The Principle of Least Action and Lagrangian Mechanics

The principle of least action is a idea that helped transition physics away from Newtonian methods and toward a more general description of physical systems.

Advertisements

Lately I’ve been studying methods of calculating path integrals, and while I would very much like to write about that, I think I should start with a more foundational topic. The principle of least action is a idea that helped transition physics away from Newtonian methods and toward a more general description of physical systems. It has been so important that it is still used across many different areas of physics today.

Principle of least action

To start, we should discuss what the principle of least action actually is. In the 1600s, Pierre de Fermat discovered that light always takes a path that takes the least amount of time to traverse. Knowing the initial point, and the final point of light, you would know what path it took by finding that path that minimized the time taken by the light. This would become known as Fermat’s principle or the principle of least time.

FermatsPrinciple.png

The light ray moves slower when passing through a material. As per Fermat’s principle, the ray is deflected in such a way that minimizes the light takes to pass from beginning to end.

This was a novel idea, since older approaches of physics first considered the present state of a system and then apply laws to determine what further point it will transition to. In Fermat’s principle, you consider the endpoints of the system (beginning and end of the light path), and then you could derive the laws based on some kind of constraint (ie: minimizing the time that light travels).

Lagrangian Mechanics

As it turns out, light isn’t the only place you can apply this kind of approach. In Lagrangian mechanics, you could define the starting point of a system, and the end point of that system, then you’d try to minimize and maximize a particular quantity known as the action (S). The action is an abstract value that depends on the path the system takes; if find the path that minimizes or maximizes the action, that path corresponds to the real path that the system takes. This sounds pretty abstract but there is a well defined system for calculating the action for a path, and it comes from formula you can build for your system called the Lagrangian function or just the Lagrangian (L).

Action.png

 The Lagrangian is defined at each point in time. Integrating along the path of the system provides the action for that path.

So by deriving the Lagrangian function and applying the principle of least action, you can describe almost any physical system. This has been a very successful approach and was later adapted to a similar technique in what we call Hamiltonian mechanics.

There are some conceptual oddities of Lagrangian mechanics. Particularly, the idea of both the past and the future influencing the behaviour of the present can be a little discordant with our idea of causality. Regardless, it has been proven as an effective approach to understanding the behaviour of a large variety of scenarios in physics.

What does it mean?

The principle of least action is an extremely handy concept that lets us tackle some problems that would otherwise be extremely difficult in physics. A commonly used example is the problem of coupled oscillators. In this problem, a series of masses are connected by springs and strung between two walls. Deriving it using tradition newtonian methods can be extremely tedious for a large number of masses. On the flip-side using lagrangian mechanics can vastly simplify the problem. If you are interested in lagrangian methods for harmonic systems, this problem can illustrate can illustrate the power of lagrangian methods:

Coupled Harmonic Oscillators by Irina Yakimenko

https://people.ifm.liu.se/irina/teaching/sem6.pdf

Keep in mind that this approach can offer more than just mathematical power, it is conceptual change in how we look at interactions in the universe. Instead seeing the universe as of a series of sequential events, we can see a system of relationships across time and space that emerge from the optimization of a few constraints.

The principle of least action also provides interesting implications for our perspective of time. As mentioned earlier, since the lagrangian derives our equations of motion from past and future constraints, it violates our intuition about causality. Some have argued that it implies an eternalist universe where the past, present and future all exist simultaneously and are separated only by the dimension of time. This is largely philosophical and as such, is difficult to demonstrate with any certainty.

 

Here are a few resources for getting a handle on Lagrangian Mechanics and the principle of least action:

PDF: Introduction to Lagrangian and Hamiltonian Mechanics, Melanie Ganz

A short pdf that introduces the main concepts and equations of Lagrangian Mechanics

http://image.diku.dk/ganz/Lectures/Lagrange.pdf

BOOK: Mechanics: Volume 1, Landau and Lifshitz

The go-to book on classical mechanics. Uses Lagrangian methods from the beginning and covers many different methods applying them to mechanical situations.

https://www.amazon.ca/Mechanics-1-L-D-Landau/dp/0750628960

VIDEO: A Special Lecture: Principle of Least Action, Kenneth Young

A fantastic introduction to the material and great speaker.

https://www.youtube.com/watch?v=IhlSqwZBW1M