Separatrix (dynamical systems)

In mathematics, a separatrix refers to the boundary separating two modes of behaviour in a differential equation.

Example

Consider the differential equation describing the motion of a simple pendulum:

:$\{d^2\; hetaover\; dt^2\}+\{gover\; l\}\; sin\; heta=0.$

where $l$ denotes the length of the pendulum, $g$ the gravitational acceleration and $heta$ the angle between the pendulum and vertically downwards. In this system there is a conserved quantity H (the Hamiltonian), which is given by

$H\; =\; frac\{dot\{\; heta\}^2\}\{2\}\; -\; frac\{g\}\{l\}cos\; heta.$

With this defined, one can plot a curve of constant H in the phase space of system. The phase space is a graph with $heta$ along the horizontal axis and $dot\{\; heta\}$ on the vertical axis - see the thumbnail to the right. The type of resulting curve depends upon the value of H.

If then no curve exist ($dot\{\; heta\}$ must be imaginary).

If

If

In this system the separatix is the curve that corresponds to $H=frac\{g\}\{l\}$. It separates (hence the name) the phase space into two distinct areas. Within the separatrix corresponds to oscillating motion back and forth, whereas the outside of the separatrix corresponds to motion with the pendulum continuously turning through circles.

External links

* [http://mathworld.wolfram.com/Separatrix.html Separatrix] from MathWorld.

Surely - better to show derivation of the constant of integration of the motion, H

using d(dx/dt)/dt = ( dx/dt )* d(dx/dt)/dx i.e. v * dv/dt [ v = dx/dt ] i.e. ( df/dt = (df/dx)*(dx/dt) and let f = dx/dt

so integrating dv/dt w.r.t. x gives (1/2) (v^2) = (1/2) [( dx/dt) ^2 ] [ briefly, dv/dt = dv/dx * dx/dt = v * dv/dxand integrating dv/dt w.r.t. x gives (1/2) v^2 ]

and to choose a datum (ZERO) for H corresponding to the pendulum at rest at the lowest point:

H' = H + (m)g/l

Then the circular (elliptical) orbits at small x are directly obtained from noting that since cos(x) ~ 1 - (1/2) x^2

then 2* H' (small) = (m)(dx/dt)^ 2 + ((m)g/l)* x^2