Hamiltonian for a Dissipative System/ Liouville's Theorem

In summary, the conversation discusses the Hamiltonian for a particle in free fall and its corresponding phase space region. It also looks at how the model can be expanded to include air friction and how this affects the Hamiltonian equations and Liouville's Theorem. The area of the region is conserved in the absence of air friction, but becomes non-conserved in the presence of it.
  • #1
kaiserwilhelm
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Homework Statement



Given is the Hamiltonian for a particle in free fall:

H(z,p) = P^2/(2m) + mgz

At time t=0 there is an region given by the constrains:

p1 less than or equal to p less than or equal to p2
E1 less than or equal to E less than or equal to E2

What is the area of the region?

This area moves in the phase space. What are the bounds at t > 0? What is the region's area at this later time?

Suppose we expand the model to include air friction of the Form F= -b(zdot), where b is a constant > 0 and zdot is the velocity. How can the Hamiltonian equations be expanded to include the air friction? Does Liousville's Theorem still apply?

Homework Equations



The area is given by integrating between the parabolas in the phase space:

(E2-E1)(p2-p1)/(mg)

The bounds for t > 0 are given by

(p1-mgt) less than or equal to p less than or equal to (p2-mgt)
E1 less than or equal to E less than or equal to E2 (unchanged)

Substitution into the above area formula yields an conserved area, as we'd expect from Liouville's Theorem.

It's the jump to the Hamiltonian for the dissipative Hamiltonian function that's got me rather stuck at the moment. I know that the phase-space trajectories will converge for the dissipative system, i.e. that the area is no longer conserved and that Liouville's Theorem no longer applies, but I'm unsure of how to show this analytically.

Any advice would be appreciated.
 
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  • #2
Try writing the Lagrange equation with the friction term included then transform to the Hamiltonian expression. Then ponder, "is this a conservative force and how does this effect Louisville's Theorem?"
 

FAQ: Hamiltonian for a Dissipative System/ Liouville's Theorem

What is a Hamiltonian for a Dissipative System?

A Hamiltonian for a dissipative system is a mathematical function that describes the dynamics of a system that is losing energy over time. It takes into account both the energy-conserving forces, described by the Hamiltonian, and the energy-dissipating forces, which are usually non-conservative and cannot be described by a Hamiltonian. This approach is used in classical mechanics to describe systems that are not in a state of equilibrium.

What is the significance of Liouville's Theorem in relation to Hamiltonian for a Dissipative System?

Liouville's Theorem states that the phase space volume occupied by a system remains constant over time. This means that the system's dynamics preserve the probability distribution of the system's possible states. In the context of Hamiltonian for a dissipative system, Liouville's Theorem shows that even though the system is losing energy, the overall dynamics of the system maintain the same probability distribution. This is important because it allows for the application of Hamiltonian methods to dissipative systems.

How is a Hamiltonian for a Dissipative System different from a Hamiltonian for a conservative system?

A Hamiltonian for a conservative system describes the dynamics of a system that conserves energy over time, meaning there are no energy-dissipating forces present. In contrast, a Hamiltonian for a dissipative system takes into account both the energy-conserving forces and the energy-dissipating forces. This results in a more complex and non-conservative Hamiltonian function.

What are the limitations of using a Hamiltonian for a Dissipative System?

One limitation of using a Hamiltonian for a dissipative system is that it assumes the system is in a state of equilibrium. This means that the system's dynamics are not affected by external factors such as fluctuations or disturbances. Additionally, the Hamiltonian approach may not accurately describe systems with strong nonlinearity or complex energy-dissipating mechanisms.

How is a Hamiltonian for a Dissipative System calculated?

A Hamiltonian for a dissipative system is typically calculated using the Hamiltonian formalism, which involves determining the system's Lagrangian function and then using the Hamiltonian equations to obtain the Hamiltonian function. The Hamiltonian function can also be obtained by applying the principle of least action to the system. However, for more complex dissipative systems, numerical methods may be needed to calculate the Hamiltonian function.

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