What is the name of this Hamiltonian?

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Discussion Overview

The discussion revolves around identifying the name of a specific Hamiltonian given by the formula H = (p² + q²)². Participants explore its classification and potential connections to known Hamiltonians in scientific literature.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant asks for the name of the Hamiltonian H = (p² + q²)², expressing uncertainty about its classification.
  • Another participant suggests it may relate to a Heisenberg group, referencing a Wikipedia article on Hamiltonian functions.
  • A participant reiterates their request for the specific name in scientific literature, clarifying they are not seeking a broader discussion on Hamiltonian theory.
  • A later reply mentions a connection to the Heisenberg group under Sub-Riemannian manifolds, indicating a specific Hamiltonian in the literature that resembles the one in question.
  • Another participant introduces the Henon–Heiles problem as an example of a polynomial Hamiltonian with chaotic solutions, suggesting it might be relevant.
  • One participant notes subtle differences between their Hamiltonian and the harmonic oscillator, proposing that their Hamiltonian can be viewed as the square of the harmonic oscillator Hamiltonian.

Areas of Agreement / Disagreement

Participants express differing views on the classification of the Hamiltonian, with no consensus reached on its specific name or relation to known Hamiltonians.

Contextual Notes

Some participants reference specific mathematical frameworks and examples, but the discussion does not resolve the classification or naming of the Hamiltonian in question.

kinichi
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Hello (and sorry for this stupid question),

Could someone tell me the name of this Hamiltonian

[tex]H = \left(\dfrac{p^2+q^2}{2}\right)^2[/tex]

Thanks in advance
 
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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.
 
kinichi said:
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.
 
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:

[tex]H = H_{HO}^2[/tex]

where

[tex]H_{HO} = \dfrac{1}{2}(p^2+q^2)[/tex].
 
Last edited:

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