Monge equation

In the mathematical theory of partial differential equations, a Monge equation, named after Gaspard Monge, is a type of first-order partial differential equation.

A Monge equation is a function of type ${\displaystyle F(u,q^{1:n},p_{1:n}):\mathbb {R} ^{2n+1}\to \mathbb {R} }$. The problem is to find solutions of type ${\displaystyle u(q^{1:n}):\mathbb {R} ^{n}\to \mathbb {R} }$, such that$${\displaystyle F\left(u,q,\partial _{q}u\right)=0.}$$In modern notation, it is an equation on a differentiable manifold ${\displaystyle M}$ defined by a function ${\displaystyle F:\mathbb {R} \times T^{*}M\to \mathbb {R} }$, where ${\displaystyle T^{*}M}$ is the cotangent bundle. The problem is to find solutions of type ${\displaystyle u(q):M\to \mathbb {R} }$, such that$${\displaystyle F\left(u,q,du\right)=0.}$$The Hamilton–Jacobi equation is a particularly important example.