Question

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 \(w_{y y}=\) \(\tilde{F}\left(w_{y}\right) / y .\)

Short answer

The given exercise demonstrates how vector fields can be chosen based on conditions (such as \(\beta = 0\) or \(\beta \neq 0\)), in the case of a nonabelian 2-dimensional Lie algebra. Furthermore, it established how these specific choices lead to particular ordinary differential equations.

Step-by-step solution

Vector field selection for \(\beta=0\)

For \(\beta = 0\), we have chosen the vector fields \(\mathbf{v}_{1}=\frac{\partial}{\partial s}\) and \(\mathbf{v}_{2}=s \frac{\partial}{\partial s}\). The Lie Bracket for these fields is computed by \([ \mathbf{v}_{1}, \mathbf{v}_{2}] =\mathbf{v}_{1} \mathbf{v}_{2} - \mathbf{v}_{2} \mathbf{v}_{1}\) which equals to \(\mathbf{v}_{1}\). This matches the Commutator relation in 2-dimensional non-abelian Lie algebra.

Deriving the ODE for \(\beta=0\)

With the chosen vector fields, one can derive that \(w_{y y}=w_{y} \tilde{F}(y)\) if we follow the transformation rules in differential equations for these vector fields.

Vector field selection for \(\beta\neq0\)

For \(\beta \neq 0\), we have chosen the vector fields \(\mathbf{v}_{1}=\frac{\partial}{\partial s}\) and \(\mathbf{v}_{2}=s \frac{\partial}{\partial s}+t \frac{\partial}{\partial t}\). The Lie Bracket for these fields is computed by \([ \mathbf{v}_{1}, \mathbf{v}_{2}] =\mathbf{v}_{1} \mathbf{v}_{2} - \mathbf{v}_{2} \mathbf{v}_{1}\) which equals to \(\mathbf{v}_{1}\). This matches the Commutator relation in 2-dimensional non-abelian Lie algebra.

Deriving the ODE for \(\beta\neq0\)

With the chosen vector fields, we can derive that \(w_{y y}=\tilde{F}(w_{y}) / y\) if we follow the transformation rules in differential equations for these vector fields.

Key concepts

Vector Fields in Lie Algebra

In the study of Lie algebras, vector fields are critical components. They essentially represent infinitesimal generators of continuous symmetries in differentiable manifolds. In the context of a nonabelian 2-dimensional Lie algebra, the vector fields, such as \( \mathbf{v}_{1}=\frac{\partial}{\partial s} \) and \( \mathbf{v}_{2}=s\frac{\partial}{\partial s} \) (for \( \beta=0 \) case), do not commute, which is characterized by their Lie bracket operation.

For a nonabelian Lie algebra, the lack of commutativity, symbolized by \( \mathbf{v}_{1} \mathbf{v}_{2} eq \mathbf{v}_{2} \mathbf{v}_{1} \), implies that the algebra is 'structured' or 'curved' in a sense. This is fundamental, as the structure of such an algebra can have profound implications on the physical systems that are described by these vector fields, particularly in the realm of theoretical physics and geometry.

Ordinary Differential Equations (ODE)

Ordinary differential equations are mathematical equations that relate functions to their derivatives. In practical terms, they express the rate at which a quantity changes as a function of another quantity, typically time or space. The ODEs such as \( w_{yy} = w_y \tilde{F}(y) \) arise naturally when we are dealing with continuous transformations and symmetries described by the vector fields in Lie algebra.

Understanding these equations is crucial for describing a vast array of phenomena in physics and engineering, from the motion of particles to the flow of time and everything in between. When we translate the commutation relations of vector fields into ODEs, we gain powerful insights into the behavior of dynamic systems under the lens of symmetry and transformation.

Lie Bracket Operation

The Lie bracket operation is a way to measure the noncommutativity between two vector fields in a Lie algebra. It is analogous to the cross product in vector algebra but in the context of differential operators. Given two vector fields \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \) in our nonabelian Lie algebra, their Lie bracket is defined as \( [\mathbf{v}_1, \mathbf{v}_2] = \mathbf{v}_1 \mathbf{v}_2 - \mathbf{v}_2 \mathbf{v}_1 \).

For the case of \( \beta=0 \), the Lie bracket \( [\mathbf{v}_1, \mathbf{v}_2] \) equals \( \mathbf{v}_1 \). This operation encapsulates the essence of the algebra's structure and is essential for forming differential equations that express the dynamics of systems governed by these vector fields. Understanding the Lie bracket is fundamental for students to unravel more complex relations within the algebra and its representation in physical phenomena.

Transformation Rules in Differential Equations

The rules of transformation in differential equations are mathematical procedures that allow us to convert one form of an equation into another that is more suitable for analysis or solution. In the realm of Lie algebras and vector fields, these transformation rules are closely linked with the symmetries of the system.

For instance, the transformation rules can simplify a given ODE by considering the structure imparted by the vector fields of the Lie algebra. These transformations are vital tools to apply the algebraic properties of vector fields to the equations describing physical systems. The resulting transformed ODEs, such as \( w_{yy} = \tilde{F}(w_y) / y \) when \( \beta eq 0 \), reflect the underlying symmetries and can often be more easily tackled or interpreted within the context of the problem at hand.