# Under which conditions Jacobi PDE system can be represented to symplectic monge Ampere equation?

We know we can reduce Symplectic monge ampere equations to Jacobi PDE system with some compatibility condition. I want to see when the vise versa is correct? and is there any theorem for it.

Here Jacobi PDE system is

$a_1+b_1\frac{\partial h_1}{\partial x_1}-c_1\frac{\partial h_1}{\partial x_2}-d_1\frac{\partial h_2}{\partial x_2}+e_1\frac{\partial h_2}{\partial x_1}+f_1\left (\frac{\partial h_1}{\partial x_1}\frac{\partial h_2}{\partial x_2}-\frac{\partial h_1}{\partial x_2}\frac{\partial h_2}{\partial x_1} \right )=0$ $a_2+b_2\frac{\partial h_1}{\partial x_1}-c_2\frac{\partial h_1}{\partial x_2}-d_2\frac{\partial h_2}{\partial x_2}+e_2\frac{\partial h_2}{\partial x_1}+f_2\left (\frac{\partial h_1}{\partial x_1}\frac{\partial h_2}{\partial x_2}-\frac{\partial h_1}{\partial x_2}\frac{\partial h_2}{\partial x_1} \right )=0$

• Can you give a reference for the definition of 'Jacobi PDE'? This is not standard terminology, as far as I know. If you give a definition, we might be able to help you. – Robert Bryant Nov 12 '12 at 15:52
• Dear Robert, I edited it – user21574 Nov 12 '12 at 16:26

Thanks for the clarification; I wasn't familiar with this terminology. I assume that the coefficients $a_i$, $b_i$, $c_i$, $d_i$, $e_i$, and $f_i$ are specified functions of $x_1,x_2,h_1,h_2$. (Let me know if this is not correct.)
Geometrically, what you have is a pair of $2$-forms on a $4$-dimensional manifold, which you have specified in coordinates $(x_1,x_2,h_1,h_2)$ as \begin{align} \Upsilon_1 &= a_1\ dx_1\wedge dx_2 + b_1\ dh_1\wedge dx_2 + \cdots + e_1\ dh_2\wedge dx_2 + f_1\ dh_1\wedge dh_2\\\\ \Upsilon_2 &= a_2\ dx_1\wedge dx_2 + b_2\ dh_1\wedge dx_2 + \cdots + e_2\ dh_2\wedge dx_2 + f_2\ dh_1\wedge dh_2\ ,\\\\ \end{align} and you are looking for surfaces that are Lagrangian with respect to both $2$-forms, as these correspond to (generalized) solutions to your PDE system.
In order for this system to be 'symplectic', what you want is to find (or, better, determine whether there exist) combinations $\Omega = \lambda^1\ \Upsilon_1 + \lambda^2\ \Upsilon_2$ that satisfy $\Omega^2\not=0$ and $d\Omega =0$. Typically, this is 4 first order linear equations for the two unknown functions $\lambda^i$, so it's overdetermined, and there are 'curvature invariants' that determine when there are solutions and how many there are.