By Edward J. Beltrami
This new version of arithmetic for Dynamic covers instruments comparable to linearization, suggestions options, using Liapunov features, and optimum keep an eye on. every one bankruptcy comprises routines, lots of which extend at the fabric within the textual content
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8 in some region Ω. 9) where υ(χχ) is the potential function f(*i) = ( /(*) ds. *0 Suppose that χλ minimizes U on Ω. Then V(x) > 0 on Ω with zero only at x. Moreover, along any orbit, = x2x2 + i x / i ^ i ) = x2[x2+f(xi)\ =0 since xx = p and x2 = p. 1, x is stable. This estabHshes the principle that any local equiUbrium of a conservative system that 46 Stable and Unstable Motion, II minimizes the potential must be stable. In all the examples so far in which U(x) was plotted, the minimum value of U indeed occurs at a stable point.
2 in which p = Θ. To begin with, it is expedient to suppose that / is independent of p. This gives P+f(p) = 0. 2. 16 as describing the motion of some body along a coordinate direction p. Inextricably bound to this idea is the concept of energy. 17) The Phase Plane 27 where U(p)= f f(s)ds. 18) Students of physics recognize \p2 as the kinetic energy of the motion, whereas U(p) is called the potential energy. 16. Note that the derivative of E is E = p(p + f(p)) = 0, which is another way of showing that E is constant, a fact known as the principle of conservation of energy.
Since V is non-increasing and bounded below, we know that V(x(t)) -> η > 0 as t -> oo. However, it is necessary to be convinced that η is actually zero and that in fact this value occurs at x = x. We regard the argument given in the previous paragraph to be sufficiently obvious that it is not necessary to fuss over details. A complete proof can be found in several of the references quoted. Liapunov Functions 43 It is an immediate corollary of the above theorem that if V < 0 everywhere on Ω except at x, then x is asymptotically stable.