# Show Lorentz invariance for Euler-Lagrange's equations- how?

• Bapelsin
In summary, the Euler-Lagrange equations are covariant with respect to the Lorentz group, which means that they remain valid under any transformation of the space-time coordinates. This is verified by showing that the equations are invariant under a change in the energy of a particle. However, I don't know how to show this mathematically.

#### Bapelsin

Hello,

I need help showing that the Euler-Lagrange equations are Lorentz invariant (if Einstein's extended energy concept is used). Is there an easy way to show this? Any help would be very much appreciated.

This and a lot more of that you can find in a book by Greiner called relativistic quantum mechanics. Good luck!

Bapelsin said:
I need help showing that the Euler-Lagrange equations are Lorentz invariant

Welcome to PF.
The E-L equations are statements derived covariantly from an invariant action integral. Therefore they are covariant with respect to Lorentz group.

(if Einstein's extended energy concept is used).

What is this, and what relation does it have with the Euler-Lagrange equations?

sam

Gigi: Thanks for your tip, but I don't think I can obtain that book soon enough. But maybe later. :-)

samalkhaiat: Thank you! First of all I want to tell you that I am more or less a layman when it comes to this (I haven't studied natural sciences for almost three years, and I'm I only half-way to obtain a MSc degree in engineering physics) so I'm not sure I'm following you completely. So covariances and tensors and even less group theory etc. isn't my strong side. This was originally supposed to serve as a short background for a philosophical paper in science theory.

My physics professor tells me I'm right concerning this, but I don't know how to show this. As I don't know how to insert formulas here, I'm attaching the relevant ones as .GIF images. By extended energy concept I simply mean the famous formula E = mc².

#### Attachments

• Euler-Lagrange.GIF
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• Relativistic-Lagrangian.GIF
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• Beta.GIF
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I forgot to mention that my main problem is that I don't know how a Lorentz' transformation in generalized coordinates look like. I've only dealt with such in Cartesian coordinates (Google etc. doesn't help me out much at all with this).

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## 1. What is Lorentz invariance?

Lorentz invariance is a principle in physics that states that the fundamental laws of nature must remain unchanged under a Lorentz transformation. This transformation involves changing the coordinates of space and time in a specific way, and it is a fundamental principle in the theory of special relativity.

## 2. How does Lorentz invariance relate to Euler-Lagrange's equations?

Euler-Lagrange's equations are used to describe the motion of a particle in a physical system. These equations are derived from the principle of least action, which is a fundamental principle in classical mechanics. Lorentz invariance is important in this context because it ensures that the equations are valid in all frames of reference, including those moving at relativistic speeds.

## 3. Can you show the Lorentz invariance of Euler-Lagrange's equations?

Yes, the Lorentz invariance of Euler-Lagrange's equations can be shown mathematically. It involves using the Lorentz transformation to change the coordinates in the equations and then showing that the resulting equations are equivalent to the original ones. This is a complex mathematical process that is beyond the scope of this brief explanation.

## 4. Why is it important to show the Lorentz invariance of Euler-Lagrange's equations?

Showing the Lorentz invariance of Euler-Lagrange's equations is important because it provides a strong mathematical foundation for the equations. It also demonstrates that the equations are valid in all frames of reference, which is crucial for understanding the behavior of particles at high speeds and in different reference frames.

## 5. Are there any real-world applications of Lorentz invariance for Euler-Lagrange's equations?

Yes, there are many real-world applications of Lorentz invariance for Euler-Lagrange's equations. For example, it is used in particle physics to study the behavior of subatomic particles at high speeds. It is also important in the field of astrophysics, where relativistic effects must be taken into account when studying the motion of objects in space.