Does a covariant version of Euler-Lagrange exist?

In summary, the conversation discusses the Euler-Lagrange equation for lagrangians with dependences on first and second derivatives. It is mentioned that in curved space, the equation is the same but with the lagrangian density multiplied by {-g}½. It is also noted that the covariant version of the equation can be found by replacing partial derivatives with covariant ones and using the formula for contraction of indices in the Christoffel symbols. There is a question about whether the last term in the equation (partial derivative of the lagrangian with respect to a double covariant derivative) can be expressed as a partial derivative of the lagrangian with respect to a double partial derivative of the field, possibly with other terms involving Christoff
  • #1
JuanC97
48
0
Hello everyone.

I've seen the usual Euler-Lagrange equation for lagrangians that depend on a vector field and its first derivatives. In curved space the equation looks the same, you just replace the lagrangian density for {-g}½ times the lagrangian density. I noticed that you can replace partial derivatives for covariant ones and easily arrive to the same result (just take into account the formula for contraction of indices in the christoffel symbols and you get this result in no more than 2 or 3 lines).

The last comment ensures there's a covariant version of Euler-Lagrange for lagrangians with dependences on first order derivatives but I may ask... what happen with dependences on second derivatives in the lagrangian? - I tried to arrive at a similar result (just changing partial derivatives for covariants) but I didn't find it and I need it for my thesis.

Guys, to be concise: Have you seen the covariant version of this equation? How can I find it?
(It is supposed to look like this https://pasteboard.co/H8AyvWS.png)
 
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  • #2
Is it possible to express the last term (partial derivative of the lagrangian wrt a double covariant derivative) as:
(partial derivative of the lagrangian wrt a double partial derivative of the field) + (other terms - maybe involving Christoffel symbols)?

Some thing like chain rule or similar?
 
  • #3
I've arrived to this conclusion:
https://pasteboard.co/HaR8xFf.png

but I was expecting just the underlined terms to be equal to zero, not with the Christoffel term.
I don't see how would the last term be identically zero (if possible) but what do you think guys?
 

1. What is the Euler-Lagrange equation?

The Euler-Lagrange equation is a fundamental equation in the field of classical mechanics. It is used to determine the equations of motion for a system by minimizing the action integral, which is the integral of the Lagrangian over time. It is a central tool in the study of variational calculus and has applications in many areas of physics, including optics, electromagnetism, and quantum mechanics.

2. What is the difference between the covariant and contravariant versions of the Euler-Lagrange equation?

The covariant version of the Euler-Lagrange equation takes into account the effects of coordinate transformations and is valid in any coordinate system. On the other hand, the contravariant version is only valid in a specific coordinate system and does not account for coordinate transformations. In other words, the covariant version is more general and applicable in a wider range of situations.

3. Why is a covariant version of the Euler-Lagrange equation necessary?

A covariant version of the Euler-Lagrange equation is necessary because it allows us to describe physical systems in any coordinate system, rather than being limited to a specific one. This is important because different coordinate systems may be more convenient for different applications, and having a covariant version ensures that the equations of motion are consistent across all coordinate systems.

4. Is the covariant version of the Euler-Lagrange equation derived differently from the contravariant version?

Yes, the covariant version of the Euler-Lagrange equation is derived using the principle of least action and the concept of covariant derivatives in differential geometry. The contravariant version, on the other hand, is derived using the principle of virtual work and the concept of contravariant derivatives. Both versions are valid and can be used depending on the specific situation.

5. What are some examples of physical systems that can be described using the covariant version of the Euler-Lagrange equation?

The covariant version of the Euler-Lagrange equation can be applied to a wide range of physical systems, including particles moving in a gravitational field, electromagnetic fields, and relativistic systems. It is also used in quantum field theory and general relativity to describe the behavior of particles in curved spacetime. Essentially, any system that can be described using the principle of least action can be analyzed using the covariant version of the Euler-Lagrange equation.

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