What is Classical Mechanics?

In the 17$^\text{th}$ century, Sir Isaac Newton published a body of work commonly known as the Principia. This was the birth of classical mechanics.

Building on the efforts of countless scientists and philosophers before him, Newton used the tools of a newly formed body of mathematics, calculus, to outline three laws of motion and a universal law of gravitation. These discoveries are collectively known as Newtonian Mechanics. This not only formed the foundation of classical mechanics, but served as a paradigm-shifting worldview. These laws were thought to describe the movement of all objects in the universe, from planets and stars to atoms.

If this were true, then classical mechanics (along with classical electromagnetism) would be our theory of everything. In hindsight, however, we know this to not be the case. The laws of classical mechanics are approximations that work extremely well at macroscopic distances, low speeds, and weak gravity (collectively known as the Newtonian or classical limit). If we wish to probe microscopic distances and describe the true behaviour of atoms, we need to use the tools of quantum mechanics. At the other extreme, if we wish to explore the fascinating regimes of high speeds and strong gravity, we rely on the tools from Relativity theory. Nevertheless, classical mechanics is the starting point, without which we would not have been able to develop our more accurate theories of the 20$^\text{th}$ century.

Newton’s picture was just the beginning. About a century after the publication of the Principia, there were efforts to reformulate Newton’s ideas in a more mathematically rigorous and elegant framework. These efforts were led, in part, by Joseph Lagrange through the framework of Lagrangian Mechanics. This was followed by the works of William Hamilton, who further developed the framework of Hamiltonian Mechanics. Although these three approaches to classical mechanics (Newtonian, Lagrangian, and Hamiltonian) are equivalent and provide identical predictions of the universe, the Lagrangian and Hamiltonian framework are the ones used to formulate our more advanced theories of physics.

Chapter 1. Newtonian Mechanics

Coming soon!

Chapter 2. Lagrangian Mechanics

•  2.1 — The Lagrangian
  • 2.1.1 — Degrees of freedom
  • 2.1.2 — Generalized coordinates
  • 2.1.3 — The Lagrangian
• 2.2 — The Principle of Stationary Action
  • 2.2.1 — The action
  • 2.2.2 — The principle of stationary action
  • 2.2.3 — Deriving the Euler-Lagrange equation
• 2.3 — The Bridge to Newtonian Mechanics
  • 2.3.1 — The connection to Newton’s second law
  • 2.3.2 — Generalized forces and momenta
  • 2.3.3 — Energy and the Hamiltonian
• 2.4 — Applying the Euler-Lagrange Equation
  • 2.4.1 — Multiple degrees of freedom
  • 2.4.2 — Two particles near the Earth
  • 2.4.3 — Simple harmonic oscillator
  • 2.4.4 — Bead sliding on a bowl
  • 2.4.5 — Driven pendulum
  • 2.4.6 — A bead, a spring, and a paraboloid walk into a bar
  • 2.4.7 — Double pendulum
  • 2.4.8 — Charged particle in an electromagnetic field
• 2.5 — Beyond Classical Mechanics: The Action in Relativistic and Quantum Theories
  • 2.5.1 — Special relativity
  • 2.5.2 — Electromagnetism and relativistic field theories
  • 2.5.3 — General relativity
  • 2.5.4 — Path integral formulation of quantum mechanics
  • 2.5.5 — Quantum field theories
• 2.6 — Noether’s Theorem

Chapter 3. Hamiltonian Mechanics

• 3.1 — The Hamiltonian
• 3.2 — Hamilton’s Equations
• 3.3 — Poisson Brackets
• 3.4 — Canonical Transformations
• 3.5 — Symmetries revisited