# Isn't working with the relativistic Lagrangian AWFUL?

• AxiomOfChoice
In summary, the relativistic electron in an external magnetic field is attempting to solve the equations of motion, but is faced with a daunting task. The Lagrangian, \mathcal L, is -mc^2 / \gamma - e \vec v \cdot \vec A, in cylindrical coordinates. However, the equations of motion are coupled and require the use of a complex e\Phi term. The relativistic electron will only be able to solve the equations of motion in one reference frame, and may need the help of cyclic coordinates.
AxiomOfChoice
I'm trying to solve a problem for a relativistic electron in an external magnetic field with vector potential $$\vec A$$ using the Lagrangian

$$\mathcal L = -mc^2 / \gamma - e \vec v \cdot \vec A$$

in cylindrical coordinates. But isn't this DREADFULLY TERRIBLE, since when I try to compute $$\dfrac{d}{dt} \dfrac{\partial \mathcal L}{\partial \dot q_i}$$ I'm going to have to take the time derivative of $$\gamma$$, which takes the form

$$\gamma = \left(1 - \frac{1}{c^2} (\dot r ^2 + r^2 \dot \phi^2 + \dot z^2) \right)^{-1/2}$$

Well, look at it this way. The experience will be good for you. I worked in general relativity before there were such things as algebraic manipulation programs. I have a notebook filled with equations a couple of pages long. It's interesting to find out how your brain can come to see patterns in the symbols that afford simplifications.

Now that I've been gentle, the teacher in me says, Don't Be Lazy!

AEM said:
Well, look at it this way. The experience will be good for you. I worked in general relativity before there were such things as algebraic manipulation programs. I have a notebook filled with equations a couple of pages long. It's interesting to find out how your brain can come to see patterns in the symbols that afford simplifications.

Now that I've been gentle, the teacher in me says, Don't Be Lazy!

Hehe, thanks for the words of encouragement. Having run $$\mathcal L = -mc^2 / \gamma - e \vec v \cdot \vec A$$ through the E-L equations, I've come up with a fairly simple EOM:

$$m \gamma^3 \frac{d \vec v}{dt} = e \frac{\partial \vec A}{\partial t} - e \nabla (\vec v \cdot \vec A)$$

Does that look right?

I should note that your Lagrangian as written is neither Lorentz-invariant, nor gauge-invariant. You need an $$e\Phi$$ term, though I forget whether it should have a minus sign or not.

So, your Lagrangian will work, but only in one reference frame.

Ben Niehoff said:
I should note that your Lagrangian as written is neither Lorentz-invariant, nor gauge-invariant. You need an $$e\Phi$$ term, though I forget whether it should have a minus sign or not.

So, your Lagrangian will work, but only in one reference frame.

I also don't see how I'm going to solve the eventual equations of motion. The $$\gamma$$ term contains first-order time derivatives of all the coordinates, so the equations are going to be mercilessly coupled...aren't they?

AxiomOfChoice said:
I also don't see how I'm going to solve the eventual equations of motion. The $$\gamma$$ term contains first-order time derivatives of all the coordinates, so the equations are going to be mercilessly coupled...aren't they?

More than likely. I might ask, how is your vector potential oriented?

It's been ages since I have done this, but my gut tells me Ben is on the right track. Write a Lagrangian that is Lorentz invariant, and at the end, set the time-component of the 4-potential to zero.

I will echo Ben and Vanadium 50. You should write your Lagrangian in 4-vector notation and then set the electric potential term equal to zero. Now, to simplify possibly ugly looking equations of motion, don't forget about the role of cyclic coordinates. Refer to Goldstein's Classical Mechanics, 3rd edition, Chapter 7 if you need to refresh your memory.

AEM said:
More than likely. I might ask, how is your vector potential oriented?

All I know about $$\vec A$$ is that it needs to be such that the azimuthal component of $$\vec B$$ is zero. So I was thinking $$\vec A = A_\phi \hat \phi$$.

Ben Niehoff said:
I should note that your Lagrangian as written is neither Lorentz-invariant, nor gauge-invariant. You need an $$e\Phi$$ term, though I forget whether it should have a minus sign or not.

So, your Lagrangian will work, but only in one reference frame.

In this case, I know $$\Phi = 0$$.

AxiomOfChoice said:
All I know about $$\vec A$$ is that it needs to be such that the azimuthal component of $$\vec B$$ is zero. So I was thinking $$\vec A = A_\phi \hat \phi$$.

I asked my question to make you think about simplifications in the dot product of v and A.

You also know that the azimuthal component of B is zero. That also tells you something about the motion. What you need to do is pick your equations apart and ask about each term: "What does this mean?" "What does this imply?"

Also, your equations of motion are expressed in terms of lab time, t. Why don't you try expressing them in terms of proper time, $\tau$? You may find something useful.

I want to thank you all for your help. I'm sacrificing my Friday night to try to grind through this problem, armed with your hints and suggestions, though I'm sure I'll be back for more help later. Thanks again!

I look forward to following this...

Hmmm...can someone tell me what, in general, you'd get from

$$\frac{\partial}{\partial \vec v} (\vec v \cdot \vec A})?$$

I'm just not sure how to work with the dot product here. Does it obey the product rule, such that

$$\frac{\partial}{\partial \vec v} (\vec v \cdot \vec A}) = \vec A + \vec v \cdot \frac{\partial \vec A}{\partial \vec v} = \vec A?$$

AxiomOfChoice said:
Hmmm...can someone tell me what, in general, you'd get from

$$\frac{\partial}{\partial \vec v} (\vec v \cdot \vec A})?$$

I'm just not sure how to work with the dot product here. Does it obey the product rule, such that

$$\frac{\partial}{\partial \vec v} (\vec v \cdot \vec A}) = \vec A + \vec v \cdot \frac{\partial \vec A}{\partial \vec v} = \vec A?$$

I've managed to convince myself (quick, trivial proof...sorry I wasted everyone's time) that my last line is correct. Nvm.

hi everyone
I'm reading 'introducing einstein's relativity' of d'inverno
I do not know how to work with the equivalent lagrangian
can anyone help me to prove 11.42 and 11.43?

## 1. What is the relativistic Lagrangian?

The relativistic Lagrangian is a mathematical function used in physics to describe the dynamics of a system in terms of its position, velocity, and time. It takes into account the effects of special relativity, which states that the laws of physics are the same for all observers in uniform motion.

## 2. Why is working with the relativistic Lagrangian considered awful?

Working with the relativistic Lagrangian can be challenging because it involves complex mathematical calculations and requires a deep understanding of both classical mechanics and special relativity. It also requires a strong grasp of mathematical concepts such as calculus and differential equations.

## 3. What are some applications of the relativistic Lagrangian?

The relativistic Lagrangian is used in a wide range of fields, including particle physics, astrophysics, and cosmology. It is used to describe the motion of particles in high-energy accelerators, the dynamics of celestial bodies in space, and the behavior of matter in extreme environments such as black holes.

## 4. Can the relativistic Lagrangian be applied to all systems?

The relativistic Lagrangian is a universal function that can be applied to any system, as long as it obeys the principles of special relativity. However, it is most commonly used for systems with high velocities or in the presence of strong gravitational fields, where the effects of relativity are significant.

## 5. How does the relativistic Lagrangian differ from the classical Lagrangian?

The relativistic Lagrangian takes into account the effects of special relativity, such as time dilation and length contraction, while the classical Lagrangian only considers the laws of classical mechanics. This makes the relativistic Lagrangian a more accurate and comprehensive tool for describing the dynamics of systems at high speeds or in extreme environments.

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