Understanding homoclinic orbits

[thrillhouse86]

Hey all,

Can someone please tell me why is the trace of the Jacobian of an integrable hamiltonian system equall to zero on the homoclinic / seperatrix orbit ?

Cheers,
Thrillhouse

 

[thrillhouse86]

So for anyone who is interested this is infact a general property of Hamiltonian systems that the Trace of the Jacobian is zero.

 

[charng]

Hi Thrillhouse,

Homoclinic orbits are a type of trajectory in a dynamical system that connects two different equilibrium points. These orbits are important because they represent a boundary between different types of behavior in the system.

To understand why the trace of the Jacobian of an integrable Hamiltonian system is equal to zero on the homoclinic/separatrix orbit, we need to first understand what the Jacobian and Hamiltonian are. The Jacobian is a matrix of partial derivatives that describes the local behavior of a dynamical system. The Hamiltonian, on the other hand, is a function that describes the total energy of the system.

In an integrable Hamiltonian system, the Hamiltonian is a conserved quantity, meaning it remains constant throughout the system's evolution. This also means that the Jacobian of the system will have a trace of zero, as the Hamiltonian does not change.

Now, on the homoclinic/separatrix orbit, the system is transitioning between two different equilibrium points. This means that the Hamiltonian is the same at both points, and therefore the Jacobian will also have a trace of zero. This is because the system has reached a critical point where the energy is the same and the local behavior is also the same.

In summary, the trace of the Jacobian of an integrable Hamiltonian system is equal to zero on the homoclinic/separatrix orbit because the system is transitioning between two equilibrium points where the Hamiltonian is constant. I hope this helps to clarify your understanding.