Question

Prove that in a sufficiently small neighborhood of any point of a Euclidean space every solution of the Hamilton-Jacobi equation \(\sum\left(\partial u / \partial x_{i}\right)^{2}=1\) is the distance to some smooth hypersurface plus a constant.

Short answer

Question: In a sufficiently small neighborhood of any point in a Euclidean space, prove that every solution of the Hamilton-Jacobi equation is equal to the distance to some smooth hypersurface plus a constant. Answer: We defined a sufficiently small neighborhood around a point in a Euclidean space, and considered a smooth hypersurface within that neighborhood. By finding the partial derivatives of the unknown function u(x) and plugging them into the Hamilton-Jacobi equation, we showed that the solution u(x) corresponds to the distance to the hypersurface plus a constant. Thus, within a small neighborhood, every solution of the Hamilton-Jacobi equation is equal to the distance to some smooth hypersurface plus a constant.

Step-by-step solution

Recap on the Hamilton-Jacobi equation

The given Hamilton-Jacobi equation is: \[ \sum_{i}\left(\frac{\partial u}{\partial x_{i}}\right)^{2} = 1. \] This equation relates the function \(u(x_1,\dots, x_n)\) to its partial derivatives, where \(x_1,\dots, x_n\) are coordinates in the Euclidean space. The unknown function \(u\) will represent the distance to a hypersurface plus a constant.

Define a sufficiently small neighborhood

Let \(x_0\) be a point in the Euclidean space. A sufficiently small neighborhood \(U\) of \(x_0\) will be an open ball centered at \(x_0\) with radius \(r > 0\) small enough so that the properties of the Euclidean space are preserved and the hypersurface does not "bend" too much within this neighborhood.

Express the solution as the distance to some smooth hypersurface

Let \(S\) be a smooth hypersurface in the Euclidean space near the point \(x_0\). Then we can define a function \(d(x)\) which represents the distance from a point \(x\) in the neighborhood \(U\) to the hypersurface \(S\). We are looking for a function \(u(x) = d(x) + c\), where \(c\) is a constant, that satisfies the Hamilton-Jacobi equation in the neighborhood \(U\).

Calculate the partial derivatives of u

To analyze the Hamilton-Jacobi equation, we need to find the partial derivatives of \(u(x)\) with respect to each coordinate \(x_i\). Since \(u(x) = d(x) + c\), taking the partial derivatives gives: \[ \frac{\partial u}{\partial x_{i}} = \frac{\partial d}{\partial x_{i}}. \]

Plug the partial derivatives into the Hamilton-Jacobi equation

Now, we can substitute the expressions for the partial derivatives of \(u\) into the Hamilton-Jacobi equation: \[ \sum_{i}\left(\frac{\partial u}{\partial x_{i}}\right)^{2} = \sum_{i} \left(\frac{\partial d}{\partial x_{i}}\right)^{2} = 1. \]

Show that the solution u(x) corresponds to the distance to the hypersurface

Since the distance function \(d(x)\) has the property that its square of gradients sum to 1 in the neighborhood \(U\), and the unknown function \(u(x)=d(x)+c\) also satisfies the Hamilton-Jacobi equation, we can conclude that the function \(u(x)\) represents the distance to the hypersurface \(S\) plus a constant. Thus, we have proven that in a sufficiently small neighborhood of any point of a Euclidean space, every solution of the Hamilton-Jacobi equation is the distance to some smooth hypersurface plus a constant.

Key concepts

Euclidean Space

Euclidean space is a mathematical framework that provides the setting for our intuitive understanding of geometry, such as points, lines, and planes. When discussing Euclidean space in mathematics, we are often referring to a space of a certain dimension, denoted as \( \mathbb{R}^n \). Here, \( n \) represents the number of dimensions, which can be any non-negative integer.

In a Euclidean space:

• Each point is represented by an \( n \)-tuple \((x_1, x_2, \ldots, x_n)\), where each \( x_i \) is a real number.
• The concept of distance between points is defined by the Euclidean metric, often seen as the familiar distance formula derived from the Pythagorean theorem.

The exercise involves exploring properties within a Euclidean space where algebra and geometry intersect to solve partial differential equations like the Hamilton-Jacobi equation.

Smooth Hypersurface

A smooth hypersurface in mathematics is essentially a surface that exists in higher-dimensional spaces, such as those defined within a Euclidean space \( \mathbb{R}^n \). It is similar to a two-dimensional surface in three-dimensional space but generalizes to \( n-1 \) dimensions in an \( n \)-dimensional space.

Smoothness implies that the hypersurface is continuously differentiable, meaning it has no sharp edges or abrupt changes in direction, which makes working with calculus particularly efficient.

• Think of smoothness as akin to a gently bending sheet of paper or the surface of a balloon.
• When we determine distances in the Euclidean space relative to these hypersurfaces, the calculus of their properties becomes relevant.

In the context of solving the Hamilton-Jacobi equation, a smooth hypersurface plays a crucial role because solutions to the equation can be expressed as the distance function to these hypersurfaces, paired with a constant.

Partial Derivatives

Partial derivatives are used when dealing with functions of multiple variables — a common occurrence in Euclidean spaces. A partial derivative measures how a function changes as one of the variables changes, while all other variables are held constant. In essence, it measures the function's response to changes in one particular direction.

The notation \( \frac{\partial u}{\partial x_i} \) represents the partial derivative of the function \( u \) with respect to the variable \( x_i \).

• Each partial derivative provides a "slice" of information about how the function behaves along one axis of the Euclidean space.
• In the Hamilton-Jacobi equation, the sum of the squares of these partial derivatives equals 1. This indicates that the gradient's magnitude is normalized.

Calculating and substituting partial derivatives into equations is crucial for understanding how functions interact with the geometric properties of the space they inhabit.

Distance Function

A distance function is a mathematical function that describes the distance from a given point in space to a specified object, such as a smooth hypersurface. This function is crucial when solving the Hamilton-Jacobi equation because it helps express the relationship between the solutions of the equation and the geometry of hypersurfaces.

In application:

• The distance from a point \( x \) to a hypersurface \( S \) can be represented as \( d(x) \), where \( d \) is the distance function.
• This concept is important because solving the Hamilton-Jacobi equation involves these distance functions to ensure that the equation holds true in the Euclidean space.

The distance function not only measures how far points are from surfaces but also helps find solutions that satisfy the conditions set by equations like those in the exercise. Ultimately, it reveals the connection between geometric objects and analytical expressions.