Hamiltonian Chaos

Hamiltonian maps

• Consider a map M : R2N → R2N ,defined as,M(x(t),t)=x(t+h)• Its differential variation is ,(∂M/∂x)δx=δx(t+h)• Now, Symplectic condition => δxT (t) Ωδ x' (t) = δxT (t+h) Ωδ x' (t+h) = ((∂M/∂x)δx(t))TΩ(∂M/∂x)δx’(t) =δxT(t)(∂M/∂x)TΩ(∂M/∂x)δx’(t)So, Ω=(∂M/∂x)TΩ(∂M/∂x).

(∂M/∂x) is symplectic in nature.

Poincare maps

• Poincare map gives an intersection of an orbit with a lower dimensional subspace of the phase space(called Poincare section).

• For periodic orbits, plotting the subsequent intersections can give a figure reflecting some of the qualitative characteristics of the dynamical system.

• For a Hamiltonian system H(q,p,t), the Poincare section is τ-periodic in time.

• Extended to (2N+1)-dimensional phase space,(q,p,ξ), where ξ=t i.e. dξ/dt=1.

• Let,ξ=� ξ mod(τ) and ξ?=to Є(0,τ) on 2N-dimensional surface of section ,then the map M(x,t0)=M(x,t0+nτ), where ,nЄZ, implies that the surface of the section map is a symplectic one.

Figure:: Poincare section

Action angle variable• Gives the tranformation relation between the points(q,p) in phase

space and points on the N-torus.• It is possible to construct new set of co-ordinates (P,Q) from the old

(q,p) such that Pi(q,p) are the constant of motion and then dPi/dt=-dH/dQi =0 implies H=H(P).

• One convenient choice to form P as P=J=(1/2π)∫гp.dq , ……….(1) where ,Г is the all the possible path around any possible angle direction on N-torus.This new variable J is called the action angle variable and it can be shown to be constant of motion by Poincare-Cartan integral theorem.

• The corresponding conjugate co-ordinate Qi=Ѳi are angles and we need to define a convenient generating function S(J,q) in order to obtain the set(J,Ѳ).

• Ѳ=∂S/∂J ………………………(2) p=∂S/∂q ………………………(3)

Action angle variable

• Now, integrating equ(3) around the path Г ,we get the change in S as,

∆iS=∫г p.dq=2πJ and ∆iѲ=(∂/∂J)(∆iS)=2πδij

• so after one circuit around the path, Ѳ value increases by 2π and the all other angles return their original value.

• So now ,the Hamiltonian is H=H(J), giving J(t)=J0 and ϴ(t)=Ѳ0+ωt , and describes the motion of the system on the torus.