Question

Let \(a_{\omega}\) be the velocity field of the points of a body rotating with angular velocity \(\omega\) about the point \(o \in R^{3}\), Find the commutator of the fields \(a_{\alpha}\) and \(a_{\beta}\).

Short answer

#tag_title#Question Compute the commutator of two velocity fields \(a_{\alpha}\) and \(a_{\beta}\) for a body rotating with angular velocity \(\omega\) about a point \(o \in R^{3}\). #tag_content#Answer The commutator of two velocity fields \(a_{\alpha}\) and \(a_{\beta}\) is given by: \([a_{\alpha}, a_{\beta}] = ( (\vec{\alpha} \times \vec{r_o})(\vec{\beta} \times \vec{r_o}) ) - ( (\vec{\beta} \times \vec{r_o})(\vec{\alpha} \times \vec{r_o}) )\).

Step-by-step solution

Recall the expression for the velocity field in terms of the angular velocity

The velocity field of the points of a rotating body can be determined using the cross product of position vector with respect to "o" and the angular velocity vector. Mathematically, it can be expressed as: \(a_{\omega} = \vec{\omega} \times \vec{r_o}\)

Calculate the actions of \(a_{\alpha}\) and \(a_{\beta}\) on each other

Given that \(a_{\alpha} = \vec{\alpha} \times \vec{r_o}\) and \(a_{\beta} = \vec{\beta} \times \vec{r_o}\), we have to compute the actions of these fields on each other. Let's start with the action of \(a_{\alpha}\) on \(a_{\beta}\), denoted as \(a_{\alpha}(a_{\beta})\). Using the properties of cross product and noting that \(a_{\alpha} = (\vec{\alpha} \times \vec{r_o})\), we have: \(a_{\alpha}(a_{\beta}) = (\vec{\alpha} \times \vec{r_o})(\vec{\beta} \times \vec{r_o})\) Next, compute the action of \(a_{\beta}\) on \(a_{\alpha}\), denoted as \(a_{\beta}(a_{\alpha})\): \(a_{\beta}(a_{\alpha}) = (\vec{\beta} \times \vec{r_o})(\vec{\alpha} \times \vec{r_o})\)

Compute the commutator \([a_{\alpha}, a_{\beta}]\)

Now, we are ready to calculate the commutator of \(a_{\alpha}\) and \(a_{\beta}\). Using the definition of the commutator as stated above, we get: \([a_{\alpha}, a_{\beta}] = a_{\alpha}(a_{\beta}) - a_{\beta}(a_{\alpha})\) Substitute the expressions for the actions of the fields and simplify: \([a_{\alpha}, a_{\beta}] = ( (\vec{\alpha} \times \vec{r_o})(\vec{\beta} \times \vec{r_o}) ) - ( (\vec{\beta} \times \vec{r_o})(\vec{\alpha} \times \vec{r_o}) )\) This is the commutator of the fields \(a_{\alpha}\) and \(a_{\beta}\).

Key concepts

Angular Velocity

Angular velocity is a crucial concept in rotational dynamics. It refers to the rate at which an object spins around a central point or axis. Imagine a spinning top or a planet rotating on its axis. Angular velocity tells us how fast this spinning occurs.

An object's angular velocity is a vector. This means it has both magnitude and direction. The magnitude indicates how fast the object spins, while the direction shows the axis around which the object rotates. In mathematical terms, angular velocity is often represented by the symbol \( \vec{\omega} \).

One fundamental formula involving angular velocity is the velocity field of a rotating body. It is given by the expression:

• \( a_{\omega} = \vec{\omega} \times \vec{r_o} \)

Here, \( \vec{r_o} \) is the position vector relative to the rotation point, and the cross product is used to obtain the actual velocity at each point of the body in rotation.

This concept helps us understand how objects behave when they rotate, including effects like centrifugal force experienced by rotating objects.

Cross Product

The cross product is an operation on two vectors in three-dimensional space. It results in a third vector that is perpendicular to both original vectors. This makes it useful for finding directions in physics and engineering problems.

To visualize the cross product, imagine two arrows representing vectors starting from the same point. The cross product creates another arrow perpendicular to both. Mathematically, the cross product \( \vec{A} \times \vec{B} \) is defined as a vector that satisfies:

• Magnitude: \( |\vec{A} \times \vec{B}| = |\vec{A}| \cdot |\vec{B}| \cdot \sin(\theta) \)
• Direction: determined by the right-hand rule

Here, \( \theta \) is the angle between \( \vec{A} \) and \( \vec{B} \).

In the context of the velocity field, the cross product helps express rotational motion. For a rotating body around a point "o", the field \( a_{\omega} \) is calculated as \( \vec{\omega} \times \vec{r_o} \).

Understanding the cross product is essential because it explains interactions between vector fields, such as forces in physics or determining the axis of rotation.

Commutator of Fields

The commutator of fields is a concept used to express how two fields or operations interact. For vector fields, the commutator provides insights into how one vector field transforms when acted upon by another.

Mathematically, the commutator of two fields \( a_{\alpha} \) and \( a_{\beta} \) is given by:

• \([a_{\alpha}, a_{\beta}] = a_{\alpha}(a_{\beta}) - a_{\beta}(a_{\alpha})\)

This expression captures the non-commutative nature of vector operations. Essentially, it shows the difference when one field acts on another, compared to the reverse.

When dealing with velocity fields from angular velocities, like \( a_{\alpha} = \vec{\alpha} \times \vec{r_o} \) and \( a_{\beta} = \vec{\beta} \times \vec{r_o} \), we compute the commutator to understand their interactions. The operations involve complex interactions among vectors.

This idea is key in advanced physics and engineering applications, where understanding how operations transform can impact the system's behavior or stability.