Critical points of a Hamiltonian system

Thread starter xeno_gear
Start date Aug 18, 2010
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Critical points Hamiltonian Points System

In summary, a Hamiltonian system is a physical system that follows the laws of classical mechanics and is named after the mathematician and physicist William Rowan Hamilton. Critical points of a Hamiltonian system are points in phase space where the gradient of the Hamiltonian function is equal to zero and can be calculated by taking the partial derivatives of the Hamiltonian function and setting them equal to zero. These points represent important physical states and play a role in determining the stability and behavior of the system. They can also change if the system's parameters or variables are altered.

Aug 18, 2010

xeno_gear

Hello. I'm studying for an ODE/PDE qualifier and I'm wondering how to do this problem. I feel like it should be pretty easy, but anyway..

Show that a Hamiltonian system in [tex] \mathbb{R}^{2n} [/tex] has no asymptotically stable critical points.

Any suggestions? Thanks..

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Aug 19, 2010

Eynstone

I think Liouville's theorem will put this through.

1. What is a Hamiltonian system?

A Hamiltonian system is a physical system that follows the laws of classical mechanics, in which the motion of particles is described by Hamilton's equations. It is named after the mathematician and physicist William Rowan Hamilton.

2. What are critical points of a Hamiltonian system?

Critical points of a Hamiltonian system are points in phase space where the gradient of the Hamiltonian function is equal to zero. They represent stationary points or equilibrium points where the system is in a state of stable or unstable equilibrium.

3. How are critical points of a Hamiltonian system calculated?

The critical points of a Hamiltonian system can be found by taking the partial derivatives of the Hamiltonian function with respect to the system's variables and setting them equal to zero. This will give a set of equations which can be solved to find the critical points.

4. What is the significance of critical points in a Hamiltonian system?

Critical points in a Hamiltonian system are important because they represent important physical states of the system, such as equilibrium points or points of maximum or minimum energy. They also play a role in determining the stability and behavior of the system.

5. Can critical points of a Hamiltonian system change?

Yes, the critical points of a Hamiltonian system can change if the system's parameters or variables are altered. This can lead to changes in the stability and behavior of the system, and can also affect the motion of particles within the system.