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Download e-book for iPad: Advanced classical mechanics: chaos by S. G. Rajeev

By S. G. Rajeev

This direction may be frequently approximately structures that can not be solved during this method in order that approximation tools are invaluable.

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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. This is the method of separation of variables.

D is a linear operator in the space of homogenous polynomials V s . It is not invertible, but is antisymmetric so that we have a decomposition V s = ker D + range D . In particular we can split H s = K s + Rs where K s ∈ ker D and Rs ∈ range D . , Γs is in normal form. This completes the proof. •The solution of the equation Dws = Rs is facilitated by diagonalizing D . Being anti-symmetric, it is diagonalized by complex linear combinations zj = qj + ipj , z¯j = qj − ipj : Dz A z¯B = i [ai − bi ]ωi z A z¯B .

Dτ dτ These are the equations for the light rays. This formulation is called Hamiltonian optics. There is a conjugate description in terms of Langrangians as well. Determining rays is helped by the existence of integral invariants which are the analogues of conserved quantities. •The eikonal equation is the approximation to the wave equation in the limit of short wavelength. Is it possible to think of the Hamilton-Jacobi equation p2 as an approximation to a wave equation? Let H(p, q) = 2m + V (q) .

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