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By S. G. Rajeev

This direction may be typically approximately structures that can not be solved during this means in order that approximation tools are helpful.

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In the case of the harmonic oscillator this frequency is even independent of energy or I ; but in general this is not true. Chapter 12 Hamilton-Jacobi Theory •Suppose we are presented a system with hamiltonian H(q, p) in some canonical co-ordinate system (q i , pi ) . If we can find a new canonical coordinate system (θi , Ii ) such that the hamiltonian depends only on the momentum variables Ii , then we have solved the equations of motion. For, in this system, the Hamiltons equations are dIi = 0, dt dθi = ω i (I) dt where the frequencies are ω i (I) = ∂H(I) .

But generically, the frequencies are not related: there 2π is no set of integers such that i ωi = 0 . In this case there is a period ωi in each variable separately, but no common period for the motion as a whole: it is quasi-periodic. •A hamiltonian for which such a normal co-ordinate system exists is said to be integrable. It is found that such hamiltonians are ‘rare’: under small 53 54 PHY411 S. G. Rajeev perturbations we will lose this property. Still much of what we know about mechanics comes from such examples.

Chapter 13 Birkhoff Normal Form •See F. G. Gustavson Astonomical Journal Vol 71, 670 (1966) for a beautiful exposition of these ideas. •We now start the study of classical systems that are not integrable. Suppose that a system has a stable equilibrium point; Near this point it is well approximated by a harmonic oscillator; there are canonical co-ordinates such that the hamiltonian is 1 H(q, p) = ωi (p2i + qi2 ) + O(3) 2 i upto terms that are cubic in the co-ordinates. •The higher order terms describes the non-linearities in the equations of motion.

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Advanced classical mechanics: chaos by S. G. Rajeev

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