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How do you decide on the sign of a Hamiltonian function?

For example, I have the following system of differential equations:

[itex]x'=y[/itex]

[itex]y'=-\dfrac{3}{2}x^{2}-2x[/itex]

With the following Hamiltonian:

[tex]H^{\oplus}=\dfrac{1}{2}x^{3}+x^{2}+\dfrac{1}{2}y^{2}[/tex]

because [itex]\dfrac{dH^{\oplus}}{dt}=0[/itex]. But if [itex]\dfrac{dH^{\oplus}}{dt}=0[/itex] then [itex]\dfrac{dH^{\ominus}}{dt}=0[/itex] is also true.

We can write [itex]H=U\left(x\right)+T\left(v\right)[/itex] with [itex]v=y[/itex]. We can then use [itex]U\left(x\right)[/itex] to construct the phase portrait of the system of differential equations.

With the use of Matlab I created the phase portrait of the system and it is obvious that the Hamiltonian with the positive sign leads to the correct plot. My question now is how do I know which Hamiltonian I should use?

Plot of [itex]U^{\oplus}[/itex]:

Plot of phase portrait:

For example, I have the following system of differential equations:

[itex]x'=y[/itex]

[itex]y'=-\dfrac{3}{2}x^{2}-2x[/itex]

With the following Hamiltonian:

[tex]H^{\oplus}=\dfrac{1}{2}x^{3}+x^{2}+\dfrac{1}{2}y^{2}[/tex]

because [itex]\dfrac{dH^{\oplus}}{dt}=0[/itex]. But if [itex]\dfrac{dH^{\oplus}}{dt}=0[/itex] then [itex]\dfrac{dH^{\ominus}}{dt}=0[/itex] is also true.

We can write [itex]H=U\left(x\right)+T\left(v\right)[/itex] with [itex]v=y[/itex]. We can then use [itex]U\left(x\right)[/itex] to construct the phase portrait of the system of differential equations.

With the use of Matlab I created the phase portrait of the system and it is obvious that the Hamiltonian with the positive sign leads to the correct plot. My question now is how do I know which Hamiltonian I should use?

Plot of [itex]U^{\oplus}[/itex]:

Plot of phase portrait:

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