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Chapter 32: Problem 8

Find the first prolongation of the generator of scaling: \(x \partial_{x}+u \partial_{u} .\)

Short Answer

Expert verified
The first prolongation of the generator of scaling \(x \partial_{x}+u \partial_{u}\) is \( pr^{(1)}X(x, u) = x \partial_{x} + u \partial_{u} + dx \wedge du \partial_{u^i)\).

Step by step solution

01

Identifying the variables

Firstly, identify the variables. Here, \(x\) and \(u\) are the existing variables.
02

Write out the prolongation formula

Write the first prolongation of a vector field \(v\) as \(pr^{(1)}v = v + \sum dx^i \wedge du \partial_{u^i}\), for a generator \(X(x,u)\) given by \(X = x \partial_{x} + u \partial_{u}\).
03

Calculate the prolongation

Plug the generator into the formula. For \(X(x, u) = x \partial_{x} + u \partial_{u}\), the first prolongation (using Cartan鈥檚 formula) is calculated as \( pr^{(1)}X(x, u) = X + dx \wedge du \partial_{u^i}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scaling Symmetry
Scaling symmetry, in the context of differential equations, refers to the property that allows transformations of variables without changing the form of a given equation. This type of symmetry is particularly useful because it helps in simplifying equations and understanding underlying structures. In simpler terms, a scaling symmetry means that you can multiply variables by a constant factor, and the equation will still hold its original essence.
  • In the exercise, the generator of scaling is expressed as \( x \partial_{x} + u \partial_{u} \).
  • This notation represents transformations involving scaling the variables \( x \) and \( u \).
Understanding scaling symmetry provides essential insights into how differential equations behave under transformations. By recognizing these symmetries, one can often reduce complexity and solve equations more elegantly.
Vector Fields
Vector fields are a fundamental concept in mathematics, specifically in calculus and differential equations. A vector field represents a vector at every point in space, which usually describes the velocity of a moving particle through the region. In our exercise, the scaling generator \( X = x \partial_{x} + u \partial_{u} \) is an example of a vector field.
  • The notation \( \partial_{x} \) and \( \partial_{u} \) refers to the partial derivatives with respect to \( x \) and \( u \), respectively.
  • This signifies that the vector field is expressed in terms of the rate of change of these variables.
By analyzing vector fields, we get a clearer picture of how the solution to differential equations could be visualized across a plane or space.
Cartan's Formula
Cartan's formula is a powerful tool used in the study of differential forms and vector fields. It provides a way to define the Lie derivative, which is essential for understanding how vector fields operate on differential forms. In our context, Cartan's formula is used to find the prolongation of a vector field.
  • The formula helps in combining the generator with differentials to extend or 'prolong' the field.
  • This prolongation is crucial for working with extended symmetries in differential equations.
Utilizing Cartan's formula allows us to systematically approach and solve more complex differential equations by knowing how the vector field interacts with associated variables.
Generator of Scaling
The generator of scaling is a specific type of vector field that represents scaling symmetry transformations. In the exercise, the generator \( X = x \partial_{x} + u \partial_{u} \) serves as a base for calculating prolongations.
  • Think of it as the starting point that 'generates' modifications to the variables \( x \) and \( u \).
  • The generator captures how transformations scale these variables without altering the original equation structure.
This concept is crucial for prolongation because it identifies the intrinsic scaling properties that can be extended to include higher-order derivatives. Understanding the generator of scaling helps simplify equations and pursue solutions more effectively.

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Most popular questions from this chapter

Show that the \(n\) th prolongation of the generator of the \(i\) th translation, \(\partial_{i}\), is the same as the original vector.

Go through the case of \(\beta=0\) in the solution of the second order ODE and, choosing \(w=s\) and \(y=t\), show that \(F\) will be a function of \(y\) alone and the original DE will reduce to \(w_{y y}=F(y)\).

Suppose \(M(x, u) d x+N(x, u) d u=0\) has a 1-parameter symmetry group with generator \(\mathbf{v}=X \partial_{x}+U \partial_{u} .\) Show that the function \(q(x, u)=\) \(1 /(X M+U N)\) is an integrating factor.

The Korteweg-de Vries equation is \(u_{t}+u_{x x x}+u u_{x}=0\). Using the technique employed in computing the symmetries of the heat and wave equations, show that the infinitesimal generators of symmetries of the Korteweg-de Vries equation are $$\begin{array}{ll}\mathbf{v}_{1}=\partial_{x}, \quad \mathbf{v}_{2}=\partial_{t}, & \text { translation } \\\\\mathbf{v}_{3}=t \partial_{x}+\partial_{u}, & \text { Galilean boost } \end{array}$$ $$\mathbf{v}_{4}=x \partial_{x}+3 t \partial_{t}-2 u \partial_{u} . \quad \text { scaling }$$

Show that in the case of the nonabelian 2 -dimensional Lie algebra, (a) the vector fields can be chosen to be $$\mathbf{v}_{1}=\frac{\partial}{\partial s}, \quad \mathbf{v}_{2}=s \frac{\partial}{\partial s}$$ if \(\beta=0 .\) (b) Show that these vector fields lead to the \(\mathrm{ODE} w_{y y}=w_{y} \tilde{F}(y)\). (c) If \(\beta \neq 0\), show that the vector fields can be chosen to be $$\mathbf{v}_{1}=\frac{\partial}{\partial s}, \quad \mathbf{v}_{2}=s \frac{\partial}{\partial s}+t \frac{\partial}{\partial t} .$$ (d) Finally, show that the latter vector fields lead to the ODE \(w_{y y}=\) \(\tilde{F}\left(w_{y}\right) / y .\)
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