Gravitational lensing formalism

In mathematics, especially in the area of mathematical analysis known as dynamical systems theory, a linear flow on the torus is a flow on the n-dimensional torus

${𝕋}^n=\underset{n}{\underbrace{S^1\times S^1\times \cdots \times S^1}}$

which is represented by the following differential equations with respect to the standard angular coordinates (θ₁, θ₂, ..., θ_n):

$\frac{d{\theta }_1}{dt}={\omega }_1,\frac{d{\theta }_2}{dt}={\omega }_2,\cdots ,\frac{d{\theta }_n}{dt}={\omega }_n.$

The solution of these equations can explicitly be expressed as

${\Phi }_{\omega }^t({\theta }_1,{\theta }_2,\dots ,{\theta }_n)=({\theta }_1+{\omega }_{1}t,{\theta }_2+{\omega }_{2}t,\dots ,{\theta }_n+{\omega }_{n}t)mod2\pi .$

If we respesent the torus as Rⁿ/Zⁿ we see that a starting point is moved by the flow in the direction ω=(ω₁, ω₂, ..., ω_n) at constant speed and when it reaches the border of the unitary n-cube it jumps to the opposite face of the cube.

For a linear flow on the torus either all orbits are periodic or all orbits are dense on a subset of the n-torus which is a k-torus. When the components of ω are rationally independent all the orbits are dense on the whole space. This can be easily seen in the two dimensional case: if the two components of ω are rationally independent then the Poincaré section of the flow on an edge of the unit square is an irrational rotation on a circle and therefore its orbits are dense on the circle, as a consequence the orbits of the flow must be dense on the torus.

See also

• Completely integrable system
• Ergodic theory
• Quasiperiodic motion

Bibliography

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