page_title: Liouville's theorem (Hamiltonian)
section_title: Damped Harmonic Oscillator
hierarchical_section_title: Liouville's theorem (Hamiltonian) / Examples / Damped Harmonic Oscillator
caption_attribution_description: English: The evolution of phase space volume for the damped harmonic oscillator.
context_page_description: In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that the phase-space distribution function is constant along the trajectories of the system—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time. This time-independent density is in statistical mechanics known as the classical a priori probability.
  There are related mathematical results in symplectic topology and ergodic theory.
  There are extensions of Liouville's theorem to stochastic systems.
context_section_description: One of the foundational assumptions of Liouville's Theorem is that the system obeys the conservation of energy. In the context of phase space, this is to say that  is constant on phase space surfaces of constant energy . If we break this requirement by considering a system in which energy is not conserved, we find that  also fails to be constant.
  As an example of this, consider again the system of  particles each in a -dimensional isotropic harmonic potential, the Hamiltonian for which is given in the previous example. This time, we add the condition that each particle experiences a frictional force. As this is a  non-conservative force, we need to extend Hamilton's equations as
  where  is a positive constant dictating the amount of friction. Following a very similar procedure to the undamped harmonic oscillator case, we arrive again at
  Plugging in our modified Hamilton's equations, we find
  Calculating our new infinitesimal phase space volume, and keeping only first order in  we find the following result.
  We have found that the infinitesimal phase space volume is no longer constant, and thus the phase space density is not conserved. As can be seen from the equation as time increases, we expect our phase space volume to decrease to zero as friction affects the system.
  As for how the phase space volume evolves in time, we will still have the constant rotation as in the undamped case. However, the damping will introduce a steady decrease in the radii of each ellipse. Again we can solve for the trajectories explicitly using Hamilton's equations, taking care to use the modified ones above. Letting  for convenience, we find
  where the values  and  denote the initial position and momentum of the  particle.
  As the system evolves the total phase space volume will spiral in to the origin. This can be seen in the figure above.
caption: The evolution of phase space volume for the damped harmonic oscillator. The same values of parameters are used as in the SHO case, with 
    
      
        
          γ
          =
          0.5
          
          (
          α
          =
          0.25
          )
        
      
      {\displaystyle \gamma =0.5\;(\alpha =0.25)}
    
  .