Hamiltonian Chaos

Indrajit MondalRatan Sarkar

Koyel Das

Hamiltonian System A Hamiltonian system is a dynamical system governed by

Hamilton’s equation of motion. The state of the system is defined by x = where p= generalized momentum and q= generalized co-ordinate. Both p and q are vectors with dimension N. Hamilton’s equations are

=

=

Hamiltonian- constant of motion

If H(q,p,t) = H(q,p) Hamiltonian is a constant of motion.Example: For Simple Harmonic Oscillator p = p, q = x So, H= + k Here the Hamiltonian does not depend on time, so the energy of the system is conserved.

Symplectic Structure

One important property of a hamiltonian system is it has a symplectic structure

= f(x,t) = Ω where Ω= I is Nindentity matrix.Liouville’s Theorem: Hamiltonian equations

preserve 2N-dimensional volume.Proof: f = + = 0So, = = 0 ( surface s is of 2N-1 dimension)

Symplectic condition: conservation of volume

Let us consider 3 orbits (p,q), (p+p, q+δq), (p+δp’, q+q’). The differential symplectic area is the sum of parallelogram area is

= = δpδq’- δqδp’ = δδx’Ω(using is symmetric, ΩΩ = -I and = I)

Integrable System Consider a function f(p(t), q(t))={f,H}If f(p,q) is a constant of motion,Then {f,H}=0 Theorem: The hamiltonian system H(p,q) is integrable if it has N

independent constants of motion (p,q),i=1,2,3……N which in involution i.e., {}=0 for all values of i and j.

N independent constants of motion restrict an integrable system on a N dimensional surface in phase space. As constants of motion are in involution, any bounded orbit on the surface is topologically equivalent to a N-dimensional torus.