What is the name of this Hamiltonian?

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In summary, the conversation discussed the name of a specific Hamiltonian and its relation to a Heisenberg group in the scientific literature. The Hamiltonian, known as the Henon-Heiles problem, is a polynomial differential equation with chaotic solutions. Figure 1 shows the regular and chaotic behavior of solutions depending on the value of the Hamiltonian. The difference between this Hamiltonian and the harmonic oscillator squared was also mentioned.
  • #1
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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  • #3
I'm just looking for the name that is called in the scientific literature. I am not looking for the Hamiltonian theory.
 
  • #4
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.
 
  • #5
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
 
  • #6
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:

What is the name of this Hamiltonian?

The name of a Hamiltonian is often dependent on the context in which it is being used. It may refer to a specific physical system, such as the Hamiltonian for a quantum harmonic oscillator, or it may be a more general term for the mathematical operator used to describe the energy of a system in classical or quantum mechanics.

What is the purpose of a Hamiltonian?

The Hamiltonian is a central concept in classical and quantum mechanics, and is used to describe the energy of a physical system. It helps to predict the behavior and evolution of a system over time by taking into account the system's potential and kinetic energy.

How is the Hamiltonian related to the energy of a system?

The Hamiltonian is a mathematical operator that represents the total energy of a system. In classical mechanics, the Hamiltonian is the sum of the system's potential and kinetic energy. In quantum mechanics, the Hamiltonian is an operator that acts on the wavefunction of a system to calculate its energy.

What is the difference between a classical and quantum Hamiltonian?

In classical mechanics, the Hamiltonian is a function of the system's position and momentum. In quantum mechanics, the Hamiltonian is an operator that acts on the wavefunction of a system to calculate its energy. Additionally, the Hamiltonian in quantum mechanics may also include terms for spin and other quantum properties.

How is the Hamiltonian used in practical applications?

The Hamiltonian is used in a wide range of practical applications, including predicting the behavior of physical systems in classical and quantum mechanics, designing and analyzing quantum algorithms, and understanding the properties of materials in condensed matter physics. It is also used in fields such as chemistry, engineering, and economics to model and solve complex problems.

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