Clarification on Lie Derivatives

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In summary, the standard definition of the lie derivative of X along Y is given by equation (*) which involves taking the limit of the difference between a pushforward of X along Y and X evaluated at a point on the flow of Y. This is equivalent to the expressions (1) and (2) for the Lie Derivative, but only when the connection is torsion-free. The torsion is defined as the difference between (2) and (1), and (2) is equivalent to (1) if and only if the torsion vanishes.
  • #1
center o bass
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The standard definition of the lie derivative of X along Y is just

$$(*) \mathcal{L}_YX = \lim_{t\to 0} \frac{X_{\phi(t)} - \phi_{t*}X}{t}$$

where ##\phi_t## is the flow generated by Y. I.e. the limit of the difference between a pushforward of X along Y and X evaluated at a point ##\phi(t)## on the flow of Y. Other than this definition I have seen the expressions

$$(1) \mathcal{L}_Y X = [Y,X]$$

and

$$(2) \mathcal{L}_YX = \nabla_Y X - \nabla_X Y$$

for the Lie Derivative, but I have not seen it stated clearly when (1) and (2) holds. Surely (1) and (2) is equivalent when the connection ##\nabla## is torsion free. However is it generally true that (*) is equivalent with (1) or (2)? What conditions on ##\nabla## must be satisfied in order to identify (*) with (1) (or (2))?
 
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  • #2
center o bass said:
The standard definition of the lie derivative of X along Y is just

$$(*) \mathcal{L}_YX = \lim_{t\to 0} \frac{X_{\phi(t)} - \phi_{t*}X}{t}$$

where ##\phi_t## is the flow generated by Y. I.e. the limit of the difference between a pushforward of X along Y and X evaluated at a point ##\phi(t)## on the flow of Y. Other than this definition I have seen the expressions

$$(1) \mathcal{L}_Y X = [Y,X]$$

and

$$(2) \mathcal{L}_YX = \nabla_Y X - \nabla_X Y$$

for the Lie Derivative, but I have not seen it stated clearly when (1) and (2) holds. Surely (1) and (2) is equivalent when the connection ##\nabla## is torsion free. However is it generally true that (*) is equivalent with (1) or (2)? What conditions on ##\nabla## must be satisfied in order to identify (*) with (1) (or (2))?

(1) and (*) are equivalent, you should be able to show this.

(2) is equivalent to (1) if and only if the torsion vanishes. In fact, the torsion is defined as the difference between (2) and (1):

[tex]T(X,Y) \equiv \nabla_X Y - \nabla_Y X - [X,Y][/tex]
 

1. What is a Lie derivative?

A Lie derivative is a mathematical concept used in differential geometry and differential equations to measure the change of a geometric object along the flow of a vector field. It is essentially a way to define how a geometric object changes as it moves along a given path.

2. How is a Lie derivative different from a regular derivative?

A Lie derivative differs from a regular derivative in that it measures the change of a geometric object along the flow of a vector field, rather than along a specific coordinate direction. It takes into account the entire vector field, rather than just a single direction.

3. What is the purpose of using Lie derivatives?

Lie derivatives are useful for understanding the dynamics and geometry of a system. They allow us to analyze how a geometric object changes over time and how it is affected by the underlying vector field. This is particularly helpful in fields such as physics and engineering.

4. Can Lie derivatives be applied to any type of geometric object?

Yes, Lie derivatives can be applied to any type of geometric object, including curves, surfaces, and higher-dimensional objects. They are a fundamental concept in differential geometry and have applications in a wide range of fields.

5. Are there any limitations or drawbacks to using Lie derivatives?

One limitation of using Lie derivatives is that they can only be applied to smooth and differentiable objects. In cases where the object is not smooth, the concept of a Lie derivative may not be well-defined. Additionally, the calculation of Lie derivatives can be complex and time-consuming for more complicated systems.

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