Manipulation of Cartesian Tensors

In summary, the conversation discusses how to prove the acceleration of a particle rotating around a point using Cartesian tensor methods. It involves breaking down the velocity and using the identity \vec u \cdot \nabla \vec u = \frac{1}{2} \nabla (\vec u \cdot \vec u) + (\nabla \times \vec u) \times \vec u. The final result is given by a = -\frac{1}{2} \nabla [(\Omega \times r)^2].
  • #1
Hoplite
51
0
I have a question relating to a particle rotating around a point with velocity [tex]u = \Omega \times r[/tex], where [tex]\Omega[/tex] is the angular velocity and r is the position relative to the pivot point.

I need to prove that the acceleration is given by,

[tex]a = -\frac{1}{2} \nabla [(\Omega \times r)^2] [/tex]

I figured it should follow from the fact that,

[tex]a = \frac{du}{dt} = \frac{\partial u}{\partial t} + u \cdot \nabla u = u \cdot \nabla u [/tex]

But I can't work out where to go from there. We are supposed to use Cartesian tensor methods to work it out.

Could anyone help me out?
 
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  • #2
[tex] \vec{a}=\frac{d\vec{v}}{dt}=\frac{d}{dt}\left(\vec{\Omega}\times \vec{r}\right) =\frac{d\vec{\Omega}}{dt}\times\vec{r} +\vec{\Omega}\times\frac{d\vec{r}}{dt}[/tex]

Daniel.
 
  • #3
[tex] \frac{d\vec{\Omega}}{dt}=\left(\vec{\Omega}\times \vec{r}\cdot\nabla\right)\vec{\Omega} [/tex]

[tex] \frac{d\vec{\Omega}}{dt}\times \vec{r}=\left[\left(\vec{\Omega}\times \vec{r}\cdot\nabla\right)\vec{\Omega}\right]\times \vec{r}= \frac{1}{2}\nabla\left[\left(\vec{\Omega}\times\vec{r}\right)\cdot\left(\vec{\Omega}\times\vec{r}\right)\right] -\vec{\Omega}\times\left(\vec{\Omega}\times\vec{r}\cdot\nabla\right)\vec{r}[/tex]

In the end i get the plus sign.

Daniel.
 
Last edited:
  • #4
Thanks, Daniel.

Yeah, I suspect the negative sign was a typo on the question sheet, especially considering the identity,

[tex]\vec u \cdot \nabla \vec u = \frac{1}{2} \nabla (\vec u \cdot \vec u) + (\nabla \times \vec u) \times \vec u [/tex]
 

Related to Manipulation of Cartesian Tensors

1. What is the purpose of manipulating Cartesian tensors?

The purpose of manipulating Cartesian tensors is to simplify complex mathematical equations and represent them in a way that is easier to work with. This can also help to reveal patterns or relationships between different physical quantities.

2. How are Cartesian tensors manipulated?

Cartesian tensors can be manipulated using tensor algebra, which involves operations such as addition, multiplication, and contraction. It also involves transformation operations such as rotation and inversion.

3. What are the applications of manipulating Cartesian tensors?

Manipulating Cartesian tensors is essential in various fields of science and engineering, such as mechanics, electromagnetics, and fluid dynamics. It is also used in computer graphics and image processing to manipulate and transform images.

4. What are the key properties of Cartesian tensors?

Cartesian tensors have several key properties, including symmetry, skew-symmetry, and orthogonality. These properties are important in determining how the tensor will behave under different operations and transformations.

5. Are there any limitations to manipulating Cartesian tensors?

One limitation of manipulating Cartesian tensors is that it can become complex and challenging to work with in higher dimensions. It also requires a good understanding of tensor algebra and vector calculus, which may be a barrier for some individuals.

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