Physics Asked on February 10, 2021
Let’s say we have a system with Hamilton equations
begin{equation}
dot{Y}^A = Omega^{AB}H_{,B}
end{equation}
where the coordinates are
begin{equation}
Y^A=big(q^i,p_jbig)
end{equation}
where $i=1,dots,N$ and $j=1,dots,N$, $A=1dots, 2N$, $H$ is the Hamiltonian of the system and $Omega$ is a symplectic 2-form, but not the canonical one, which is $Omega_0 = begin{bmatrix}0&mathbb{I}_{Ntimes N}-mathbb{I}_{Ntimes N}&0 end{bmatrix}$
The question is: Can we find new coordinates $Z^A$ (and probably new hamiltonian $K$) such that the equations of motion are written in the canonical form
begin{equation}
dot{Z}^A=Omega_0 ^{AB}K_{,B}
end{equation}
and if so, are there any conditions on $Omega$ for it to happen? Is there a way to obtain such coordinate transformation?
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