point] for related results).

If there are expansions of the form (37View Equation) – (38View Equation), we have seen that va can be computed according to Equation (39View Equation). As a consequence, it is of interest to ask if the limit on the right-hand side of this equation always exists. Due to [79Jump To The Next Citation Point, Corollary 6.9, p. 1009], the answer is yes. As a consequence, we are naturally led to the following definition.

Definition 4 Let (Q, P) be a solution to Equations (16View Equation)–(17View Equation) and let 𝜃0 ∈ S1. Then we define the asymptotic velocity at 𝜃 0 to be

[ ]1∕2 v∞ (𝜃0) = lτim→∞ 𝒦 (τ,𝜃0) .


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