Uses for Lie derivative

[christianpoved]

Main Question or Discussion Point

Hello everybody, Im an undergrad physics student and I'm taking some "Geometry and Topology for physicist" course. We saw Lie Derivative some time ago and I still dont know how can I use it on physics, can anyone give me some examples? thanks

 

[D H]

As a starter, one that you are already very well familiar with is angular velocity.

 

[deiki]

You can use it to prove the Poincaré Lemma and compute the local potential of a closed differential smooth form. You can use it when you are computing the derivative of an integral on a manifold that depends on a the parameter you're conducting a derivative with, like the general case shown in this page : http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign.

 

[atyy]

The Lie derivative is the commutator of two flows, ie. whether the order of doing two different operations matters. The commutator is called the Lie bracket, and the traditional example is how you can parallel park using two operations.
http://www.math.cornell.edu/~goldberg/Talks/Flows-Olivetti.pdf
http://rigtriv.wordpress.com/2007/10/01/parallel-parking/

The Lie derivative is also used to define Killing vectors in general relativity. For a free falling test particle (ie. under no forces except gravity = spacetime curvature), the Killing vectors give conserved quantities along the particle's trajectory.
http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll5.html

 

[WannabeNewton]

In general relativity, a vector field $\xi$ in a space-time $(M,g)$ that preserves the metric tensor $g$ is exactly one which satisfies $\mathcal{L}_{\xi}g = 0$ where $\mathcal{L}_{\xi}$ is the Lie derivative along $\xi$; such a vector field is called a killing field.

As an example, take $M = \mathbb{R}^{4}$ and $g = \eta$ where $\eta$ is the Minkowski metric i.e. we are in Minkowski space-time. Then the vector fields solving $\mathcal{L}_{\xi}\eta = 0$ are given by $\xi = AX + T$. It can be shown that $T$ is the generators of translations in Minkowski space-time and that $A$ is the generator of Lorentz transformations (i.e. Lorentz boosts and spatial rotations).

Further more, in general relativity a freely falling particle satisfies the equations of motion $\nabla_u u = 0$ where $\nabla$ is the Levi-Civita connection associated with the metric tensor $g$ of the space-time, and $u$ is the 4-velocity of the particle. If $\xi$ is a killing field then $\nabla_u (u \cdot \xi) = \xi \cdot \nabla_u u + u \cdot \nabla_u \xi = 0$ so $u \cdot \xi$ is a conserved quantity along the freely falling particle's worldline. For example if $\xi$ generates time translations then $u \cdot \xi$ represents the conserved energy of the freely falling particle.

 

[christianpoved]

Wow, thanks everybody for the examples, everything seems useful now!

 

[WannabeNewton]

If you're interested, also look into the exterior derivative. There are a myriad of fundamental uses of the exterior derivative in physics. Generally a course in differential topology which touches on cohomology will introduce the theory of exterior calculus.

 

[christianpoved]

Oh yeah, I also have some questions about the exterior derivative and Maxwell equations but I'll ask this another time