Classical Mechanics

The foundation of all physics: Lagrangian and Hamiltonian formulations that extend from particles to fields.

🎯 Why Classical Mechanics is Essential

Classical mechanics is not just about particles - it provides the mathematical framework and conceptual foundation for all of modern physics:

⚛️Quantum Mechanics: Hamilton's formulation → Schrödinger equation, canonical quantization
🌌Quantum Field Theory: Lagrangian field theory, path integrals, Noether's theorem
📊Statistical Mechanics: Phase space, Liouville's theorem, ergodicity, ensembles
⚡Plasma Physics: Particle orbits, adiabatic invariants, Hamiltonian chaos
🪐General Relativity: Variational principles, geodesics, Einstein-Hilbert action

Course Overview

This course goes far beyond Newton's F = ma. We develop the Lagrangian and Hamiltonian formulations of classical mechanics - elegant mathematical frameworks that reveal deep connections between symmetries and conservation laws, and provide the conceptual foundation for quantum mechanics, field theory, and statistical physics.

The Lagrangian approach uses the principle of least action - nature chooses paths that minimize (or extremize) the action integral. This profound principle extends from classical particles to quantum fields and even general relativity.

The Hamiltonian formulation introduces phase space - the arena where statistical mechanics lives. Understanding phase space is absolutely critical for quantum mechanics (canonical quantization), statistical mechanics (Liouville's theorem), and chaos theory.

Key Topics Covered

Lagrangian Mechanics

• Generalized coordinates and constraints
• Principle of least action (Hamilton's principle)
• Euler-Lagrange equations
• Symmetries and Noether's theorem
• Conservation laws (energy, momentum, angular momentum)
• Applications: pendulum, central forces, rigid bodies

Hamiltonian Mechanics

• Legendre transformation: L → H
• Hamilton's equations of motion
• Phase space and Poisson brackets
• Canonical transformations
• Liouville's theorem (phase space conservation)
• Connection to quantum mechanics (canonical quantization)

Central Force Problems

• Kepler problem (planetary orbits)
• Effective potential and orbital mechanics
• Virial theorem
• Scattering theory and Rutherford scattering
• Small oscillations and stability

Rigid Body Dynamics

• Inertia tensor
• Euler angles and rotational kinematics
• Euler's equations for rigid body rotation
• Spinning tops and gyroscopes
• Precession and nutation

Fundamental Equations

Euler-Lagrange Equations

The fundamental equation of motion derived from the principle of least action:

$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0$

where L = T - V (kinetic energy - potential energy), and q_i are generalized coordinates.

Hamilton's Equations

First-order equations in phase space (q, p):

$\dot{q}_i = \frac{\partial H}{\partial p_i}, \quad \dot{p}_i = -\frac{\partial H}{\partial q_i}$

where H(q, p) is the Hamiltonian (total energy), and p_i = ∂L/∂q̇_i are conjugate momenta.

Principle of Least Action

Nature chooses the path that extremizes the action:

$\delta S = \delta \int_{t_1}^{t_2} L(q, \dot{q}, t)\, dt = 0$

This principle extends to quantum mechanics (path integrals) and field theory (Lagrangian densities).

Noether's Theorem

Profound connection between symmetries and conservation laws:

Every continuous symmetry of the action corresponds to a conserved quantity:

• Time translation symmetry → Energy conservation
• Space translation symmetry → Momentum conservation
• Rotation symmetry → Angular momentum conservation

This theorem is central to quantum field theory, where gauge symmetries lead to conserved currents.

🔗 Connections to Advanced Physics

→ Quantum Mechanics

Canonical Quantization: Replace Poisson brackets with commutators:

$\{q_i, p_j\}_{PB} = \delta_{ij} \quad \longrightarrow \quad [\hat{q}_i, \hat{p}_j] = i\hbar\delta_{ij}$

Hamilton's H(q, p) becomes the Hamiltonian operator Ĥ. The Schrödinger equation follows from applying the Hamiltonian operator to the wave function.

→ Quantum Field Theory

Lagrangian Field Theory: Replace discrete coordinates q_i(t) with fields φ(x, t):

$S = \int \mathcal{L}(\phi, \partial_\mu\phi)\, d^4x$

Euler-Lagrange equations become field equations. Noether's theorem gives conserved currents. Path integrals extend the action principle to quantum fields.

→ Statistical Mechanics

Phase Space Foundation: Statistical mechanics lives in phase space (q, p):

• Liouville's theorem: phase space volume preserved → foundation of statistical ensembles
• Microcanonical ensemble: constant energy surface in phase space
• Ergodic hypothesis: time averages = ensemble averages (phase space exploration)
• Boltzmann's H-theorem uses phase space trajectories

→ Plasma Physics

Single Particle Orbits: Charged particles in electromagnetic fields:

• Hamiltonian: H = (p - qA)²/2m + qφ (includes vector potential)
• Adiabatic invariants from slowly-varying Hamiltonians
• Magnetic moment conservation: guiding center approximation
• Chaos theory: when particle orbits become unpredictable

📺 Video Lecture Series

Leonard Susskind - Stanford University

10 comprehensive lectures covering the complete Lagrangian and Hamiltonian formulations of classical mechanics. Susskind's clear explanations and physical intuition make these advanced concepts accessible while maintaining mathematical rigor.

Topics covered:

• Principle of least action and Lagrangian mechanics
• Euler-Lagrange equations and applications
• Symmetries, conservation laws, and Noether's theorem
• Hamiltonian mechanics and phase space
• Poisson brackets and canonical transformations
• Connection to quantum mechanics

Watch Susskind Lectures →

📚 Recommended Textbooks

Goldstein, Poole & Safko

Classical Mechanics (3rd Edition)

The classic graduate text. Comprehensive, rigorous, with excellent problems. Standard reference for graduate classical mechanics courses.

Landau & Lifshitz

Mechanics (Vol. 1 of Course of Theoretical Physics)

Elegant, concise, profound. Emphasizes the principle of least action from the start. More advanced perspective that rewards careful study.

John R. Taylor

Classical Mechanics

Excellent undergraduate text. Clear explanations, many examples. Good bridge from introductory physics to Lagrangian/Hamiltonian formulations.

V.I. Arnold

Mathematical Methods of Classical Mechanics

Mathematically sophisticated. Emphasizes geometric and modern perspectives. Excellent for understanding deep mathematical structure.

Prerequisites

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Calculus & Differential Equations:
Multivariable calculus, vector calculus, ordinary differential equations. Understanding of partial derivatives and variational calculus is very helpful.
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Linear Algebra:
Vectors, matrices, eigenvalues/eigenvectors. Essential for rigid body dynamics and advanced topics.
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Introductory Physics:
Newton's laws, kinematics, energy, momentum. We'll reformulate these in Lagrangian/Hamiltonian language.

Recommended Learning Path

1
Watch Susskind Lectures 1-10
Complete all 10 lectures to get the full picture. Take notes on key derivations.
2
Work Through Textbook Problems
Use Taylor (undergraduate) or Goldstein (graduate). Problems are essential for mastery.
3
Master Key Examples
Harmonic oscillator, central force problems, rigid body rotation. These appear everywhere in physics.
4
Connect to Other Courses
Apply to quantum mechanics (canonical quantization), statistical mechanics (phase space), or field theory (Lagrangian densities).