Advanced classical mechanics: chaos by S. G. Rajeev

By S. G. Rajeev

This path might be frequently approximately structures that can't be solved during this means in order that approximation equipment are important.

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H ¯2 2 ∇ ψ + V (q)ψ = Eψ. 2m This is just the Schrodinger equation; h ¯ is Planck’s constant!. Hamilton had some inkling of the existence of such a wave mechanics. But at that time there was no experimental reason to doubt the absolute validity of Newtonian mechanics. • Example Let us solve the Kepler problem this way. In spherical polar co-ordinates − p2φ 1 p2θ 2 H= p + + + V (r, θ, φ). 2m r r2 r2 sin2 θ PHY411 S. G. Rajeev 57 As long as the potential is of the form V = a(r) + c(φ) b(θ) + r2 r2 sin2 θ a solution of the form W = R(r) + Θ(θ) + Φ(φ) exists.

90 years. 6 days. 13 Lagrange thought that these special solutions were artificial and that they would never be realized in nature. But we now know that there are asteroids ( Trojan asteroids) that form an equilateral triangle with Sun and Jupiter. 14 Lagrange discovered something even more astonishing: the equilateral triangle is an exact solution for the full three body problem, not assuming one of the bodies to be infinitesimally small. 15 If the change of variables from x, y to r1 , r2 were invertible everywhere, there would have been no other extrema.

Thus the variable I is just the area of the ellipse upto a factor of π . •Why do we get such a geometrical answer? The point is that ∂(θ, I) {θ, I} = ∂(q, p) is just the Jacobian of the transformation (q, p) → (θ, I) . Thus canonical transformations preserve the area. Indeed, the area of a curve can be written either as γ pdq or γ Idθ as long as ther two co-ordinates are canonically related. •Thus we can generalize the above ideas to an arbitrary conservative system with one degree of freedom. The curves H(p, q) = E enclose an area I(E) = γE pdq .

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