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. 

This first lecture is a general discussion of the nature of the laws of physics and in particular classical mechanics. The notions of configuration, reversibility, determinism, and conservation law are introduced for simple systems with a finite number of states.

Topics

• 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

References

• State diagram

• Coordinate systems

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

References

• Newton’s laws of motion

• Inertial reference frames

• Conservation of momentum

• Potential energy

• Phase space

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

References

• Principle of Least Action

• Calculus of variations

• Euler Lagrange equations

• Lagrangian mechanic

Topics 

• 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

References

• Symmetry and conservation laws

• Noether’s theorem

• Harmonic oscillator

• Generalized coordinates

• Angular momentum conservation

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

References

• Hamiltonian mechanics

• Active and passive transformation

Topics: 

• 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

References:

• Double pendulum

• Hamiltonian mechanics

• Phase space

Topics

• 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

References

• Liouville’s theorem

• Poisson bracket

Topics

• 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

References

• Poisson bracket

• Energy conservation

• Gyroscope

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

References

• Field theory

• Lorentz force.

• Electromagnetism

• Vector calculus

• Levi-Civita symbol

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

References

• Vector potential

• Canonical coordinates

• Gauge invariance

• Lorentz force