**Copyright Information:**All rights reserved to Prof. Leonard Susskind, Stanford University.

### Lecture Description

Leonard Susskind discusses the some of the basic laws and ideas of modern physics. In this lecture, he focuses on Liouville's Theorem, which he describes as one of the basis for Hamiltonian mechanics. He works to prove the reversibility of classical mechanics. He analyses the flow in phase space of multiple systems, and that flow is shown to be incompressible. Poisson brackets are introduced as yet another way to express classical mechanics formally. Topics covered: - Liouville's famous theorem - Review of Hamiltonian and energy conservation - Energy conservation and surfaces in phase space - Concept of flow in phase space - Compressible and incompressible flows, the divergence - Demonstration of Liouville's theorem - Liouville using a toy Hamiltonian. Topology of evolving phase space elements - The damped harmonic oscillator as a counterexample of Liouville - Definition of Poisson bracket - Poisson bracket and time derivative of any quantity Recorded on November 7, 2011.

### Course Index

- State diagrams and the Nature of Physical Laws
- Newton's Law, Phase Space, Momentum and Energy
- Lagrangian, Least Action & Euler-Lagrange Equations
- Symmetry and Conservation Laws
- The Hamiltonian
- Hamilton's Equations
- Liouville's Theorem & Phase Spaces
- Poisson Brackets
- Lagrangian of Static Electric and Magnetic Fields
- Particles in Static Electric and Magnetic Fields

### Course Description

This is the first course in a collection of 6 core physics courses by renowned physicist Leonard Susskind's series, The Theoretical Minimum. Our exploration of the theoretical underpinnings of modern physics begins with classical mechanics, the mathematical physics worked out by Isaac Newton (1642--1727) and later by Joseph Lagrange (1736--1813) and William Rowan Hamilton (1805--1865). We will start with a discussion of the allowable laws of physics and then delve into Newtonian mechanics. We then study three formulations of classical mechanics respectively by Lagrange, Hamiltonian and Poisson. Throughout the lectures we will focus on the relation between symmetries and conservation laws. The last two lectures are devoted to electromagnetism and the application of the equations of classical mechanics to a particle in electromagnetic fields.