# What is the name of this Hamiltonian?

Hello (and sorry for this stupid question),

Could someone tell me the name of this Hamiltonian

$$H = \left(\dfrac{p^2+q^2}{2}\right)^2$$

## Answers and Replies

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I'm just looking for the name that is called in the scientific literature. I am not looking for the Hamiltonian theory.

jedishrfu
Mentor
I'm just looking for the name that is called in the scientific literature. I am not looking for the Hamiltonian theory.
In the article, it shows a specific hamiltonian similar to yours and relates it to a "Heisenberg group" under the topic Sub-Riemannian manifolds.

Its unfortunate that the URL didn't directly jump there as expected.

jedishrfu
Mentor
Or maybe this one:

Example 5 (Henon–Heiles problem)
The polynomial Hamiltonian in two degrees of freedom
is a Hamiltonian differential equation that can have chaotic solutions.

Figure 1 shows a regular behaviour of solutions when the value of the Hamiltonian is small,and a chaotic behaviour for large Hamiltonian.
http://www.unige.ch/~hairer/poly_geoint/week1.pdf

Ok, Thank you jedishrfu.

However, there are subtle differences between these two Hamiltonians. "My" can be thought of as the Hamiltonian of the harmonic oscillator squared:

$$H = H_{HO}^2$$

where

$$H_{HO} = \dfrac{1}{2}(p^2+q^2)$$.

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