In this video, I introduce you to the principle of least action which is the basis of modern physics. We will look at very simple examples together, starting with free fall. We will see how we find that trajectories are parabolas using a balance of forces (like in high school!), then we will find this result by minimizing a certain Action. It is this action which then takes center stage! We will finally make this principle more general and abstract, forming what is called Lagrangian analytical mechanics, with nice mathematical structures. In particular, we will need to use the tangent bundle structure, on which the Lagrangian function is defined. As an application, we will see a version of Noether's theorem which links symmetries and conserved quantities. For the level, the beginning will be approximately bac+1, then bac+3 for the introduction of the Lagrangian and bac+5 for the final mathematical formalism. The notes are available here: http://www.antoinebourget.org/attachm... The video I'm referring to at 3:16:42 is • Lagrangian and Hamiltonian Mechanics ... You can also watch, for the historical aspects, this documentary from the Poincaré Institute: • The principle of least action and the m... ------------------------------------------------------------------- My name is Antoine Bourget, I am a theoretical physicist, and I try to convey in video what I find elegant in mathematics and physics. To follow the news of the channel, and contact me, you can join the Discord server or follow me on social networks. If you want to donate, I also have a Tipeee Discord account: / discord Twitter: / antoinebrgt My personal website: http://www.antoinebourget.org Tipeee: https://fr.tipeee.com/scientia-egregia/ ------------------------------------------------------------------- 00:00 Start 04:50 Plan Part 1: Physics 08:30 Example 1: Free fall, parabolic trajectory 21:40 Example 2: The simple pendulum 34:10 Essential ideas, intuitive version 43:00 Introduction to the Lagrangian and application to free fall 1:04:00 Principle of least action 1:11:00 Why use the Lagrangian? 1:15:00 The pendulum with the Lagrangian, Lagrange multipliers 1:25:10 Other physical examples 1:31:53 Rotating frame, inertial forces, Coriolis forces Part 2: Mathematics 1:47:50 Physical system, configuration space 1:56:00 Paths, variations, derivatives 2:06:25 Action and principle of least action 2:12:40 Tangent space, tangent vectors 2:22:50 Tangent bundle and derivative 2:30:45 Standard coordinates 2:38:20 Euler-Lagrange equations 2:50:30 Conservation laws 2:52:10 Conservation of energy 3:00:20 Symmetries and Noether's theorem 3:16:10 Summary and conclusion ------------------------------------------------------------------- References: The content of this video is quite classic, but there is no real source where you can find all this at once. For a very intuitive introduction to physics, you can look at David Tong's notes: https://www.damtp.cam.ac.uk/user/tong... For a more mathematical treatment, there is Arnold, Mathematical Methods of Classical Mechanics Takhtajan, Quantum Mechanics for Mathematicians (chapter 1). For details on tangent bundles, you can look at these lecture notes by Marco Gualtieri: http://www.math.toronto.edu/mgualt/co...