Can a Lie Derivative be Taken in the Direction of a Scalar Function?

In summary, Thirring's Classical Mathematical Physics defines and uses the lie derivative on a vector field, but later on, he also uses it on the Hamiltonian, which is a scalar function. However, this does not mean taking the lie derivative in the direction of the Hamiltonian itself, but rather the vector field induced by the Hamiltonian. It is not possible to have a lie derivative in the direction of a scalar function, as it can only be taken with respect to a vector.
  • #1
redrzewski
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I'm working thru Thirring's Classical Mathematical Physics. The lie derivative is defined and used on a vector field. I.e. L(x)f where x is a vector field. () = subscript

However, later on, he uses the lie derivative of the hamiltonian, which is a scalar function. I.e. L(H)f () = subscript

I'm assuming that this means the vector field induced by the hamiltonian, and not the lie derivative in the direction of the hamiltonian itself. However, usually Thirring is careful to call out this distinction (i.e. notating X(H) as the vector field induced by the hamiltonian).

My question: is it possible to have a lie derivative in the direction of a scalar function?

thanks
 
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  • #2
No, you can only take Lie derivatives with respect to a vector.
 

What is a Lie derivative?

A Lie derivative is a mathematical operation used in differential geometry and differential topology to measure how a geometric object changes along a given direction. It is a generalization of the concept of a derivative in calculus, but can be applied to more abstract spaces such as manifolds.

How is a Lie derivative calculated?

The Lie derivative is calculated using the Lie bracket, which is a binary operation between two vector fields. It represents the difference between the two vector fields when they are transported along each other. The Lie derivative is then defined as the directional derivative along the flow of a vector field generated by the Lie bracket.

What is the significance of Lie derivatives?

Lie derivatives are important in differential geometry because they allow us to study the geometry of a space by considering how it changes along different directions. They also have applications in physics, particularly in the study of symmetry and conservation laws.

What is the difference between a Lie derivative and a partial derivative?

A partial derivative measures how a function changes with respect to one variable while holding all other variables constant. In contrast, a Lie derivative measures how a geometric object changes along a given direction in a more general space. While a partial derivative can be thought of as a special case of a Lie derivative, the two are not equivalent.

Are there any limitations to using Lie derivatives?

One limitation of Lie derivatives is that they can only be defined for certain types of geometric objects, such as vector fields and tensors. Additionally, they can only be calculated on certain types of spaces, such as smooth manifolds. Furthermore, the calculation of Lie derivatives can become quite complicated and computationally intensive for higher dimensional spaces.

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