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In the [[mathematics|mathematical]] theory of [[partial differential equation]]s (PDE), the '''Monge cone''' is a geometrical object associated with a first-order equation. It is named for [[Gaspard Monge]]. In two dimensions, let


:<math>F(x,y,u,u_x,u_y) = 0\qquad\qquad (1)</math>


be a PDE for an unknown real-valued function ''u'' in two variables ''x'' and ''y''.  Assume that this PDE is non-degenerate in the sense that <math>F_{u_x}</math> and <math>F_{u_y}</math> are not both zero in the domain of definition. Fix a point (''x''<sub>0</sub>, ''y''<sub>0</sub>, ''z''<sub>0</sub>) and consider solution functions ''u'' which have
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:<math>z_0 = u(x_0, y_0).\qquad\qquad (2)</math>
 
Each solution to (1) satisfying (2) determines the [[tangent plane]] to the graph
 
:<math>z = u(x,y)\,</math>
 
through the point (''x''<sub>0</sub>,''y''<sub>0</sub>,''z''<sub>0</sub>). As the pair (''u''<sub>''x''</sub>, ''u''<sub>''y''</sub>) solving (1) varies, the tangent planes [[Envelope (mathematics)|envelope]] a cone in '''R'''<sup>3</sup> with vertex at (''x''<sub>0</sub>,''y''<sub>0</sub>,''z''<sub>0</sub>), called the '''Monge cone'''.  When ''F'' is [[Differential equations#Types of differential equations|quasilinear]], the Monge cone degenerates to a single line called the '''Monge axis'''. Otherwise, the Monge cone is a proper cone since a nontrivial and non-coaxial one-parameter family of planes through a fixed point envelopes a cone. Explicitly, the original partial differential equation gives rise to a scalar-valued function on the [[cotangent bundle]] of ''R''<sup>3</sup>, defined at a point (''x'',''y'',''z'') by
:<math>adx+bdy+cdz \mapsto F(x,y,z,-a/c,-b/c).</math>
Vanishing of ''F'' determines a curve in the [[projective plane]] with [[homogeneous coordinates]] (''a'':''b'':''c''). The [[dual curve]] is a curve in the projective [[tangent space]] at the point, and the affine cone over this curve is the Monge cone. The cone may have multiple branches, each one an affine cone over a simple closed curve in the projective tangent space.
 
 
As the base point (''x''<sub>0</sub>,''y''<sub>0</sub>,''z''<sub>0</sub>) varies, the cone also varies.  Thus the Monge cone is a cone field on '''R'''<sup>3</sup>. Finding solutions of (1) can thus be interpreted as finding a surface which is everywhere tangent to the Monge cone at the point.  This is the [[method of characteristics]].
 
The technique generalizes to scalar first-order partial differential equations in ''n'' spatial variables; namely,
 
:<math>F\left(x_1,\dots,x_n,u,\frac{\partial u}{\partial x_1},\dots,\frac{\partial u}{\partial x_n}\right) = 0.</math>
 
Through each point <math>(x_1^0,\dots,x_n^0, z^0)</math>, the Monge cone (or axis in the quasilinear case) is the envelope of solutions of the PDE with <math>u(x_1^0,\dots,x_n^0) = z^0</math>.
 
==Examples==
;Eikonal equation
The simplest fully nonlinear equation is the [[eikonal equation]].  This has the form
:<math>|\nabla u|^2 = 1,</math>
so that the function ''F'' is given by
:<math>F(x,y,u,u_x,u_y) = u_x^2+u_y^2-1.</math>
The dual cone consists of 1-forms ''adx''+''bdy''+''cdz'' satisfying
:<math>a^2+b^2-c^2=0.</math>
Taken projectively, this defines a circle.  The dual curve is also a circle, and so the Monge cone at each point is a proper cone.
 
==See also==
* [[Tangent cone]]
 
==References==
* {{cite book|title=Methods of mathematical physics, Volume 2|author=[[David Hilbert]] and [[Richard Courant]]|publisher=Wiley Interscience|year=1989}}
* {{springer|id=m/m064630|title=Monge cone|author=Ivanov, A.B.}}
* {{cite book|author=Monge, G.|title=Application de l'analyse à la géométrie|publisher=Bachelier|year=1850}} {{fr}}
 
[[Category:Partial differential equations]]

Latest revision as of 09:35, 24 December 2014


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