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By S. G. Rajeev

This direction can be as a rule approximately structures that can't be solved during this means in order that approximation equipment are worthwhile.

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Extra resources for Advanced classical mechanics: chaos

Example text

8 Sometimes it is convenient to use cartesian co-ordinates, in which the hamiltonian and lagrangian are, 1 1 H = x˙ 2 + y˙ 2 + V (x, y), 2 2 1 1 L = x˙ 2 + y˙ 2 + Ω [xy˙ − y x] ˙ − V (x, y). There are two ways this can happen: the asteroid can form an equilateral triangle with the Sun and Jupiter on either side of the line joining them. These are the Lagrange points L4 and L5 . These are actually maxima of the potential. In spite of this fact, they correspond to stable equilibrium points because of the effect of the velocity dependent forces.

Now in order that ω be real, the eignvalues of K should be positive: it must be a positive matrix. •The the eigenvectors of the pair (g, K) are the normal modes of the system: they form a co-ordinate system in which each axis has a definite frequency of vibration. •If the normal frequencies ωj are all integer multiples of some underlying real number ω all solutions will be periodic. In general, the motion is quasi-periodic: each normal mode is periodic with a different period. •Let us complete the story by studying velocity dependent forces such as magnetic fields.

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) . Such a system will be called ∂Ii normal co-ordinate system; they are also called action-angle variables. •The motion is on surfaces of constant Ii , on which the n variables θi are co-ordinates; they depend linearly on time. Often this is a surface of finite volume and the co-ordinates θi are some sort of angles: the surface is a torus. Indeed, there is a vector field (the velocity) which is everywhere nonzero and has constant components in these co-ordinates.

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