Calculating Length of Closed Orbits: Gutzwiller Formula & Hamiltonian Systems"

[zetafunction]

given a Hamiltonian $$H=p^2 + V(x)$$ how can you calculate the length of the closed orbits ? , i mean in gutzwiller formula you must perform a summation over the length of the closed orbits to calculate density of states g(E) but how can you know what the lenghts are ?? .. of course for Harmonic Oscillator orbits are circles so we can calculate their length but how about for other Hamiltonian systems ?

 

[alxm]

The Gutzwiller formula is semi-classical. It uses classical periodic orbits.

 

[Entropia]

The Gutzwiller formula is a powerful tool in calculating the density of states for Hamiltonian systems with closed orbits. However, as you mentioned, it requires knowledge of the lengths of these closed orbits in order to perform the summation.

In order to determine the lengths of closed orbits for a given Hamiltonian system, one approach is to use numerical methods. This involves solving the equations of motion for the system and tracing the trajectory of a particle over a period of time. The length of the closed orbit can then be calculated by integrating the distance traveled by the particle over the period. This process can be repeated for different initial conditions to obtain a range of closed orbit lengths.

Another approach is to use analytical techniques, such as perturbation theory or canonical transformations, to approximate the lengths of closed orbits for a given Hamiltonian system. These methods can provide insight into the general behavior of the orbits and can be used to calculate their lengths in certain cases.

It is important to note that for more complex Hamiltonian systems, the lengths of closed orbits may not have a simple analytical expression and may need to be approximated using numerical methods. Additionally, the Gutzwiller formula may need to be modified for systems with non-circular closed orbits.

In conclusion, the calculation of the lengths of closed orbits for a given Hamiltonian system can be done using a combination of numerical and analytical techniques. These lengths are necessary for the application of the Gutzwiller formula in calculating the density of states.