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 ↓


Conservation laws


In the field of classical mechanics and, by extension, Newtonian mechanics, conservation laws are fundamental principles that describe the stability of certain physical quantities as an isolated system evolves over time. These laws form the basis of the theoretical framework of physics and reveal the symmetric nature of physical laws. At the core of these conservation laws are three key quantities: momentum, energy, and angular momentum.

Conservation of momentum

Momentum is a measure of the motion of an object and is defined as the product of an object's mass and its velocity. In mathematical terms, it is expressed as:

P = M * V

Where p is momentum, m is mass, and v is velocity. The principle of conservation of momentum states that within a closed and isolated system, if no external forces act on the system, the total momentum remains constant.

To illustrate this, imagine two smooth, identical ice skaters initially at rest. If one skater pushes the other, the two will move in opposite directions. According to Newton's third law, the forces they exert on each other are equal and opposite, and thus the total momentum before and after the push is zero, so momentum is conserved.

Before the PushAfter the push

Quantitatively, if skater 1 with mass m_1 moves with velocity v_1, and skater 2 with mass m_2 moves with velocity v_2, then momentum conservation can be written as:

m_1 * v_1 + m_2 * v_2 = 0

Energy conservation

Energy is a more abstract concept than momentum, involving the capacity of a system to do work. The principle of conservation of energy in Newtonian mechanics states that the total energy within a closed system remains constant over time. Energy can change forms, such as kinetic or potential energy, but it cannot be created or destroyed.

For example, consider a pendulum swinging. At its highest point, the pendulum has maximum potential energy and zero kinetic energy. As it swings downward, the potential energy turns into kinetic energy, which is maximum at the lowest point. Then, as it swings upward, the kinetic energy turns back into potential energy.

Maximum Potential EnergyMaximum Potential EnergyMaximum kinetic energy

The mathematical expression for the total mechanical energy in a closed system, which is the sum of kinetic energy (KE) and potential energy (PE), remains constant:

KE + PE = constant

When considering a pendulum, the kinetic energy is given by:

KE = 0.5 * m * v^2

And the gravitational potential energy is given by:

PE = M * G * H

Where m is the mass, v is the velocity, g is the acceleration due to gravity, and h is the height.

Conservation of angular momentum

Angular momentum is a scalar quantity associated with bodies in rotational motion, just as linear momentum is associated with translational motion. It is defined as the product of a body's rotational inertia and its angular velocity. In mathematical terms:

L = I * ω

Where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity. The principle of conservation of angular momentum states that if no external torque acts on an object or system, its angular momentum remains constant.

Consider a figure skater doing a spin. When her arms are extended, the skater's moment of inertia is large, and thus the spin speed is slow. As the skater pulls her arms in, the moment of inertia decreases, causing the spin speed to increase in order to conserve angular momentum.

arms extendedarms bent inward

This effect can be felt when a person sits on a rotating chair and rotates, pulling in or extending the arms can control their rotation speed. In a closed system without external torque, the angular momentum remains the same before and after:

I * ω_initial = I * ω_final

Applications and implications

Conservation laws play a key role in predicting the outcomes of collisions, analyzing mechanical systems, and solving complex physics problems. For example, understanding the conservation of linear momentum helps predict the outcomes of vehicle collisions, while the conservation of angular momentum is important in cosmic phenomena such as the dynamics of galaxies and the rotation of planets.

In all these cases, conservation laws provide powerful tools for understanding how systems evolve and predicting possible future states based only on initial conditions. They embody the symmetry and consistency of physical laws, as expressed in Noether's theorem, which correlates symmetry and conservation laws.

Conclusion

Conservation laws in Newtonian mechanics reveal the fascinating symmetry and stability of our universe. By understanding these principles, we acknowledge the interconnected dance of forces and motions that allow the universe to operate with unwavering precision. Through momentum, energy, and angular momentum conservation, physicists can pull back the curtain on the beautiful ballet of matter and motion that defines our reality.


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Conservation laws

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