Let us consider a manifold and a hypersurface . Given a smooth function , we are interested in solving for the eikonal equation
where the initial condition is for smooth.
(The phrase “eikonal equation” seems to refer to a specific case of this, but I’ll follow Taylor in applying the term to this more general situation.)
Using the story already discussed about symplectic forms, we can do this locally in the noncharacteristic case.
To define this, we pick local coordinates around a fixed such that is given by . Let the corresponding local coordinates for be together with . Choose a cotangent vector such that . Suppose , and . This is the noncharacteristic hypothesis.
Theorem 1 Under the noncharacteristic hypothesis, the eikonal equation can be locally solved—that is, there exists a neighborhood containing and a smooth function on satisfying (*) with . Moreover, .
The idea of the proof of this existence theorem is to construct by constructing , which in turn is done by constructing its graph—which after all must satisfy a specific equation—as a submanifold of . At first it will not be obvious that the graph actually corresponds to any . This will follow from the analysis of lagrangian submanifolds.
Lagrangian submanifolds of the cotangent bundle
Given a symplectic vector space (i.e. a vector space equipped with a nondegenerate skew-symmetric form ), a subspace is said to be lagrangian if and . There is an easy example if we are given a symplectic basis with . Then the subspaces spanned by the (resp. ) are lagrangian.
A submanifold of a symplectic manifold is lagrangian iff for , is a lagrangian subspace.
Proposition 2 Let be a section (i.e. 1-form) embedding as a submanifold of the cotangent bundle, which is endowed with the canonical form. Then is lagrangian iff is closed.
We will prove this by using local coordinates for and corresponding which together form a system of local coordinates on . The symplectic form on is then locally
Suppose in this local coordinates the map is defined by
or in other words
Then a basis for the tangent space of is given by
and we can compute
Comparing this with the local expression for shows that this is precisely the closedness condition.
Incidentally, I’m curious if there is an invariant proof here. Any ideas?
Constructing a surface in the cotangent bundle
We are now going to define the surface that will be the graph of , where is the solution of the eikonal equation (*). We start with the fiber over . Let be the inclusion.
In words, consists of cotangent vectors lying over that annihilate and which correspond to the differential of the boundary condition plus possibly something extra in another direction. Near , is a surface of dimension by the implicit function theorem and the noncharacteristic hypothesis. Take a neighborhood and intersect it with to get . So is a surface over the intersection of with some neighborhood of then.
However, we need to extend this surface over a neighborhood of in . We want to be zero on this surface, and we know that is constant on the integral curves of the Hamiltonian vector field because . So, we take the union of with the integral curves of through points of . I now claim that this is a surface indeed. As before, choose local coordinates about and let be the local coordinates for . The symplectic form is locally given as . Then because , includes a nonzero multiple of . So this is indeed a surface of dimension , which can be represented in some neighborhood of as a section of into . And on this surface, .
I claim now that the surface thus constructed is lagrangian, so we will be able to apply the previous result. So if and , we must prove then .
We first do this when . If we actually have tangency to , not simply , i.e. , then the fact that follows from Proposition 2 with replacing that , since the portion of the tangent vectors pointing orthogonally to cancel each other out. (I.e. use local coordinates so is given by ; then the portions of with in them annihilate each other. The other parts, which consist of for , do so as well by what has already been proved.) Next, if one of is a multiple of at the point , then
where denotes the application of the tangent vector to . If are both multiples of at , then trivially . So the symplectic form vanishes on if . The general case because the flows of the Hamiltonian vector field preserve the symplectic form , and every point on is obtained by such a flow from .
So, given the lagrangianness of the surface , we have thus found with satisfying the eikonal PDE above. Also, , so by modifying by a constant we get the appropriate boundary condition.
Next, I’ll try to say something about uniqueness, and more generally first-order nonlinear PDEs. After that, I’m thinking of possibly a post or two on integrable systems and perhaps the Arnold-Liouville theorem. Then perhaps some tools from analysis (e.g., distributions, Fourier transforms, Sobolev spaces, etc.) to get to the general theory of linear PDEs.