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 ↓


Newtonian mechanics


Newtonian mechanics, also known as classical mechanics, is a branch of physics that deals with the motion of physical objects and the forces that act upon them. This field of study is based on the work of Sir Isaac Newton and requires an understanding of Newton's three laws of motion as well as concepts such as force, mass, and energy. In this lesson, we will explore the foundations of Newtonian mechanics, providing numerous examples and explanations to illustrate these principles.

Understanding momentum

The study of motion can be divided into two broad categories: kinematics, which describes motion without considering the causes of motion, and dynamics, which deals with the forces and torques that cause motion. Newtonian mechanics covers both of these areas, providing a comprehensive view of the motion of objects in our universe.

Dynamics: description of motion

Kinematics focuses on the geometry of motion. It describes the position, velocity, and acceleration of objects over time, without reference to the forces causing the motion. In essence, kinematics asks "what is moving?" and "how is it moving?" However, it does not answer "why is it moving?"

Key concepts in dynamics include:

  • Distance: A scalar quantity that represents the total path length traveled by an object.
  • Displacement: A vector quantity that represents the change in the position of an object. It considers only the initial and final positions, not the path taken.
  • Velocity: A vector quantity that describes the rate of change of displacement. It has both magnitude and direction.
  • Speed: A scalar quantity that describes how fast an object is moving, viewed without regard to direction.
  • Acceleration: A vector that describes the rate of change of velocity with time.

Dynamics: force and motion

Dynamics explores the causes of motion, specifically, the forces that cause an object to move. It is essential for understanding not only how objects move, but also why they move the way they do.

Newton's laws of motion

At the core of Newtonian mechanics are Newton's three laws of motion. These laws form the basis for analyzing and predicting the motion of objects. Let's take a deeper look at each law with textual and graphical examples:

First law: Law of inertia

An object at rest remains at rest, and an object in motion remains in motion, unless some external force is applied on it.

This law introduces the idea of inertia, which is the resistance of an object to any change in its state of motion. Objects will maintain their current state of motion unless a net external force is applied.

Example: Consider a puck sliding on an ice rink. Once it is pushed, it will continue to slide in a straight line at a constant speed until friction or some other force slows it down or changes its direction.

Second law: Law of acceleration

The acceleration of an object is proportional to the net force acting on it and inversely proportional to its mass.

This law is often expressed by the equation:

F = ma

Where F is the net force acting on the object, m is its mass, and a is the resultant acceleration.

Example: Imagine pushing two boxes on a frictionless surface, one of which is twice as heavy as the other. When the same force is applied to both, the lighter box will move faster.

M 2m

Third law: The law of action and reaction

Every action has an equal and opposite reaction.

This principle states that forces always come in pairs. Whenever one object exerts a force on another object, the second object also exerts a force of equal magnitude and opposite direction on the first object.

Example: When you sit on a chair, your body exerts a downward force due to gravity, and the chair exerts an equal upward force to support you.

Chair

Applications of Newtonian mechanics

The principles of Newtonian mechanics are essential for solving a wide range of physical problems, from simple motion to complex systems. Here are some applications:

Projectile motion

Projectile motion describes the trajectory of an object under the influence of gravity only. This is a common situation that is handled using Newton's laws.

The equations involved in projectile motion are:

x = v_0 * t * cos(θ) 
y = v_0 * t * sin(θ) - 0.5 * g * t^2

where v_0 is the initial velocity, θ is the launch angle, and g is the acceleration due to gravity.

Example: A soccer player kicks a ball at a 45 degree angle at a speed of 20 m/s. The motion of the ball can be analyzed using the equations above to determine its trajectory.

Simple harmonic motion

Many systems, which when displaced from their equilibrium position experience a restoring force proportional to the displacement, undergo simple harmonic motion.

F = -kx

where k is the spring constant and x is the displacement from equilibrium.

Example: Consider a mass attached to a spring. When released from the displaced position, it will oscillate back and forth in simple harmonic motion.

Gravitational force

The force of gravity is a universal force of attraction acting between all substances. Newton described it with his law of universal gravitation:

F = G(m_1*m_2)/r^2

where G is the gravitational constant, m_1 and m_2 are the masses of the two objects, and r is the distance between the centers of the two masses.

This force keeps the planets revolving around the stars and the moons revolving around the planets.

Conservation laws in mechanics

Newtonian mechanics also includes conservation principles that play an important role in understanding and solving complex problems:

Conservation of momentum

In a closed system the total momentum is conserved. Momentum is calculated as:

p = mv

Example: In a perfectly elastic collision, the total momentum before collision is equal to the total momentum after collision.

Energy conservation

Energy cannot be created or destroyed, it can only be converted from one form to another. In mechanics, we often deal with kinetic and potential energy.

KE = 0.5 * m * v^2 
PE = m * g * h

Example: Consider a swinging pendulum. At its highest point, it has maximum potential energy and zero kinetic energy. At its lowest point, it has maximum kinetic energy and zero potential energy.

Limitations of Newtonian mechanics

Newtonian mechanics is incredibly powerful, but it has its limitations. It accurately describes motion at everyday speeds and sizes, but fails at very high speeds (close to the speed of light), at very small scales (quantum level), or in strong gravitational fields.

This led Albert Einstein to develop relativistic mechanics, which describes phenomena at high speeds, and quantum mechanics, which explains physics at the atomic and subatomic levels.

Despite these limitations, Newtonian mechanics remains a cornerstone of physics. Its principles are foundational, providing a gateway to more complex and subtle areas of physics.


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Newtonian mechanics

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