## Relativistic Lagrangians

I’m a part of a cool group of folks interested in infusing computation into undergraduate physics curriculum. One of the projects is called “relativistic dynamics” and it really got me thinking. I thought I’d get my thoughts down here.

### Lagrangian

I’ve used a Lagrangian approach a ton in my work with students and my posts here. It’s a great way to model the dynamics of a system because you just have to parametrize the kinetic and potential energy of the system and you’re off. No vectors, no free body diagrams, just fun 🙂

Here’s the idea in a nutshell:

Hold a ball in your hand. In 2 seconds it needs to be back in your hand. What should you do with the ball during those two seconds to minimize the time integral of the kinetic energy minus the potential energy during the journey?

It’s a fun exercise to do with students. You’re asking them to minimize this integral over two seconds:

$\int_0^2 \text{KE}(\vec{r}, t)-\text{PE}(\vec{r},t)\,dt$

When I do this their first guess is to leave the ball in your hand. They like to define the gravitational potential energy there to be zero, and then know the kinetic energy is zero if it doesn’t move so they’ve found an easy way to get a total of zero for the integral. So I challenge them to find a path who’s answer would be negative! It’s a pretty fun exercise, especially if you actually calculate the integrals for their crazy ideas.

The point is that the winner is to throw the ball up so that it’s trajectory, responding simply to gravity, takes 2 seconds (ie throw it up 1.225 meters). The kinetic energy is positive during the whole journey (except for an instant at the top, of course) but the potential energy is positive during the whole journey too.

Calculus of variations teaches us that if you want to minimize an integral like this:

$\int_\text{start}^\text{finish}f(x, \dot{x}, t)\,dt$

(where $\dot{x}$ is shorthand for the x-velocity) you really just need to integrate this differential equation over the same time integral:

$\frac{\partial f}{\partial x}-\frac{d}{dt}\frac{\partial f}{\partial \dot{x}}=0$

What’s cool is that if the function is KE-PE the equation above becomes Newton’s second law! That’s why this works. You use scalar energy expressions and you get the force equation for every component of motion! Now there are some other cool things like not needing to worry about constraint forces but I won’t worry about that in this post.

### Relativity

Ok, so what happens when you consider relativistic speeds (ie close to the speed of light)? Well, the first thing I did (which, spoiler, didn’t work) was to wrack my brain for an expression for the kinetic energy and plug away. When teaching relativity you get to a point when you’re making the argument with your students that KE isn’t just $1/2 m v^2$ anymore but is really $mc^2(\gamma-1)$ where gamma is given by:

$\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

If you take the limit of that expression for small v’s you get the usual expected result, and that’s certainly what we do right away with our students to make them feel better.

Ok, so I plugged it in and got a relativistic version of Newton’s 2nd law:

Note how the second term on the left side looks a little like “ma” while the right hand side is just the force from a conservative potential energy (U). The extra term on the left hand side is the weird stuff.

Without really thinking about whether that was the right equation, I modeled a constant force system and got this for the velocity

(I set the speed of light to 1). You can see that the speed is forced to obey the cosmic speed limit.

But here’s the problem. The equation above is wrong. That is not the correct relativistic Newton’s 2nd law equation.

So what happened? I plugged in the correct relativistic kinetic energy and the Lagrangian trick (minimizing KE-PE) gave a trajectory that doesn’t match what actually happens! So something’s wrong. Here’s a few possibilities (one is right, see if you can guess before reading the next paragraph):

• I’m using the wrong expression for kinetic energy
• The Lagrangian trick has some non-relativistic bias in it
• I’m minimizing the wrong function

It turns out it’s the last one. It took me a while of digging around, but this wikipedia article set me straight. The gist of what’s talked about there is this:

• Andy’s hard work above just doesn’t work
• But we know the right relativistic expression for momentum ($\gamma m v$ which, interestingly enough is a crazy thing that’s conserved in all frames of reference during collisions so we tell our students that since it’s conserved we should call it momentum).
• Let’s differentiate that momentum to get what the force should be and then search for a functional (that’s what f above is) that works out in the calculus of variations

Yeah, weird, I know. It’s like “hey, I know what the answer in the back of the book is so I’m going to futz with my early equations until they give me the right answer. So what is the right functional to use? This:

$-\frac{m c^2}{\gamma}$

Yep, it’s negative. Yep, it’s not an expression you’ve ever seen before if you’ve studied special relativity. But, guess what, it works! When you plug it in and do the calculus of variations trick you get the right dynamics. Surprise, surprise, given that it was built to do just that.

Here’s the same graph as above but not comparing that prediction with the right dynamics (in red):

It also asymptotes to the cosmic speed limit, just at a different rate.

### So what’s being minimized?

That’s the question I was really wondering about. Luckily google came to the rescue with this great wikibook article that it found for me. It points out that the kinetic energy portion of the functional you use to make the relativistic dynamics work is really just proportional to the invariant space-time interval:

$ds=\sqrt{c^2dt^2-dx^2}$

This is an expression for the “distance” between two distinct events in space-time that is the same for all inertial observers. It’s really cool given all the weird time dilation and length contraction that can go on in the various inertial frames.

So basically the trajectories that actual things follow is designed to make the space-time “jumps” add up to the smallest number. That’s super cool

Your thoughts? Here are a few starters for you:

• I like how you talk about teaching the Lagrangian. What I would add is . . .
• I hate how you talk about teaching the Lagrangian. What I would rip, burn, and bury is . . .
• Why would you even think that the Lagrangian formalism, which clearly treats space and time differently, could so easily be co-opted into a relativistic treatment?
• Why is one of your equations an gif instead of WordPress’ built in $\LaTeX$?
• I can tell you used Mathematica’s TeXForm command. You are really lazy.
• You did a simple constant force. What would something connected to a spring do?
• Non-relativistic (red) and relativistic (blue) mass on a spring

• What do you mean when you say that “it’s conserved so let’s just call it momentum”?
• Why didn’t you put that last question mark inside the quotation marks?
• What planet were you on when you figured out the 1.225 meter throw?