Energy and Hamiltonian Mechanics
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In this video we will talk about energy in mechanics, and the associated formalism, the Hamiltonian formalism of analytical mechanics. We will see how this generalizes the Lagrangian formalism, and provides a more general principle of least action than the one seen in the last video. We will see the fundamental notion of phase space appear, and we will be able to touch on some applications related to the long-time behavior of systems and chaos. Finally we will establish the mathematical tools allowing us to understand the structure that is hidden behind all this: Legendre transformation, cotangent bundle and symplectic geometry. Link to the notes taken during the video: http://www.antoinebourget.org/attachm... ------------------------------------------------------------------- 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/ ------------------------------------------------------------------- References: - Landau, L., & Lifchitz, E. (1966). Mécanique, Editions Mir. Moscou - Arnol'd, VI (2013). Mathematical methods of classical mechanics (Vol. 60). Springer Science & Business Media. - Takhtajan, LA (2008). Quantum mechanics for mathematicians Graduate Studies in Mathematics vol 95 (Providence, RI: American Mathematical Society). ------------------------------------------------------------------- Plan: 00:00 Start 8:20 Example 1: the pendulum 20:22 Introduction to phase space 29:43 Example 2: harmonic oscillator 34:53 Changes of variables and units 43:30 Changes of variables in phase space 54:37 Hamiltonian formalism 1:11:55 Legendre transform 1:24:15 Off-topic on thermodynamics 1:36:10 Hamiltonian flow and Liouville's theorem 1:48:55 Consequences of the conservation of volume 1:59:03 Principle of Least Action 2:13:10 Mathematical structure of the cotangent bundle 2:31:30 Symplectic form in standard coordinates 2:48:46 Transition from the Lagrangian to the Hamiltonian 2:56:07 Poisson bracket 3:02:52 Action as a function of coordinates and Hamilton-Jacobi 3:08:04 Canonical transformations and generating functions 3:16:55 Zero Hamiltonian transformation 3:23:40 Example of the harmonic oscillator 3:32:07 Asymmetric Keplerian problem 3:44:00 Summary
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