Cauchy Problem for Hamilton-Jacobi Equations

Abstract

In classical terms, the boundary/initial value problem, the Cauchy problem for H-J equations, can be recast as follows

Appendices

Appendix 1: Envelopes, a (Very) Brief Introduction

Given a family of n − 1-dimensional submanifolds $G_{u} \subset \mathbb{R}^{n},\ u \in \mathbb{R}^{k}$, defined in implicit way by

$$\displaystyle{x \in G_{u}\qquad \text{if}\qquad f(x,u) = 0}$$

where $f: \mathbb{R}^{n} \times \mathbb{R}^{k} \rightarrow \mathbb{R}$, we say that G is its (smooth) envelope, whenever exists, if

$$\displaystyle{\forall x \in G}$$

• (i)  $\exists u \in \mathbb{R}^{k}\qquad u =\tilde{ u}(x)$ such that

  $$\displaystyle{x \in G_{\tilde{u}(x)}\qquad \mathrm{or}\qquad f(x,\tilde{u}(x)) = 0}$$

• (ii)  The tangent space at x to G is precisely the tangent space at x to $G_{\tilde{u}(x)}$:

  $$\displaystyle{T_{x}G = T_{x}G_{\tilde{u}(x)}}$$

  Setting $F(x):= f(x,\tilde{u}(x))$, we have that F(x) = 0 is a representation of G. Analytically, the above relation (ii) between tangent spaces can be read

  $$\displaystyle{\mathrm{ker}\,dF(x) = \mathrm{ker}\,d_{x}f(x,u)\big\vert _{\tilde{u}(x)}}$$

  in other words, v ∈ T _x G, that is,

  $$\displaystyle{\Big( \frac{\partial f} {\partial x_{i}}(x,u)\big\vert _{u=\tilde{u}(x)} + \frac{\partial f} {\partial u_{\alpha }}(x,u)\big\vert _{u=\tilde{u}(x)} \frac{\partial \tilde{u}_{\alpha }} {\partial x_{i}}(x)\Big)v^{i} = 0}$$

  iff $v \in T_{x}G_{\tilde{u}(x)}$,

  $$\displaystyle{ \frac{\partial f} {\partial x_{i}}(x,u)\big\vert _{u=\tilde{u}(x)}v^{i} = 0}$$

  and a recipe in order to realize this is

  $$\displaystyle{ x \in G\quad \mathrm{if,\ for\ some\ }u \in \mathbb{R}^{k},\quad \quad \left \{\begin{array}{rl} f(x,u) = 0,&\ \\ \\ \frac{\partial f} {\partial u_{\alpha }}(x,u) = 0.&\ \end{array} \right. }$$

  (4.31)

  But, at the end, given a family G _u , in general we cannot find a everywhere global smooth envelope G as in (i) and (ii), and we define simply as envelope of G _u precisely the subset of the points x of $\mathbb{R}^{n}$ such that (4.31) are holding for some $u \in \mathbb{R}^{k}$.

We see that the above k + 1 equations involving n + k variables, whenever suitable rank conditions are satisfied, give us a n − 1 dimensional locus G (in Fig. 4.2 some example of envelop).

Fig. 4.2

Appendix 2: Computation of Caustics via Projective Duality

In this section, based on [10], we show how to use the Projective Duality to compute the support of caustics related to geometrical solutions (i.e. Lagrangian submanifolds) of the geometrical Cauchy problem for the eikonal equation, a special case of the Hamilton-Jacobi equation. Although the computation is carried out for the simple Hamiltonian function $H(q,p) = \frac{1} {2}p^{2}$ on $T^{{\ast}}\mathbb{R}^{2}$, we will deal with arbitary C ² initial data $\sigma:\varSigma \rightarrow \mathbb{R}$, assigned on the initial manifold (curve) Σ embedded in $\mathbb{R}^{2}$.

4.2.1 Projective Duality and Plücker Coordinates

In order to be self-contained, in this section we review some basic facts about projective geometry. We recall that $\mathbb{P}^{2}(\mathbb{R})$ can be identified with $\mathbb{R}^{3} -\{ 0\}/ \sim$, where the equivalence relation ∼ means that two points (x, y, z) and (x′, y′, z′) in $\mathbb{R}^{3}$ are identified iff there exists a $\lambda \in \mathbb{R}^{{\ast}}$, such that (x, y, z) = (λ x′, λ y′, λ z′). Under this identification, a point $Q \in \mathbb{P}^{2}(\mathbb{R})$ corresponds to a straight line through the origin in $\mathbb{R}^{3}$.

Given two distinct points Q and Q′ in $\mathbb{P}^{2}(\mathbb{R})$, which correspond to lines l and l′ in $\mathbb{R}^{3}$ respectively, we construct the line π _QQ′ in $\mathbb{P}^{2}(\mathbb{R})$, joining Q and Q′. If the lines $l = (\bar{x},\bar{y},\bar{z}) = (\lambda x_{0},\lambda x_{1},\lambda x_{2})$ and $l' = (x',y',z') = (\mu x_{0}^{'},\mu x_{1}^{'},\mu x_{2}^{'})$ correspond to Q = [x ₀, x ₁, x ₂] and Q′ = [x ₀ ^′, x ₁ ^′, x ₂ ^′] respectively, then the equation of the plane Π in $\mathbb{R}^{3}$ containing l and l′ is of the form

$$\displaystyle{\mathit{ax} + \mathit{by} + \mathit{cz} = 0,}$$

where the coefficients (a, b, c) are obtained imposing the conditions l ⊂ Π and l′ ⊂ Π. This implies that

$$\displaystyle{ (a,b,c) = (x_{0},x_{1},x_{2}) \wedge (x_{0}^{'},x_{ 1}^{'},x_{ 2}^{'}), }$$

(4.32)

where ∧ means vector product. Therefore a vector $v = (\bar{x},\bar{y},\bar{z}) \in (\mathbb{R}^{3} -\{ 0\})$ belongs to Π iff $a\bar{x} + b\bar{y} + c\bar{z} = 0$. This last equation can actually be read as an equation in $\mathbb{P}^{2}(\mathbb{R})$, recalling that v identifies a unique line l through the origin and so a unique point Q in $\mathbb{P}^{2}(\mathbb{R})$. Under this identification, we have that a point $S = [\tilde{x}_{0},\tilde{x}_{1},\tilde{x}_{2}] \in \mathbb{P}^{2}(\mathbb{R})$ belongs to π _QQ′ iff

$$\displaystyle{ a\tilde{x}_{0} + b\tilde{x}_{1} + c\tilde{x}_{2} = 0,\quad (a,b,c)\stackrel{(<InternalRef RefID="Equ36">4.32</InternalRef>)}{=}(x_{1}x_{2}^{'} -x_{ 2}x_{1}^{'},x_{ 2}x_{0}^{'} -x_{ 0}x_{2}^{'},x_{ 0}x_{1}^{'} -x_{ 1}x_{0}^{'}). }$$

(4.33)

In the literature, the coefficients (a, b, c) are called Plücker coordinates of the lines π _QQ′. By construction, these coordinates are defined up to multiplication by a common nonzero constant. In this way, the coordinates (a, b, c) can be thought as homogenous coordinates [a, b, c] on a different projective space: under the identification between lines and their Plücker coordinates, we obtain that lines in $\mathbb{P}^{2}(\mathbb{R})$ become the elements (points) of a two-dimensional projective space, called dual projective space $\mathbb{P}^{2}(\mathbb{R})^{{\ast}}$. Therefore, associated to $\mathbb{P}^{2}(\mathbb{R})$ there is $\mathbb{P}^{2}(\mathbb{R})^{{\ast}}$ whose points are lines (hyperplanes) of $\mathbb{P}^{2}(\mathbb{R})$.

It is natural to ask what are the lines of $\mathbb{P}^{2}(\mathbb{R})^{{\ast}}$ (i.e. points in the double dual $\mathbb{P}^{2}(\mathbb{R})^{{\ast}{\ast}}$.) Projective duality states that $\mathbb{P}^{2}(\mathbb{R})^{{\ast}{\ast}}\equiv \mathbb{P}^{2}(\mathbb{R})$, so that lines in $\mathbb{P}^{2}(\mathbb{R})^{{\ast}}$ can be identified with points in $\mathbb{P}^{2}(\mathbb{R}).$

Now we are going to review how to apply projective duality in order to compute the dual curve C ^∗ to a given curve C. Suppose we are given a parametric curve C in $\mathbb{P}^{2}(\mathbb{R})$, whose homogeneous coordinates are described by $[x_{0}(s),x_{1}(s),x_{2}(s)] \in \mathbb{P}^{2}(\mathbb{R})$. Fix a point $Q = [x_{0}(\bar{s}),x_{1}(\bar{s}),x_{2}(\bar{s})]$ in C: we want to compute the equation of the tangent line τ _Q to C in Q. Let Q′ be a different point belonging to C. By the results so far developed in this subsection, we know how to compute the equation in $\mathbb{P}^{2}(\mathbb{R})$ of the line π _QQ′ secant to C through Q and Q′:

$$\displaystyle\begin{array}{rcl} & \pi _{\mathit{QQ}'}: \quad h(x_{0},x_{1},x_{2}):= x_{0}(x_{1}(\bar{s})x_{2}(s^{'}) - x_{2}(\bar{s})x_{1}(s^{'})) & {}\\ & +x_{1}(x_{2}(\bar{s})x_{0}(s^{'}) - x_{0}(\bar{s})x_{2}(s^{'})) + x_{2}(x_{0}(\bar{s})x_{1}(s^{'}) - x_{1}(\bar{s})x_{0}(s^{'})) = 0.& {}\\ \end{array}$$

Then the tangent line τ _Q can be thought as obtained “by letting Q′ approach Q”, so that

$$\displaystyle{ \tau _{Q}: \quad \lim _{s^{'}\rightarrow \bar{s}}\frac{h(x_{0},x_{1},x_{2})} {s^{'} -\bar{ s}}, }$$

(4.34)

that is

$$\displaystyle{ \begin{array}{ll} \tau _{Q}:&\quad x_{0}(x_{1}(\bar{s})\dot{x}_{2}(\bar{s}) - x_{2}(\bar{s})\dot{x}_{1}(\bar{s})) + x_{1}(x_{2}(\bar{s})\dot{x}_{0}(\bar{s}) - x_{0}(\bar{s})\dot{x}_{2}(\bar{s})) \\ &\quad + x_{2}(x_{0}(\bar{s})\dot{x}_{1}(\bar{s}) - x_{1}(\bar{s})\dot{x}_{0}(\bar{s})) = 0. \end{array} }$$

(4.35)

By Eq. (4.35) we have obtained that the Plücker coordinates $(a,b,c)_{\tau _{Q}}$ of the tangent line to a curve C at a point $Q = [x_{0}(\bar{s}),x_{1}(\bar{s}),x_{2}(\bar{s})]$ are simply given by:

$$\displaystyle{ (a,b,c)_{\tau _{Q}} = (x_{0}(\bar{s}),x_{1}(\bar{s}),x_{2}(\bar{s})) \wedge (\dot{x}_{0}(\bar{s}),\dot{x}_{1}(\bar{s}),\dot{x}_{2}(\bar{s})). }$$

(4.36)

Assuming for simplicity that for any point in C there is a unique tangent line to C at this point and repeating this construction for every point of the curve C, we get a new curve C ^∗ in $\mathbb{P}^{2}(\mathbb{R})^{{\ast}}$, called the dual curve to C. The homogenous coordinates of points belonging to C ^∗ are exactly Plücker coordinates of tangent lines to C, so the homogenous parametric equation of C ^∗ are simply:

$$\displaystyle{ [x_{0}^{{\ast}}(s),x_{ 1}^{{\ast}}(s),x_{ 2}^{{\ast}}(s)] = [(x_{ 0}(\bar{s}),x_{1}(\bar{s}),x_{2}(\bar{s})) \wedge (\dot{x}_{0}(\bar{s}),\dot{x}_{1}(\bar{s}),\dot{x}_{2}(\bar{s}))]. }$$

(4.37)

As expected, if now we compute the dual curve to C ^∗ we come back to our original curve C, as a straightforward calculation can prove.

4.2.2 Computation of Caustics

In this section we determine the support of the caustic corresponding to a geometric solution Λ of a geometric Cauchy problem for the Hamilton-Jacobi equation when:

• 1. The Hamiltonian function $H: T^{{\ast}}\mathbb{R}^{2} \rightarrow \mathbb{R}$ is given by $H(p,q) = \frac{1} {2}p^{2}$;

• 2. The plane curve Σ is represented by $\varSigma \hookrightarrow Q \equiv \mathbb{R}^{2}$, s ↦ (x(s), y(s)), where $s \in I = (a,b) \subset \mathbb{R}$ and Σ is regular, that is the functions x(s) and y(s) are smooth and

  $$\displaystyle{\dot{x}(s)^{2} +\dot{ y}(s)^{2}\neq 0\quad \forall s \in I;}$$

• 3. The initial datum σ is an arbitrary smooth function on Σ (actually C ² is already sufficient), which can be identified with a function $\sigma: I \ni s\mapsto \sigma (s) \in \mathbb{R}$.

Theorem 4.3

Under the hypotheses described above on H, Σ and σ, the homogeneous coordinates [x ₀ ,x ₁ ,x ₂ ] in $\mathbb{P}^{2}(\mathbb{R})$ of the caustic associated to the corresponding geometric solution of the Hamilton-Jacobi problem (for a fixed value e of the Hamiltonian function) are given respectively by:

$$\displaystyle\begin{array}{rcl} x_{0} = \left [2e(\dot{x}\ddot{y} -\ddot{ x}\dot{y})\,\mathrm{sgn}(\dot{x})(A\dot{B} - B\dot{A})\right ],& &{}\end{array}$$

(4.38)

$$\displaystyle\begin{array}{rcl} x_{1} = \left [\mathit{AB}\vert \dot{x}\vert + 2\mathit{ex}(\dot{x}\ddot{y} -\ddot{ x}\dot{y}) - B^{2}\dot{y}x\,\mathrm{sgn}(\dot{x})(A\dot{B} - B\dot{A})\right ],& &{}\end{array}$$

(4.39)

$$\displaystyle\begin{array}{rcl} x_{2} = \left [\mathit{AB}\dot{y} + 2\mathit{ey}(\dot{x}\ddot{y} -\ddot{ x}\dot{y}) + B^{2}\dot{x}y\,\mathrm{sgn}(\dot{x})(A\dot{B} - B\dot{A})\right ],& &{}\end{array}$$

(4.40)

where A and B are given by:

$$\displaystyle\begin{array}{rcl} A =\dot{\sigma },& &{}\end{array}$$

(4.41)

$$\displaystyle\begin{array}{rcl} B = \sqrt{2e(\dot{x}^{2 } +\dot{ y}^{2 } ) -\dot{\sigma } ^{2}},& &{}\end{array}$$

(4.42)

and $\mathrm{sgn}(\dot{x}) = \frac{\dot{x}} {\vert \dot{x}\vert }$ ,

where dot denote derivative with respect to the parameter s.

Proof

It is based on projective duality in the following sense. Suppose we have to determine in $\mathbb{P}^{2}(\mathbb{R})$ a curve C (the caustic) and of it we know only the family of straight lines tangent to it. More precisely, we assume that a set of three functions is given

$$\displaystyle{ f(s) = (f_{0}(s),f_{1}(s),f_{2}(s)), }$$

(4.43)

depending on the parameter s on the curve C in such a way that, for $\bar{s}$ fixed the expression (4.43) gives the Plücker coordinates of the line τ _Q , tangent to the curve C at the point $Q = [x_{0}(\bar{s}),x_{1}(\bar{s}),x_{2}(\bar{s})]$. As Q varies in C the Plücker coordinates of τ _Q , seen as homogeneous coordinates in the dual projective space, define a new curve in $\mathbb{P}^{2}(\mathbb{R})^{{\ast}}$, the dual curve C ^∗. By hypothesis we know the support of this new curve C ^∗ and we want to recover C. It is easy to determine the Plücker coordinates of the tangent lines to C ^∗ at each of its points. Indeed, using Eq. (4.36) we get immediately:

$$\displaystyle{ \left (f_{1}(s)\dot{f}_{2}(s) - f_{2}(s)\dot{f}_{1}(s),f_{2}(s)\dot{f}_{0}(s) - f_{0}(s)\dot{f}_{2}(s),f_{0}(s)\dot{f}_{1}(s) - f_{1}(s)\dot{f}_{0}(s)\right ). }$$

(4.44)

Due to projective duality, we can think of expression (4.44) as identifying homogeneous coordinates for a curve in $\mathbb{P}^{2}(\mathbb{R})$, which is exactly the curve C we are searching for.

Let us come back to the original problem. We fix a positive values of the energy e and consider the submanifold determined by the constraint H ⁻¹(e) in $T^{{\ast}}\mathbb{R}^{2}$. Associating to each point of the curve Σ = { (x(s), y(s))} _s ∈ I the tangent vector $(\dot{x}(s),\dot{y}(s))$, from the equations which describe the submanifold of initial data Λ _(Σ, σ), we obtain that it has to be satisfied the following equation:

$$\displaystyle{ \dot{x}p_{x} +\dot{ y}p_{y} =\dot{ s}. }$$

(4.45)

Equation (4.45) means that the derivative of σ is equal to the scalar product between the tangent vector and p. Obviously, because it has to be true that $\varLambda _{(\varSigma,\sigma )} \cap H^{-1}(e)\neq \varnothing,$ it has to be true that p ² = 2e and so it has to be satisfied not only (4.45), but at the same time the following equation

$$\displaystyle{ p_{x}^{2} + p_{ y}^{2} = 2e. }$$

(4.46)

To determine the direction between the straight line escaping from Σ at the point P (this line is the projection of the corresponding phase curve on $\mathbb{R}^{2}$) and the tangent τ _P to Σ in P we use Eqs. (4.45) and (4.46). Observe that it is useless to determine the angle between the unit tangent vector and p, since we are interested only in the homogeneous coordinates (0, p _x , p _y ) which give the direction of the straight line emanating from Σ. Indeed, solving (4.45) with respect to p _x and substituting in (4.46), we find the following equation for p _y :

$$\displaystyle{ p_{y}^{2}(\dot{x}^{2} +\dot{ y}^{2}) - 2\dot{\sigma }\dot{y}p_{ y} +\dot{\sigma } ^{2} - 2e\dot{x}^{2}. }$$

(4.47)

Using the definitions of the quantities A and B introduced above, we get immediately

$$\displaystyle{ p_{x} = \frac{A -\dot{ y}p_{y}} {\dot{x}}, }$$

(4.48)

and

$$\displaystyle{ p_{y} = \frac{A\dot{y}\vert \dot{x}\vert B} {\dot{x}^{2} +\dot{ y}^{2}}. }$$

(4.49)

In order to use the projective duality, we embed the plane $\mathbb{R}^{2}$ in the projective plane $\mathbb{P}^{2}(\mathbb{R})$ in the usual way: that is if (x(s), y(s)) are the parametric equations of Σ in $\mathbb{R}^{2}$, then the corresponding homogeneous coordinates will be

$$\displaystyle{(x_{0}(s),x_{1}(s),x_{2}(s)) = (1,x(s),y(s)),}$$

while the homogeneous coordinates which define the direction of the straight line escaping from Σ at the point (1, x(s), y(s)) are given by (see Eqs. (4.48) and (4.49) and assume $\dot{x}\neq 0$):

$$\displaystyle{ (0,A -\dot{ y}p_{y},\dot{x}p_{y}), }$$

(4.50)

or more explicitly

$$\displaystyle{ (0,A\dot{x}^{2} \mp \vert \dot{x}\vert \dot{y}B,\dot{x}\dot{y}A\dot{x}\vert \dot{x}\vert B). }$$

(4.51)

To obtain Eq. (4.51) we have multiplied by $\dot{x}^{2} +\dot{ y}^{2}$ which is never zero by hypothesis.

Now we can find the Plücker coordinates of the corresponding straight lines (in the ordinary projective plane) which define the caustic as their envelope. We find:

$$\displaystyle{ \left ((A\dot{x}^{2} \mp \vert \dot{x}\vert \dot{y}B)y - x(\dot{x}\dot{y}A\dot{x}\vert \dot{x}\vert B),\quad \dot{x}\dot{y}A\dot{x}\vert \dot{x}\vert B,\quad \vert \dot{x}\vert \dot{y}B - A^{2}\right ). }$$

(4.52)

Now, to prove Eqs. (4.38)–(4.40) we have simply to identify (4.52) with (4.43) and then apply relation (4.44), which gives the homogeneous coordinates in $\mathbb{P}^{2}(\mathbb{R})$ of the envelope curve (which is the caustic in our case). Computations are rather lengthy, but straightforward. □ 

Let us remark that in the case the initial datum σ is constant, from Eqs. (4.38) to (4.40) we can find immediately the already known expression for the coordinates (in $\mathbb{R}^{2}$) of the caustic:

$$\displaystyle\begin{array}{rcl} x_{\mathrm{caus}} = x + \frac{\dot{y}(\dot{x}^{2} +\dot{ y}^{2})} {\ddot{x}\dot{y} -\ddot{ y}\dot{x}},& &{}\end{array}$$

(4.53)

$$\displaystyle\begin{array}{rcl} y_{\mathrm{caus}} = y -\frac{\dot{x}(\dot{x}^{2} +\dot{ y}^{2})} {\ddot{x}\dot{y} -\ddot{ y}\dot{x}}.& &{}\end{array}$$

(4.54)

Equations (4.53) and (4.54), describe the locus given by the centers of the circles osculating Σ. In this case, the caustic coincides with the evolute of Σ.