PHD
  1. Classical mechanics  1.1. Newtonian mechanics  1.1.1. Laws of motion in Newtonian mechanics  1.1.2. Dynamics of particles and systems  1.1.3. Conservation laws  1.1.4. Central force motion  1.2. Lagrangian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Constraints and normalized coordinates  1.2.4. Noether's theorem  1.3. Hamiltonian mechanics  1.3.1. Hamilton's equations  1.3.2. Canonical conversion  1.3.3. Poisson bracket  1.3.4. Action-angle variable  1.4. Rigid body mobility  1.4.1. Rotation of rigid bodies  1.4.2. Moment of inertia tensor  1.4.3. Euler's equations in rigid body dynamics  1.4.4. Gyroscopic motion  1.5. Chaos and nonlinear dynamics  1.5.1. Stage space and attractive  1.5.2. Bifurcation and chaos theory  1.5.3. Hamiltonian Chaos  1.5.4. Lyapunov exponent
  2. Electrodynamics  2.1. Maxwell's equations  2.1.1. Gauss's law for electricity  2.1.2. Gauss's law for magnetism  2.1.3. Faraday's Law  2.1.4. Ampere's Law  2.1.5. Boundary conditions in Maxwell's equations  2.2. Electromagnetic waves  2.2.1. Wave equation  2.2.2. Polarization  2.2.3. Reflection and Refraction  2.2.4. Waveguides and resonators  2.3. Special relativity  2.3.1. Lorentz transformations  2.3.2. Relativistic energy and momentum  2.3.3. Relativistic electrodynamics  2.3.4. Minkowski spacetime  2.4. Radiation and scattering  2.4.1. Dipole radiation  2.4.2. Multipole expansion  2.4.3. Compton scattering  2.4.4. Thomson and Rayleigh scattering  2.5. Plasma Physics  2.5.1. Debye Screening  2.5.2. Magnetohydrodynamics  2.5.3. Plasma instability  2.5.4. Fusion Plasma
  3. Quantum mechanics  3.1. Foundations of quantum mechanics  3.1.1. Principles of quantum mechanics  3.1.2. Wave function and probability interpretation  3.1.3. Uncertainty principle  3.1.4. Quantum Tunneling  3.2. Schrödinger Equation  3.2.1. Time-dependent Schrödinger equation  3.2.2. Time-independent Schrödinger equation  3.2.3. Eigenvalues and eigenfunctions in the Schrödinger equation  3.2.4. Path integrals in quantum mechanics  3.3. Quantum operators  3.3.1. Commutators and observables  3.3.2. Angular momentum operator  3.3.3. Ladder Operator  3.3.4. Pauli matrices  3.4. Quantum entanglement and measurement  3.4.1. Bell's theorem  3.4.2. Quantum Teleportation  3.4.3. Quantum entanglement and quantum decoherence in measurement  3.4.4. Quantum entanglement and measurement in quantum mechanics  3.5. Relativistic quantum mechanics  3.5.1. Klein–Gordon equation  3.5.2. Dirac equation  3.5.3. Quantum field theory  3.5.4. Feynman path integral
  4. Statistical mechanics and thermodynamics  4.1. Classical thermodynamics  4.1.1. Laws of Thermodynamics  4.1.2. Carnot cycle  4.1.3. Entropy and free energy  4.1.4. Thermodynamic efficiency  4.2. Kinetic theory of gases  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Transport phenomenon  4.2.3. Mean free path  4.2.4. Non-equilibrium systems in the kinetic theory of gases  4.3. Statistical mechanics  4.3.1. Microstates and Macrostates  4.3.2. Partition function  4.3.3. Bose–Einstein and Fermi–Dirac statistics  4.3.4. Fluctuations and correlations  4.4. Phase transition  4.4.1. Important events  4.4.2. Landau theory  4.4.3. Ising model  4.4.4. Renormalization group theory
  5. Quantum field theory  5.1. Second Quantization  5.1.1. Quantum harmonic oscillator in second quantization  5.1.2. Creation and annihilation operators in second quantization  5.1.3. Path Integral in Second Quantization  5.1.4. Fock space  5.2. Quantum Electrodynamics  5.2.1. Feynman diagrams  5.2.2. Renormalization in quantum electrodynamics  5.2.3. Gauge invariance in quantum electrodynamics  5.2.4. Vacuum polarization  5.3. Quantum Chromodynamics  5.3.1. Quarks and gluons  5.3.2. Color range  5.3.3. Lattice QCD  5.3.4. Asymptotic freedom  5.4. Standard model of particle physics  5.4.1. Electroweak theory  5.4.2. Higgs mechanism  5.4.4. Grand Unified Theories
  6. General relativity and gravity  6.1. Einstein's field equations  6.1.1. Ricci tensor and scalar curvature  6.1.2. Schwarzschild solution  6.1.3. Kerr metric  6.1.4. Gravitational waves  6.2. Cosmology  6.2.1. Friedmann equation  6.2.2. Cosmic inflation  6.2.3. Dark matter and dark energy  6.2.4. Large scale structure  6.3. Black holes and wormholes  6.3.1. Event horizon in black holes and wormholes  6.3.2. Hawking radiation  6.3.3. Penrose process  6.3.4. Information paradox in black holes and wormholes
  7. Condensed matter physics  7.1. Crystal structure and lattice  7.1.1. Bravais lattices  7.1.2. Band theory in crystal structure and lattices  7.1.3. Phonons in crystal structure and lattice  7.1.4. Block's theorem  7.2. Superconductivity  7.2.1. BCS principle  7.2.2. Meissner effect  7.2.3. High Temperature Superconductor  7.2.4. Josephson effect

PHD ↓


Classical mechanics


Classical mechanics is a branch of physics that deals with the motion of macroscopic objects. The principles of classical mechanics are based on intuitively sound concepts such as force, energy, and momentum. These principles are based on observations and experiments that date back to the time of Aristotle and were later formalized by scientists such as Isaac Newton.

The study of classical mechanics involves understanding various physical concepts and applying them to describe how objects move. This includes understanding concepts such as velocity, acceleration, force, and energy. In this detailed explanation, we will understand these concepts with detailed explanation and examples that are easy to understand.

Basic concepts and definitions

1. Dynamics

Dynamics is the discipline that describes motion, but does not consider the forces that cause it. It includes the following terms:

  • Position - This refers to the location of an object in space. It is often described using coordinates, e.g., (x, y, z).
  • Velocity - The rate of change of position with respect to time. It is a vector quantity.
  • Acceleration - The rate of change of velocity relative to time.

Consider the motion of a car on a straight road. If we plot the motion of the car, the position changes with time. The slope of the position-time graph gives the velocity, while the slope of the velocity-time graph gives the acceleration.

v = frac{d}{dt}x(t)

2. Mobility

Dynamics studies the causes of motion and the effects it produces. It includes the following concepts:

Force: Force is an interaction that changes the motion of an object without opposition. It is described by Newton's second law of motion:

F = ma

Where F is the force, m is the mass, and a is the acceleration produced.

3. Newton's laws of motion

Newton's laws form the foundation of classical mechanics:

  • First Law (Inertia): An object at rest remains at rest and an object in motion continues to move at a constant velocity unless an external force is applied on it.
  • Second Law (F=ma): The acceleration of an object is proportional to the net force acting on it and inversely proportional to its mass.
  • Third Law (Action-Reaction): For every action there is an equal and opposite reaction.

Imagine you are pushing a book on a table. According to the first law, the book will remain at rest unless you apply a force. The harder you push (applying the second law), the faster the book will move. The force you apply to the book is equal to the force applied in the opposite direction (third law).

4. Work and energy

The concepts of work and energy explain how forces cause displacement. They also explain the transformation and conservation of energy.

Work: Work is done when a force causes an object to move. If F is the force and d is the displacement, then the work is expressed as:

W = F cdot d cdot cos(theta)

Kinetic energy: It is the energy of a moving object and is expressed as follows:

KE = frac{1}{2}mv^2

Potential energy: This is the energy stored in an object due to its position or arrangement. For example, gravitational potential energy is:

PE = mgh

5. Conservation laws

Conservation laws state that certain properties of isolated physical systems do not change over time. The most important of these are:

Conservation of Energy: Energy can neither be created nor destroyed, but it can be converted from one form to another.

Conservation of Momentum: The total momentum of a closed system remains constant provided no external force is applied.

Detailed explanation of concepts

Position, velocity and acceleration

Let's start by using a practical example. Imagine you are standing at a train station and watching a train pass by. The train's position is a vector that tells you where the train is on its track. During its journey, the train starts moving from a station, accelerates, reaches a maximum speed, and finally slows down as it approaches the next station.

station a passing train station

The velocity of a train is how fast it is moving in a particular direction. If the train is moving at 80 km/h, this is its velocity. Acceleration is the rate at which the train changes its velocity. If the train goes from 40 km/h to 80 km/h in 10 seconds, its acceleration is positive.

Acceleration Velocity

Application of Newton's laws

Let us take the example of a ball rolling on the ground. Initially it is at rest, but when you kick it, it moves; this demonstrates Newton's first law. However, it eventually stops due to friction, which is a force opposing its motion.

If you kick a heavier ball with the same force, it will have less acceleration according to Newton's second law, since for a given force, acceleration is inversely proportional to mass.

When you walk, you push back on the ground, and yet you move forward. This is because the ground pushes forward on you in an equal and opposite direction (Newton's third law).

Work and energy in practice

Climbing a hill involves doing work against gravity. The higher you climb, the more gravitational potential energy you gain. Once you descend, this potential energy slowly turns into kinetic energy.

Imagine a cyclist climbing a hill. The cyclist's muscles do work to climb up, converting chemical energy (from food) into gravitational potential energy. Upon reaching the top and descending, the bike's speed increases, which converts potential energy back into kinetic energy.

Hill Bottom Bottom

Conservation principle in action

Consider two ice skaters pushing against each other. Initially, they are at rest, meaning they have zero momentum. As they push, they gain speed, yet their total momentum remains zero, which satisfies conservation of momentum, since they push in opposite directions.

Skater 1 Skater 2

In another scenario, consider a swinging pendulum. At its highest point, the energy is entirely potential. As it swings downward, it is converted into kinetic energy. At the lowest point, all the energy is kinetic, perfectly demonstrating energy conservation.

Conclusion

Classical mechanics provides the tools and framework for understanding the physical world at the macroscopic scale. Despite its long history, the principles of classical mechanics remain integral elements in a variety of fields, including engineering, astronomy, and everyday life applications. Understanding these fundamental principles through practical examples and straightforward visual models not only deepens understanding but also enhances the ability to effectively apply these concepts to a wide range of situations.


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