Question

Based on dynamics, describe the behavior of a phase portrait of a n-body problem.

Answer 1

The state of the system at any time t is represented by n-real variables
{x1, x2, x3, . . . , xn} ⇒ ~r
as coordinates of a vector ~r in an abstract n− dimensional space. We refer to
this space as the state space of simply phase space1
in keeping with its usage
in Hamiltonian dynamics. Thus the state of the system at any given time is a
point in this phase space.
2. The time evolution of the system, motion is represented by a set of first order
equations- the so called “equations of motion”:
dx1/dt = v1(x1, x2, . . . , xn, t)
dx2/dt = v2(x1, x2, . . . , xn, t)
. . .
dxn/dt = vn(x1, x2, . . . , xn, t).

Now we will find all the fixed points , like stable nodes, unstable nodes, Hyperbolic fixed points and stable spiral fixed points etc.

phase velocity is given by
~v = (∂H/∂p , −∂H/∂q )

The above portrait is of a pendulum.

1.Elliptic fixed point: For small φ - the pendulum just oscillates about the fixed
point at the origin. Close to the origin linear stability analysis shows that that
this is an elliptic fixed point which corresponds to E = −mgl. The motion
in the region −mgl < E < mgl is usually referred to as Libration which
is characterised by an average momentum: hpφi = 0. The phase curves are
approximate ellipses around the fixed point.
2. Unstable hyperbolic fixed point: The hyperbolic point corresponds to φ = ±π-
the point where the pendulum is held vertically upwards corresponding to (E =mgl); This is an unstable situation since a small displacement forces the mass to

move away from the fixed point unlike the stable case. The E = mgl curve in the
phase space passing through the hyperbolic fixed point is called the separatrix
whose equation is given by
1
2
mgl cos2(φ/2) =
p
2
φ
2m
3. The motion in the region E > mgl is rotation. Here pφ does not change sign
and hpφi 6= 0.
4. The separatrix divides the phase space into two disconnected regions: inside the
separatrix the phase curves are closed and may be continuously deformed into one another by changing energy. Outside the separatrix the motion is characterised by rotations with pφ having a definite sign through out the phase curve and hence open. Thus there are two distinct homotopy classes2
separated by
the separatrix.