How to derive a symmetric tensor?

In summary, the conversation involves discussing a symmetrical tensor, ##Q_{ij}##, which is derived using the formula ##Q_{ij}= \frac{m}{2} \dot x_i \dot x_j + \frac{k}{2} x_i x_j##. The purpose of this tensor and its physical interpretation, as well as the need for adding indices to create a symmetrical form, are also discussed.
  • #1
Cathr
67
3
Let ##Q_ik## be a symetric tensor, so that ##Q_ik= \frac{m}{2} \dot x_i \dot x_j + \frac{k}{2} x_i x_j## (here k is also a sub, couldn't do it better with LaTeX).
How do we derive such a tensor, with respect to time? And what could such a tensor mean in a physical sense? It really looks like the tensor for the total energy, except that I don't understand the need for adding indices to create a symmetrical form.
 
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  • #2
I apologize, the formula is actually ##Q_{ij}= \frac{m}{2} \dot x_i \dot x_j + \frac{k}{2} x_i x_j## .
 

1. What is a symmetric tensor?

A symmetric tensor is a mathematical object that represents a linear transformation between vector spaces, where the order of the input vectors does not affect the result. In other words, it is a tensor that remains unchanged when its indices are swapped.

2. How do you derive a symmetric tensor?

To derive a symmetric tensor, you can start by defining a tensor with a specific number of indices and then apply the symmetry property to reduce the number of independent components. This can be done by setting up a system of equations and solving for the unknown components.

3. What is the importance of symmetric tensors in science?

Symmetric tensors have many applications in science, especially in physics and engineering. They are used to describe physical quantities such as stress, strain, and elasticity. They also play a crucial role in the study of fluid mechanics, electromagnetism, and general relativity.

4. Can you give an example of a symmetric tensor?

One example of a symmetric tensor is the stress tensor, which describes the distribution of forces within a solid material. It has six independent components, and its symmetry property ensures that the order of the input vectors does not affect the resulting stress values.

5. How does a symmetric tensor differ from an asymmetric tensor?

A symmetric tensor has the property of symmetry, meaning that it remains unchanged when its indices are swapped. On the other hand, an asymmetric tensor does not have this property and can change when its indices are swapped. Asymmetric tensors are commonly used to describe physical quantities that have a direction, such as velocity or angular momentum.

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