Graduate
  1. Classical mechanics  1.1. Advanced Kinematics  1.1.1. Generalized coordinate systems  1.1.2. Parametric equations of motion  1.1.3. Non-inertial frames and fictitious forces  1.1.4. Covariant formulation of momentum  1.1.5. Action-angle variable  1.2. Lagrangian and Hamiltonian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Noether's theorem and conservation laws  1.2.4. Normalized Speed  1.2.5. Canonical conversion  1.2.6. Poisson bracket and Hamilton–Jacobi theory  1.2.7. Hamiltonian chaos and integrability  1.3. Rigid body mobility  1.3.1. Euler's equations of motion  1.3.2. Principal moments of inertia  1.3.3. Gyroscopic motion and precession  1.3.4. Inertia tensor  1.3.5. Stability of rotational speed  1.4. Nonlinear dynamics and chaos  1.4.1. Phase space and stability analysis  1.4.2. Poincaré sections and bifurcation theory  1.4.3. Strange Attractors and Fractals  1.4.4. Lyapunov exponent  1.4.5. KAM theorem and quasi-periodic motion
  2. Electromagnetism  2.1. Advanced Electrodynamics  2.1.1. Green's function and potential theory  2.1.2. Multipole expansion  2.1.3. Image charging method  2.1.4. Boundary conditions and uniqueness theorem  2.2. Electromagnetic wave propagation  2.2.1. Plane waves in dielectrics and conductors  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarization and Jones calculus  2.3. Relativistic electrodynamics  2.3.1. Covariant formulation of electrodynamics  2.3.2. Lienard–Wiechert potentials  2.3.3. Synchrotron radiation and bremsstrahlung  2.3.4. Electromagnetic field tensor and Lorentz transformations  2.4. Plasma physics  2.4.1. Magnetohydrodynamics  2.4.2. Debye shielding and plasma oscillations  2.4.3. Alfvén waves and tokamak confinement  2.4.4. Plasma instabilities and turbulence
  3. Statistical mechanics and thermodynamics  3.1. Advanced Thermodynamics  3.1.1. Legendre transforms and thermodynamic potentials  3.1.2. Fluctuation–dissipation theorem  3.1.3. Onsager interpersonal relations  3.1.4. Critical events and phase transitions  3.2. Quantum statistical mechanics  3.2.1. Density Matrix and Ensemble Theory  3.2.2. Bose–Einstein condensates  3.2.3. Quantum phase transition  3.2.4. Fermi–Dirac and Bose–Einstein statistics  3.2.5. Partition functions and grand canonical ensembles
  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
  6. Condensed matter physics  6.1. Band structure and transport theory  6.1.1. Block theorem and the Kronig–Penney model  6.1.2. Fermi surface topology  6.1.3. Quantum Hall Effect  6.2. Superconductivity  6.2.1. BCS theory and Cooper pairs  6.2.2. Meissner effect and flux quantization  6.2.3. The Josephson Effect and SQUIDs  6.3. Topological phases of matter  6.3.1. Topological Insulators  6.3.2. Majorana fermions in topological phases of matter
  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

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Classical mechanics


Classical mechanics is a branch of physics that deals with the motion of bodies based on Isaac Newton's laws of motion. It forms the foundation of all other areas of physics and provides a comprehensive description of the behavior of objects under various circumstances in the world. In this document, we will explore the various components and principles of classical mechanics, providing examples to illustrate these concepts.

Newton's laws of motion

The foundation of classical mechanics rests on Isaac Newton's three laws of motion, presented in his seminal work "Philosophiae Naturalis Principia Mathematica." Let's take a look at each of these laws with examples:

First law of motion

The first law of motion states that a body at rest stays at rest, and a body in uniform motion stays in uniform motion unless an external force is applied. This law is also called the law of inertia.

F_net = 0 implies v = constant

For example, imagine a soccer ball lying on a grass field. It will remain at rest until someone kicks it, applying an external force and moving it.

Second law of motion

The second law of motion quantifies the effect of force on the motion of an object. It states that the acceleration of an object is proportional to the net force applied to it and inversely proportional to its mass. Mathematically, it is expressed as:

F = ma

Where:

  • F is the net force applied,
  • m is the mass of the object, and
  • a is the acceleration produced.

Suppose you push a shopping cart with a force of 10 N, and the cart has a mass of 2 kg. The acceleration can be found by rearranging the formula to a = F/m. Thus:

a = 10 N / 2 kg = 5 m/s²

Third law of motion

The third law of motion states that for every action there is an equal and opposite reaction. This means that forces always occur in pairs.

For example, if you sit on a chair, your body exerts a downward force on the chair. At the same time, the chair exerts an upward force of equal magnitude and opposite direction on your body. This is why you remain stationary on the chair.

Dynamics

Kinematics is the branch of classical mechanics that describes the motion of objects without considering the forces that cause them. It includes concepts such as displacement, velocity, acceleration and time. Some of the formulas used in kinematics are:

v = u + at 
s = ut + 0.5at² 
v² = u² + 2as

Here:

  • v is the final velocity,
  • u is the initial velocity,
  • a is the acceleration,
  • s is the displacement, and
  • t is the time.

Initial Last

Dynamics and forces

Dynamics deals with how forces cause motion. Forces can be contact forces, such as friction, tension, and normal forces, or field forces, such as gravitational force.

Friction

Friction is a force that opposes motion between two surfaces in contact. It can be static or kinetic, the former resisting motion and the latter opposing motion already taking place.

F_friction = μN

Where:

  • μ is the coefficient of friction, and
  • N is the normal force.

Force Friction

Tension

Tension is the force transmitted through a rope, wire, cable, or similar object when it is pulled by forces acting from opposite ends.

Consider a block suspended from the ceiling by a rope. The tension in the rope is equal to the force of gravity acting on the block, assuming there is no acceleration:

T = mg

Here:

  • T is the tension in the rope,
  • m is the mass of the block, and
  • g is the acceleration due to gravity (about 9.8 m/s² on Earth).

Energy and work

Energy is the capacity to do work. Work is done when a force moves an object a certain distance. The equation for work is:

W = Fd cos(θ)

Where:

  • W is the work done,
  • F is the applied force,
  • d is the distance traveled, and
  • θ is the angle between the force direction and displacement.

Energy can be potential or kinetic. Kinetic energy is the energy of motion, which is given as follows:

KE = 0.5mv²

Where:

  • KE is the kinetic energy,
  • m is the mass, and
  • v is the velocity.

Potential energy, specifically gravitational potential energy, is the energy stored due to the position of an object. It is given as:

PE = mgh

Where:

  • PE is the potential energy,
  • m is the mass,
  • g is the acceleration due to gravity, and
  • h is the height above the reference point.

MGH

Conservation laws

Conservation laws are very important in physics, providing powerful constraints on the behavior of systems. The principle of conservation of energy states that energy in a closed system cannot be created or destroyed, only transformed.

Conservation of momentum

The principle of conservation of momentum states that the total momentum of a closed system is conserved. Momentum is the product of an object's mass and velocity:

p = mv

For a system of particles, the total momentum is the sum of the individual momenta:

p_total = Σ mᵢvᵢ

In the absence of external forces, this total momentum remains constant. Consider two ice skaters who are initially at rest. When they push each other, they move in opposite directions. The velocity and mass of each skater may change, but the combined momentum remains zero.

Rotational motion

Just as linear motion involves parameters such as position, velocity, and acceleration, rotational motion involves angular position, angular velocity, and angular acceleration. For example, consider a spinning wheel or the rotation of the Earth.

Moment of inertia

Moment of inertia is the rotational analogue of mass in linear motion. It measures the resistance of an object to a change in its rotational motion. For a point mass m located at a distance r from the axis of rotation, the moment of inertia I is:

I = mr²

For extended bodies, the moment of inertia is the sum of the moments of the individual point masses that make up the system. For example, the moment of inertia of a solid cylinder rotating around its longitudinal axis is:

I = 0.5MR²

where M is the mass and R is the radius of the cylinder.

Torque

Torque is the rotational analog of force. It is the result of a force rotating an object. Torque τ is:

τ = rF sin(θ)

Where:

  • τ is the torque,
  • r is the distance from the axis of rotation to the point where the force is applied,
  • F is the magnitude of the force, and
  • θ is the angle between the force vector and the lever arm.

For example, when you open a door, you apply force to the handle, which is away from the hinges. This creates a torque that causes the door to turn and open.

Conclusion

Classical mechanics provides a comprehensive framework for understanding the motion and interaction of objects in our universe. From Newton's laws of motion to the principles of energy and momentum conservation, these concepts are essential for explaining and predicting physical phenomena. By exploring these principles and their applications, we gain a deeper understanding of the rules that govern our natural world.


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Classical mechanics

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