  # The Theoretical Minimum I: Classical Mechanics

## Video Lectures

Displaying all 10 video lectures.
 Lecture 1 Play Video State diagrams and the Nature of Physical LawsLeonard Susskind gives a brief introduction to the mathematics behind physics including the addition and multiplication of vectors as well as velocity and acceleration in terms of particles. The notions of configuration, reversibility, determinism, and conservation law are introduced for simple systems with a finite number of states. Topics covered: - Allowable laws of physics - Configurations of a coin - Equations of motion of a coin and a die - The concept of conserved quantities and conservation laws - Systems with infinite number of configurations - Laws of physics that are not allowable - Non reversibility - Predictability in the real world - Determinism in the past and future - System of point particles in space - Mathematics: coordinate systems, vector algebra, vector dot-product, triangle law of cosines - Motion of a particle: position, velocity and acceleration - Circular motion and centripetal acceleration Recorded on September 26, 2011. Lecture 2 Play Video Newton's Law, Phase Space, Momentum and EnergyLeonard Susskind focuses on classical mechanics expressed using Newton's 2nd law. The notions of phase space, momentum and energy are introduced. He also discusses some of the basic laws and ideas of modern physics. In this lecture, he focuses on some of the incorrect laws of motion that were first proposed by Aristotle. While they are invalid they provide some insight into how modern physics has developed to the state it is at today. Topics: - Aristotle incorrect laws of motion - Newton's law (the 2nd law) - Inertial reference frames - Newton's determinism and the need of position and velocity - Momentum and Newton's law - Phase space - Newton and reversibility - Newton's law and conserved quantities - Newton's 3 laws - Proof of conservation of momentum for an isolated system of particles - Potential energy - Energy conservation for a system of particles - Harmonic oscillator and energy Recorded on October 3, 2011. Lecture 3 Play Video Lagrangian, Least Action & Euler-Lagrange EquationsThis lecture introduces Lagrange's formulation of classical mechanics. That formulation is formal and elegant; it is based on the Least Action Principle. The concepts introduced here are central to all modern physics. Prof. Susskind discusses Lagrangian functions as they relate to coordinate systems and forces in a system. The lecture ends with angular momentum and coordinate transforms. Topics: - Principle of Least Action (“stationary action”) - Equilibrium points of a function - Trajectories - Calculus of variations - Light in a refractive media and hanging chain catenary - Lagrangian and Action - Euler Lagrange equations of motion - Newton equations from the Lagrangian of a system of particles - Importance of the Lagrange formulation of physics - Lagrangian and coordinate changes - Rotating frame, centrifugal and Coriolis forces - Polar coordinates and angular momentum conservation - Lagrangian, conservation and cyclic coordinates Recorded on October 10, 2011. Lecture 4 Play Video Symmetry and Conservation LawsLeonard Susskind focuses on symmetry and conservation laws, including the principle of least action and Lagrangian methods. He lectures on the relation between continuous symmetries of the Lagrangian and conserved quantities. Generalized coordinates and canonical conjugate momentum are introduced. Topics covered: - Symmetry and conservation laws: they are always related - Review of the principle of least action (stationary action) - Generalized coordinates and their canonical conjugate momentum - Conserved quantities and translation and rotation symmetry - Noether theorem concept and outline - Momentum conservation as a consequence of translation symmetry - Angular momentum conservation as a consequence of rotational symmetry - The harmonic oscillator - Discrete symmetries have no associated conserved quantities in classical mechanics Recorded on October 17, 2011. Lecture 5 Play Video The HamiltonianLeonard Susskind discusses different particle transformations as well as how to represent and analyze them using tools like the Lagrangian. The lecture starts with a thorough review of symmetries and conservation laws. Energy conservation is shown to be a consequence of time translation symmetry and the Hamiltonian is introduced. Topics: - Recommended books - Superluminal neutrinos in the news - Review of symmetries and conservation laws - Active vs passive transformations - Review of momentum and angular momentum conservation and associated symmetries - Energy conservation as a consequence of time translation symmetry - Hamiltonian and energy conservation Recorded on October 17, 2011. Lecture 6 Play Video Hamilton's EquationsLeonard Susskind discusses the some of the basic laws and ideas of modern physics. In this lecture, he focuses on the motion of objects. He starts with a general example of a wedge on a frictionless plane and uses it as the building block for more complicated theory. - Motion of a ball on a wedge as an example of Euler-Lagrange equations - The ball on a wedge: Associated symmetries and conservation laws, conjugate momentum - Double pendulum example treated in detail. Associated symmetries and conservation - Hamiltonian, forbidden laws, reversibility, convergent and divergent paths in state space - Hamilton's equations of motion - Harmonic oscillator using Hamilton's equations and energy conservation - Phase space Recorded on November 1, 2011. Lecture 7 Play Video Liouville's Theorem & Phase SpacesLeonard 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. Lecture 8 Play Video Poisson BracketsIn this lecture, Prof. Susskind looks closely at the concept of angular momentum and Poisson Brackets. He derives the basic angular momentum equations and shows how they can describe this fundamental type of motion. Poisson brackets are another formal formulation of classical mechanics. They help make the connection between symmetries and conservation laws more explicit. The Poisson bracket of the x,y,z components of angular momentum are derived. Topics covered: - Poisson brackets and angular momentum - Review of Poisson brackets - The algebra of Poisson brackets - Angular momentum conservation, rotation symmetry and Poisson brackets as tools to compute the generators of rotation - Momentum conservation, translation symmetry and Poisson brackets as tools to compute the generators of translation - Energy conservation, time shift symmetry and Poisson bracket as a tool to compute the time shift generator - General relation between symmetry and conservation law expressed with Poisson bracket. - Poisson brackets of the x, y, z components of angular momentum. - The gyroscope equations of motion as an example of the power of Poisson brackets Recorded on November 14, 2011. Lecture 9 Play Video Lagrangian of Static Electric and Magnetic FieldsIn this lecture, Leonard Susskind dives into the topics of magnetic and electrostatic forces. He derives these forces to show their relationship to magnetic fields and potential. He introduces the static electric and magnetic fields with the associated Lagrangian and the Lorentz force. The vector potential, it's gauge field and gauge invariance are also introduced. Topics: - Magnetic and electric fields - The concept of field - The “del” or “nabla” symbol - Vector calculus: Gradient, Divergence and Curl - The Levi-Civita symbol - Algebra: div curl and curl grad vanish - The vector potential and why it's needed - Gauge field: "Gauge" is a misnomer - Lorentz force. Lorentz force compared with the Coriolis force - Lagrangian for charged particles in a electro-static and magneto-static fields - Gauge invariance of the equations of motions associated to the electro-magneto-static Lagrangian Recorded on November 21, 2011. Lecture 10 Play Video Particles in Static Electric and Magnetic FieldsLeonard Susskind wraps up the lecture series by finishing his talk on particles and both electric and magnetic fields and how they relate to physics. Topics: - Review of the vector potential, concept of gauge and gauge invariance - Lorentz force law - Example of different vector potentials for a constant magnetic field and the gauge transformation that relate them - Importance of gauge invariance and choice of gauge - Lagrangian of a particle in a static magnetic field. Review of the related action gauge invariance - Distinction between mechanical and canonical momentum: only the canonical momentum is related to symmetries and invariance - Derivation of the Euler-Lagrange equation of motion from the magneto-static Lagrangian and rediscovery of the Lorentz force - Justification of the vector potential as an essential tool for the least action principle - Derivation of the magneto-static Hamiltonian - Smart choice of gauge and derivation of the Lorentz force from symmetry arguments only, “cyclic coordinates” - Circular motion of a charged particle in a static magnetic field - Monopoles discussion as part of the questions session - Brief Quaternions discussion as part of the questions session Recorded on November 28, 2011.