Original playlist name: "PHYSICS 68 ADVANCED MECHANICS: LAGRANGIAN MECHANICS"

Author: Michel van Biezen.

High school classical mechanics material, no mention of the key continuous symmetry part.

But does have a few classic pendulum/pulley/spring worked out examples that would be really wise to get under your belt first.

As mentioned on the Wikipedia page en.wikipedia.org/w/index.php?title=Stationary_Action_Principle&oldid=1020413171, "principle of least action" is not accurate since it could not necessarily be a minima, we could just be in a saddle-point.

A function that takes input function and outputs a real number.

These are the final equations that you derive from the Lagrangian via the Euler-Lagrange equation which specify how the system evolves with time.

The function that fully describes a physical system in Lagrangian mechanics.

This experiment seems to be really hard to do, and so there aren't many super clear demonstration videos with full experimental setup description out there unfortunately.

Wikipedia has a good summary at: en.wikipedia.org/wiki/Double-slit_experiment#Overview

For single-photon non-double-slit experiments see: single photon production and detection experiments. Those are basically a pre-requisite to this.

photon experiments:

electron experiments: single electron double slit experiment.

Non-elementary particle:

Basically: S.

When we particles particles, the action is obtained by integrating the Lagrangian over time:

In the case of field however, we can expand the Lagrangian out further, to also integrate over the space coordinates and their derivatives.

Since we are now working with something that gets integrated over space to obtain the total action, much like density would be integrated over space to obtain a total mass, the name "Lagrangian density" is fitting.

E.g. for a 2-dimensional field $f(x,y,t)$:

Of course, if we were to write it like that all the time we would go mad, so we can just write a much more condensed vectorized version using the gradient with $x\in {\mathbb{R}}^2$:

And in the context of special relativity, people condense that even further by adding $t$ to the spacetime Four-vector as well, so you don't even need to write that separate pesky $t$.

The main point of talking about the Lagrangian density instead of a Lagrangian for fields is likely that it treats space and time in a more uniform way, which is a basic requirement of special relativity: we have to be able to mix them up somehow to do Lorentz transformations. Notably, this is a key ingredient in a/the formulation of quantum field theory.

physics.stackexchange.com/questions/614757/intuitive-explanation-for-why-time-symmetry-implies-conservation-of-energy on Physics Stack Exchange

Equivalent to Lagrangian mechanics but formulated in a different way.

Motivation: Lagrangian vs Hamiltonian.

TODO understand original historical motivation, www.youtube.com/watch?v=SZXHoWwBcDc says it is from optics.

Intuitively, the Hamiltonian is the total energy of the system in terms of arbitrary parameters, a bit like Lagrangian mechanics.

Bibliography:

Analogous to what the Euler-Lagrange equation is to Lagrangian mechanics, Hamilton's equations give the equations of motion from a given input Hamiltonian:So once you have the Hamiltonian, you can write down this system of partial differential equations which can then be numerically solved.

The "taxon cycle" is a concept used in biogeography and ecology to describe the natural evolutionary and geographical progression of certain species or taxa over time. It outlines how populations of species move through a predictable series of stages as they adapt to changing environments, often in response to factors such as habitat availability, climate change, or other ecological pressures.

This is how you transform the Lagrangian into the Hamiltonian.