More physics: Action and Lagrange

My last post on physics, about forces, momentum and energy, left a few open questions that I didn't want to dive into, lest I never finish the post. One that seemed obvious to me was "if you get momentum and energy by integrating force over time and distance, what happens if you integrate over both?". I carefully avoided researching this to avoid going down a rabbit hole.

A more pressing gap was Lagrangians, and the "principle of least action". Lots of university-level discussions of physics use Lagrangians, and if I wanted to push any further I thought I really needed to understand how they work.

The TL;DR of the principle of least action is that there is a function of the paths of objects in the system over time, called the "action", and the path actually taken by the system is the minimum of this function. The paths taken, somehow know the route to minimise this quantity. How mysterious!

The action itself is an integral over time of a function of position and time. That function is the fabled "Lagrangian" (or rather, a Lagrangian is a generalisation of this function). The Lagrangian for the basic case is a function of the system's energy. That is, the action is an integral of energy over time, but it can also be calculated as an integral of momentum over the path: It is precisely that integral of force over time and space that I was wondering about earlier!

So, that's the principle of least action, but why is it equivalent to Newton's equations of motion? Awesomely, there's a Feynman Lecture on Physics on this specific topic - I never really got on that well with these lectures (his view of physics somehow doesn't resonate with me, but I found Penrose's The Road to Reality very intuitive), but I found this chapter very accessible.

Why does the Principle of Least Action work? Well, the mysterious "whole path is minimised" property is actually just a way of saying that every point along the path is locally optimised. The solution is not necessarily a global minimum, or even just a minimum, it's a local extremum along the route. For it to be a local extremum, small pertubations must not change the integral. In some sense, the local derivative of the action with respect to tweaking the path must be zero. And the function that describes how the action changes with path tweaks is an equation that is zero when Newton's laws of motion are satisfied! (Roughly.)

Anyway, go read the original for details. It includes an introduction to the calculus of variations, which is how you optimise along paths, and is a very neat tool. He provides a muuuuch more rigorous explanation of it all than I do (though still physicist-level rigour :p ).

The Principle of Least Action seems a weird way to understand Newtonian Physics to me. As a simulation-minded kind of person, a Newtonian universe would clearly follow his laws of motion step-by-step, rather than be the solution to an integral minimisation problem. Practically, using a Lagrangian has some advantages. It allows you to constrain the motion of a system conveniently, with the aid of Lagrange multipliers. So, it's just a convenient calculation tool, right?

Yet this "solution of path integrals" approach seems meaningful in other areas of physics. In optics, light takes the minimum time route between points. This minimum time route, as a local minimum, is a stationary point in the phase of the light at the destination - in other words, it's a point of constructive interference as a wave. Quantum mechanics takes this further, with the probability of a particle being at a point being based on an integral over all the possible paths.

What I felt was a computational convenience for the "real" equations of motion might in some sense be more physically meaningful than those "real" equations!

Around this point, I had another realisation, and it's about there I got stuck.

Going back to Newtonian mechanics, and extending it in the direction of special relativity, the action is an integral of force over time and space... but in special relativity there is only space-time! In special relativity, momentum and energy are the space and time components of momenergy. Perhaps action in SR can be viewed as something else? Some kind of thing where 4-forces get integrated to momenergy, and momenergy gets integrated to whatever the SR version of action is?

I tried to work out the maths behind how this would work in SR, and... failed. I tried a few approaches to generalise the action in a way that becomes Newtonian at low speeds, but couldn't find the right way to break into it, and then got stuck.

Fortunately, the Wikipedia Relativistic Lagrangian mechanics page exists, and does the work for me. Fascinatingly, it's the path that minimises the proper time of the object. I was hoping it'd come out as something like this, and apparently it does!

I can also see this as a step towards general relativity, which I must admit I don't have a mathematical understanding of. This "shortest path in spacetime, as determined by proper time" feels very metric-like, and you can see how you'd end up dropping the forces and just end up with this metric that corresponds to curved space-time. Yet more fun!

And there's some similarity between how I find the principle of least action a weird way to express e.g. Newtonian mechanics, and how I see people viewing curved space-time as a special thing for GR. Reading Penrose's Road to Reality (see above), it mentions that Newtonian gravity can also be modelled through curved space-time. I believe optics can also be handled be distorting space - assume light travels at a constant speed, and insert more space into the regions where light is travelling more slowly (lens material etc.), although handling the discontinuities is an exercise for the reader.

So this seems to be leading me to try to think about GR, but that's a whole different post!

Posted 2024-04-06.