- #1

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Could someone tell me the name of this Hamiltonian

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

Thanks in advance

- Thread starter kinichi
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- #1

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Could someone tell me the name of this Hamiltonian

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

Thanks in advance

- #2

jedishrfu

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http://en.wikipedia.org/wiki/Hamiltonian_function#Mathematical_formalism#Sub-Riemannian manifolds

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- #4

jedishrfu

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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

jedishrfu

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http://www.unige.ch/~hairer/poly_geoint/week1.pdfExample 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.

- #6

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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].

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].

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