Under which conditions Jacobi PDE system can be represented to symplectic monge Ampere equation? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T01:04:59Z http://mathoverflow.net/feeds/question/112173 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112173/under-which-conditions-jacobi-pde-system-can-be-represented-to-symplectic-monge-a Under which conditions Jacobi PDE system can be represented to symplectic monge Ampere equation? Hassan Jolany 2012-11-12T13:35:10Z 2012-11-12T18:59:36Z <p>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.</p> <p>Here Jacobi PDE system is </p> <p>$a_1+b_1\frac{\partial h_1}{\partial x_1}-c_1\frac{\partial h_1}{\partial x_2}-d_1\frac{\partial h_2}{\partial x_2}+e_1\frac{\partial h_2}{\partial x_1}+f_1\left (\frac{\partial h_1}{\partial x_1}\frac{\partial h_2}{\partial x_2}-\frac{\partial h_1}{\partial x_2}\frac{\partial h_2}{\partial x_1} \right )=0$ $a_2+b_2\frac{\partial h_1}{\partial x_1}-c_2\frac{\partial h_1}{\partial x_2}-d_2\frac{\partial h_2}{\partial x_2}+e_2\frac{\partial h_2}{\partial x_1}+f_2\left (\frac{\partial h_1}{\partial x_1}\frac{\partial h_2}{\partial x_2}-\frac{\partial h_1}{\partial x_2}\frac{\partial h_2}{\partial x_1} \right )=0$</p> http://mathoverflow.net/questions/112173/under-which-conditions-jacobi-pde-system-can-be-represented-to-symplectic-monge-a/112208#112208 Answer by Robert Bryant for Under which conditions Jacobi PDE system can be represented to symplectic monge Ampere equation? Robert Bryant 2012-11-12T18:59:36Z 2012-11-12T18:59:36Z <p>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.)</p> <p>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 &amp;= 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 &amp;= 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. </p> <p>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.</p> <p>These are what are often called 'conservation laws' in some literature on this subject, and one place you could look (at least in the hyperbolic case; the elliptic case and parabolic cases are somewhat similar, though, at least in the formal setting) is in our paper with Phillip Griffiths and Lucas Hsu, <em>Hyperbolic exterior differential systems and their conservation laws, I &amp; II</em>, which appeared in Selecta Mathematica (New Series) <strong>1</strong> (1995), pp. 21–112 and 265–323. (.dvi files of these papers are available at <a href="http://fds.duke.edu/db/aas/math/faculty/bryant/publications.html" rel="nofollow">http://fds.duke.edu/db/aas/math/faculty/bryant/publications.html</a>)</p>