Quasiperiodic motion

[1][2][3][4] In Hamiltonian mechanics with n position variables and associated rates of change it is sometimes possible to find a set of n conserved quantities.

One then has new position variables called action-angle coordinates, one for each conserved quantity, and these action angles simply increase linearly with time.

This gives motion on "level sets" of the conserved quantities, resulting in a torus that is an n-manifold – locally having the topology of n-dimensional space.

[5] The concept is closely connected to the basic facts about linear flow on the torus.

[6] Quasiperiodic motion does not exhibit the butterfly effect characteristic of chaotic systems.

[4] Rectilinear motion along a line in a Euclidean space gives rise to a quasiperiodic motion if the space is turned into a torus (a compact space) by making every point equivalent to any other point situated in the same way with respect to the integer lattice (the points with integer coordinates), so long as the direction cosines of the rectilinear motion form irrational ratios.

In higher dimensions it means the direction cosines must be linearly independent over the field of rational numbers.

Assuming the dimension of T is at least two, these can be thought of as one-parameter subgroups of the torus given group structure (by specifying a certain point as the identity element).

[7] There are n "internal frequencies", being the rates at which the n angles progress, but as mentioned above the set is not uniquely determined.

See Kronecker's theorem for the geometric and Fourier theory attached to the number of modes.

[9] The theory of almost periodic functions is, roughly speaking, for the same situation but allowing T to be a torus with an infinite number of dimensions.

[10] Ian Stewart wrote that the default position of classical celestial mechanics, at this period, was that motions that could be described as quasiperiodic were the most complex that occurred.

[11] For the Solar System, that would apparently be the case if the gravitational attractions of the planets to each other could be neglected: but that assumption turned out to be the starting point of complex mathematics.

[12] The research direction begun by Andrei Kolmogorov in the 1950s led to the understanding that quasiperiodic flow on phase space tori could survive perturbation.