example of rewriting a differential equation as a Pfaffian system
The first step is to rewrite the equation as a system of first-order equations
To translate these equations into the language of differential forms, we shall use the fact that
from which it follows that
We can do likewise with or or in the place of ; there is no point in repeating the formulas for each of these variables.
Multiplying the differential equations through by the form and using the above identities to eliminate partial derivatives, we obtain the following system of differential forms:
From the way these forms were constructed, it is clear that a three dimensional surface in the seven dimensional space with coordinates which solves Pfaff’s problem and can be parameterized by corresponds to the graph of a solution to the system of differential equations, and hence to a solution of the wave equation.
Note: These considerations are purely local. The global topology of the seven-dimensional space will depend on the domain on which the original wave equation was formulated and on the boundary conditions.
|Title||example of rewriting a differential equation as a Pfaffian system|
|Date of creation||2013-03-22 14:38:49|
|Last modified on||2013-03-22 14:38:49|
|Last modified by||rspuzio (6075)|