By George D. Birkhoff

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Not to be obsessive about it however, for the rest of this discussion the parentheses will be left off, but the summation signs will be left explicit. 31) where, in this case each “mass” mr is necessarily positive because T is positive definite. By judicious choice of the scale of the yr each “mass” can be adjusted to 1. We will assume this has already been done. T= 1 n 2 y˙ r . 33) r =1 defines a surface (to be called a hypersphere). From now on we will consider only points y = (y1 , . . , yn ) lying on this sphere.

Zn , each maximizing V consistent with being orthogonal to its predecessors, is determined, and that the sequence of values, V (zr ) = 12 κr , r = 1, 2, . . , n, is monotonically nonincreasing. 33) but with ζ otherwise arbitrary. Next assume this point is “close to” z1 in the sense that is arbitrarily small (and not necessarily positive). Since z1 maximizes V it follows that V (z1 + ζ , z1 + ζ ) ≤ 0. 4 Linear, Multiparticle Systems Show therefore that V (z1 , ζ ) = 0. 42) because, other than being orthogonal to z1 , ζ is arbitrary.

105) and the wave speed is given by v=λ ω = 2π k a. 106) (In this low frequency approximation) since this speed is independent of ω, low frequency pulses will propagate undistorted on the beaded string. Replacing the index i by a continuous variable x = ia, our conjectured solution therefore takes the form η ( x, t) = x sin . 107) These equations form the basis for the so-called “lumped constant delay line,” especially when masses and springs are replaced by inductors and capacitors. 2 The Continuum Limit Propagation on a continuous string can be described by appropriately taking a limit N → ∞, a → 0, while holding Na = L0 + ∆L.