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Outline of physics
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Outline of physics
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The following outline is provided as an overview of and topical guide to physics:

Part of a series on
Physics
Branches
Research

Physics – natural science that involves the study of matter[1] and its motion through spacetime, along with related concepts such as energy and force.[2] More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.[3][4][5]

What type of subject is physics?

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Physics can be described as all of the following:

  • An academic discipline – one with academic departments, curricula and degrees; national and international societies; and specialized journals.
  • A scientific field (a branch of science) – widely recognized category of specialized expertise within science, and typically embodies its own terminology and nomenclature. Such a field will usually be represented by one or more scientific journals, where peer-reviewed research is published.

Branches

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  • Astronomy – studies the universe beyond Earth, including its formation and development, and the evolution, physics, chemistry, meteorology, and motion of celestial objects (such as galaxies, planets, etc.) and phenomena that originate outside the atmosphere of Earth (such as the cosmic background radiation).
    • Astrodynamics – application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft.
    • Astrometry – the branch of astronomy that involves precise measurements of the positions and movements of stars and other celestial bodies.
    • Astrophysics – the study of the physical aspects of celestial objects
    • Celestial mechanics – the branch of theoretical astronomy that deals with the calculation of the motions of celestial objects such as planets.
    • Extragalactic astronomy – the branch of astronomy concerned with objects outside our own Milky Way Galaxy
    • Galactic astronomy – the study of our own Milky Way galaxy and all its contents.
    • Physical cosmology – the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution.
    • Planetary science – the scientific study of planets (including Earth), moons, and planetary systems, in particular those of the Solar System and the processes that form them.
    • Stellar astronomy – natural science that deals with the study of celestial objects (such as stars, planets, comets, nebulae, star clusters, and galaxies) and phenomena that originate outside the atmosphere of Earth (such as cosmic background radiation)
  • Atmospheric physics – the study of the application of physics to the atmosphere
  • Atomic, molecular, and optical physics – the study of how matter and light interact
    • Optics – the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it.
  • Biophysics – interdisciplinary science that uses the methods of physics to study biological systems
    • Neurophysics – branch of biophysics dealing with the nervous system.
    • Polymer physics – field of physics that studies polymers, their fluctuations, mechanical properties, as well as the kinetics of reactions involving degradation and polymerization of polymers and monomers respectively.
    • Quantum biology – application of quantum mechanics to biological phenomenon.
  • Chemical physics – the branch of physics that studies chemical processes from physics.
  • Computational physics – study and implementation of numerical algorithms to solve problems in physics for which a quantitative theory already exists.
  • Condensed matter physics – the study of the physical properties of condensed phases of matter.
  • Electricity – the study of electrical phenomena.
  • Electromagnetism – branch of science concerned with the forces that occur between electrically charged particles.
  • Geophysics – the physics of the Earth and its environment in space; also the study of the Earth using quantitative physical methods
  • Magnetism – the study of physical phenomena that are mediated by magnetic field.
  • Mathematical physics – application of mathematics to problems in physics and the development of mathematical methods for such applications and the formulation of physical theories.
  • Mechanics – the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment.
    • Aerodynamics – study of the motion of air.
    • Biomechanics – the study of the structure and function of biological systems such as humans, animals, plants, organs, and cells using the methods of mechanics.
    • Classical mechanics – one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces.
    • Continuum mechanics – the branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles.
    • Dynamics – the study of the causes of motion and changes in motion
    • Fluid mechanics – the study of fluids and the forces on them.
    • Statics – the branch of mechanics concerned with the analysis of loads (force, torque/moment) on physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at a constant velocity.
  • Medical Physics – the branch of physics that deals with the application of physics in medicine – such as imaging exams (NMR, PET scans, and so on), radiotherapy and nuclear medicine.
  • Statistical mechanics – the branch of physics which studies any physical system that has a large number of degrees of freedom.
  • Thermodynamics – the branch of physical science concerned with heat and its relation to other forms of energy and work.
  • Nuclear physics – field of physics that studies the building blocks and interactions of atomic nuclei.
  • Particle physics – the branch of physics that studies the properties and interactions of the fundamental constituents of matter and energy.
  • Psychophysics – quantitatively investigates the relationship between physical stimuli and the sensations and perceptions they affect.
  • Plasma physics – the study of plasma, a state of matter similar to gas in which a certain portion of the particles are ionized.
  • Quantum physics – branch of physics dealing with physical phenomena where the action is on the order of the Planck constant.
    • Quantum field theory – the application of quantum theory to the study of fields (systems with infinite degrees of freedom).
    • Quantum information theory – the study of the information-processing capabilities afforded by quantum mechanics.
    • Quantum foundations – the discipline focusing in understanding the counterintuitive aspects of the theory, including trying to find physical principles underlying them, and proposing generalisations of quantum theory.
  • Quantum gravity – the search for an account of gravitation fully compatible with quantum theory.
  • Relativity – theory of physics which describes the relationship between space and time.
  • Other
    • Agrophysics – the study of physics applied to agroecosystems
      • Soil physics – the study of soil physical properties and processes.
    • Cryogenics – cryogenics is the study of the production of very low temperature (below −150 °C, −238 °F or 123 K) and the behavior of materials at those temperatures.
    • Econophysics – interdisciplinary research field, applying theories and methods originally developed by physicists to solve problems in economics
    • Materials physics – use of physics to describe materials in many different ways such as force, heat, light, and mechanics.
    • Vehicle dynamics – dynamics of vehicles, here assumed to be ground vehicles.
  • Philosophy of physics – deals with conceptual and interpretational issues in modern physics, many of which overlap with research done by certain kinds of theoretical physicists.

History

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History of physics – history of the physical science that studies matter and its motion through space-time, and related concepts such as energy and force

History of fields of physics

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General concepts

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Basic principles

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Physics – branch of science that studies matter[9] and its motion through space and time, along with related concepts such as energy and force.[10] Physics is one of the "fundamental sciences" because the other natural sciences (like biology, geology etc.) deal with systems that seem to obey the laws of physics. According to physics, the physical laws of matter, energy and the fundamental forces of nature govern the interactions between particles and physical entities (such as planets, molecules, atoms or the subatomic particles). Some of the basic pursuits of physics, which include some of the most prominent developments in modern science in the last millennium, include:

Gravity, light, physical system, physical observation, physical quantity, physical state, physical unit, physical theory, physical experiment

Theoretical concepts: mass–energy equivalence, elementary particle, physical law, fundamental force, physical constant

Fundamental concepts

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Measurement

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Motion

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Overview

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This is a list of the primary theories in physics, major subtopics, and concepts.

Note: the Theory column below contains links to articles with infoboxes at the top of their respective pages which list the major concepts.
Theory Major subtopics Concepts
Classical mechanics Newton's laws of motion, Lagrangian mechanics, Hamiltonian mechanics, kinematics, statics, dynamics, chaos theory, acoustics, fluid dynamics, continuum mechanics Density, dimension, gravity, space, time, motion, length, position, velocity, acceleration, mass, momentum, force, energy, angular momentum, torque, conservation law, harmonic oscillator, wave, work, power
Electromagnetism Electrostatics, electrodynamics, electricity, magnetism, Maxwell's equations, optics Capacitance, electric charge, electric current, electrical conductivity, electric field, electric permittivity, electrical resistance, electromagnetic field, electromagnetic induction, electromagnetic radiation, Gaussian surface, magnetic field, magnetic flux, magnetic monopole, magnetic permeability
Theory of relativity Special relativity, general relativity, Einstein field equations Covariance, Einstein manifold, equivalence principle, four-momentum, four-vector, general principle of relativity, geodesic motion, gravity, gravitoelectromagnetism, inertial frame of reference, invariance, length contraction, Lorentzian manifold, Lorentz transformation, metric, Minkowski diagram, Minkowski space, principle of relativity, proper length, proper time, reference frame, rest energy, rest mass, relativity of simultaneity, spacetime, special principle of relativity, speed of light, stress–energy tensor, time dilation, twin paradox, world line
Thermodynamics and statistical mechanics Heat engine, kinetic theory Boltzmann constant, conjugate variables, enthalpy, entropy, equation of state, equipartition theorem, first law of thermodynamics, free energy, heat, ideal gas law, internal energy, irreversible process, partition function, pressure, reversible process, second law of thermodynamics, spontaneous process, state function, statistical ensemble, temperature, thermodynamic equilibrium, thermodynamic potential, thermodynamic processes, thermodynamic state, thermodynamic system, third law of thermodynamics, viscosity, zeroth law of thermodynamics
Quantum mechanics Path integral formulation, scattering theory, Schrödinger equation, quantum field theory, quantum statistical mechanics Adiabatic approximation, correspondence principle, free particle, Hamiltonian, Hilbert space, identical particles, matrix mechanics, Planck constant, operators, quanta, quantization, quantum entanglement, quantum harmonic oscillator, quantum number, quantum tunneling, Schrödinger's cat, Dirac equation, spin, wavefunction, wave mechanics, wave–particle duality, zero-point energy, Pauli exclusion principle, Heisenberg uncertainty principle

Concepts by field

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Field Subfields Major theories Concepts
Particle physics Accelerator physics, nuclear physics, nuclear astrophysics, particle astrophysics, particle physics phenomenology Standard Model, quantum field theory, quantum chromodynamics, electroweak theory, effective field theory, lattice field theory, lattice gauge theory, gauge theory, supersymmetry, Grand Unified Theory, superstring theory, M-theory Fundamental force (gravitational, electromagnetic, weak, strong), elementary particle, spin, antimatter, spontaneous symmetry breaking, brane, string, quantum gravity, theory of everything, vacuum energy
Atomic, molecular, and optical physics Atomic physics, molecular physics, atomic and molecular astrophysics, chemical physics, optics, photonics Quantum optics, quantum chemistry, quantum information science Atom, molecule, diffraction, electromagnetic radiation, laser, polarization, spectral line, Casimir effect
Condensed matter physics Solid state physics, high pressure physics, low-temperature physics, nanoscale and mesoscopic physics, polymer physics BCS theory, Bloch's theorem, Fermi gas, Fermi liquid, many-body theory Phases (gas, liquid, solid, Bose–Einstein condensate, superconductor, superfluid), electrical conduction, magnetism, self-organization, spin, spontaneous symmetry breaking
Astrophysics Cosmology, gravitation physics, high-energy astrophysics, planetary astrophysics, plasma physics, space physics, stellar astrophysics Big Bang, Lambda-CDM model, cosmic inflation, general relativity, law of universal gravitation Black hole, cosmic background radiation, cosmic string, cosmos, dark energy, dark matter, galaxy, gravity, gravitational radiation, gravitational singularity, planet, Solar System, star, supernova, universe, nova

Lists

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Index of physics articles

See also

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Notes

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Outline of physics

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Physics is the most fundamental natural science, dedicated to understanding the behavior of matter, energy, space, and time through observation, experimentation, and mathematical modeling. The outline of physics serves as a structured framework for this discipline, systematically categorizing its core principles, historical milestones, major branches, and interdisciplinary connections to provide a comprehensive roadmap for studying the underlying laws governing the universe. At its core, the outline delineates , which encompasses (the study of motion and forces, as formulated by ), electromagnetism (interactions of electric and magnetic fields, unified by James Clerk Maxwell), and (principles of heat, work, and energy transfer). These foundational areas, developed primarily between the 17th and 19th centuries, explain macroscopic phenomena and form the basis for engineering and everyday technologies like engines and electrical systems. In contrast, extends the outline to 20th-century revolutions, including special and (Albert Einstein's theories on space-time and ) and (the probabilistic behavior of particles at atomic scales). Subfields such as (structure and reactions of atomic nuclei), (fundamental particles and forces probed by accelerators), (properties of solids and liquids, enabling semiconductors and superconductors), and (celestial phenomena and cosmology) further branch out, addressing scales from subatomic to cosmic. The importance of physics, as highlighted in its outline, lies in its universal laws—such as and —that underpin all natural processes and enable predictions across sciences, from chemical reactions in to structural designs in and innovations like MRI machines in . Historically, physics has progressed from ancient inquiries into motion and elements to the scientific method's refinement during the , culminating in quantum and relativistic paradigms that reshaped human understanding of reality. This structured outline not only aids education and research but also underscores physics' role as the cornerstone of technological progress and scientific inquiry.

Nature and Scope

Definition and Objectives

Physics is the natural science that investigates the fundamental laws and principles governing the universe, spanning scales from subatomic particles to vast cosmic structures. It focuses on the behavior of matter, energy, motion, and their interactions across space and time, seeking to uncover the underlying mechanisms that explain observable phenomena. Through systematic observation and experimentation, physics aims to formulate general principles applicable to diverse natural processes. The term "physics" originates from the ancient Greek word physis, meaning "nature," which underscores its dedication to studying the natural world and its inherent properties. The core objectives of physics include explaining natural phenomena via testable theories, predicting outcomes of physical events based on empirical data, and pursuing unification to integrate disparate forces and principles into coherent frameworks. These goals are achieved through the development of mathematical models that encapsulate empirical observations, enabling precise simulations and forecasts. A key distinction between physics and lies in physics's insistence on falsifiable hypotheses—propositions that can be empirically tested and potentially refuted—coupled with quantitative predictions verifiable through experimentation. This empirical rigor sets physics apart from philosophical inquiry, which may explore conceptual foundations without requiring direct observational confirmation. As a result, physics prioritizes reproducible results and measurable outcomes over purely speculative reasoning.

Interdisciplinary Connections

Physics provides the foundational principles for understanding atomic structure, which underpins much of modern chemistry. The of the atom, introduced in 1913, described electrons orbiting the nucleus in quantized energy levels, laying the groundwork for by explaining spectral lines and chemical bonding through physical laws. Similarly, in , applies physical models to cellular processes, such as the Hodgkin-Huxley equations from 1952, which mathematically describe action potentials in neurons via dynamics, revealing how electrical signals propagate in living cells. In engineering, drives advancements in and by leveraging physical properties to design functional materials. For instance, physics enables the development of transistors and integrated circuits, where quantum effects and band theory dictate flow and device performance. This interdisciplinary application transforms into practical technologies, such as photovoltaic cells in systems. Physics overlaps significantly with mathematics, employing tools like differential equations to model dynamic systems and to analyze symmetries. Ordinary and partial differential equations describe phenomena from wave propagation to , bridging mathematical rigor with physical predictions. , pioneered in by Wigner's 1927 work on conservation laws, classifies particle states and symmetries, essential for understanding atomic spectra and fundamental interactions. Emerging interdisciplines further extend physics' reach, including , which integrates algorithms to simulate complex systems like , and , which uses physical principles such as and to probe Earth's interior in earth sciences. A prominent example is , which bridges physics and ; Richard Feynman's 1982 proposal demonstrated that could efficiently simulate physical processes intractable for classical computers, founding the field by exploiting superposition and entanglement.

Historical Development

Ancient and Classical Foundations

The foundations of physics trace back to ancient civilizations, where early observations of natural phenomena laid the groundwork for later theoretical developments. In , Babylonian astronomers made significant contributions through systematic skywatching, recording planetary positions, eclipses, and other celestial events on clay tablets dating from the BCE onward. These records enabled the creation of a and predictive models for astronomical phenomena, influencing subsequent Greek and Hellenistic astronomy. In , philosophical inquiries into the nature of matter and motion emerged around the 5th century BCE. Democritus proposed , theorizing that the universe consists of indivisible particles called atoms moving in a void, explaining observable changes through their combinations and rearrangements without invoking divine intervention. This materialist view contrasted with earlier elemental theories but lacked empirical testing. , in his outlined in works like Physics and (circa 350 BCE), described motion as tied to the four elements—earth, water, air, and fire—each seeking its natural place, with terrestrial objects falling due to heaviness and celestial bodies moving in perfect circles. His framework emphasized qualitative explanations over quantitative measurement, dominating Western thought for centuries. Key Greek figures advanced specific principles that bridged observation and theory. , in his treatise (circa 250 BCE), formulated the principle of buoyancy, stating that an object immersed in a fluid experiences an upward force equal to the weight of the displaced fluid, derived from hydrostatic balance considerations during his analysis of crown density for King Hieron II. In astronomy, Claudius Ptolemy's (circa 150 CE) synthesized geocentric models, placing at the universe's center with planets orbiting via epicycles and deferents to account for observed retrograde motions, providing predictive tables used for over a millennium. Medieval Islamic scholars built on these Greek foundations, integrating observation with mathematical rigor. (Alhazen), in his (Kitāb al-Manāẓir, completed circa 1021 CE), rejected emission theories of vision, arguing instead that light rays emanate from objects to the eye, and conducted experiments on , reflection, and the to demonstrate how images form through pinholes. In , astronomers like (circa 499 CE) proposed Earth rotates on its axis to explain and calculated planetary periods with trigonometric methods in , while (circa 628 CE) refined gravitational concepts and interpolation techniques for ephemerides in Brahmasphutasiddhanta. These works emphasized empirical data over purely metaphysical speculation. The marked a shift toward , prioritizing controlled observations. , in early 17th-century experiments, used inclined planes to study , rolling bronze balls down grooves to measure distance-time relations, demonstrating that speed increases uniformly with time rather than Aristotelian claims of constant velocity for natural motion. , analyzing Tycho Brahe's precise observations, formulated laws of planetary motion in (1609), including the third law stating that the square of a planet's TT is proportional to the of its average distance RR from the Sun (T2∝R3T^2 \propto R^3), derived from harmonic ratios in elliptical orbits. This era's emphasis on repeatable experiments and mathematical description over a priori metaphysics heralded the transition to the modern .

Scientific Revolution to 19th Century

The marked a pivotal shift in physics, emphasizing mathematical rigor and empirical experimentation over qualitative descriptions from ancient traditions, such as Galileo's experiments demonstrating uniform . This era culminated in Isaac Newton's , published in 1687, which laid the foundations of through three laws of motion. The first law states that an object remains at rest or in uniform motion unless acted upon by an external , establishing the concept of . The second law quantifies the relationship between , , and as F=ma\mathbf{F} = m \mathbf{a}, where is the rate of change of . The third law asserts that for every action, there is an equal and opposite reaction. These laws provided a deterministic framework for understanding motion on Earth and in the heavens. In the same work, Newton introduced the law of universal gravitation, positing that every particle attracts every other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them: F=Gm1m2r2F = G \frac{m_1 m_2}{r^2}, where GG is the . This unified terrestrial and , explaining planetary orbits as conic sections under inverse-square attraction and resolving discrepancies in Kepler's laws. Newton's , detailed in (1704), portrayed light as streams of particles to account for reflection, , and color dispersion through prisms, challenging wave theories and influencing for over a century. The extended Newtonian principles to and . , using a torsion balance, formulated in 1785 the law of electric force between charged particles, analogous to gravitation: F=kq1q2r2F = k \frac{q_1 q_2}{r^2}, where kk is Coulomb's constant, establishing as an interaction. advanced in Mécanique Céleste (1798–1825), applying to predict planetary orbits with high precision, confirming the stability of the solar system under Newtonian gravity and incorporating tidal effects. The integrated these fields through and . Michael Faraday's experiments in 1831 demonstrated , showing that a changing induces an in a circuit, laying the groundwork for electric generators and transformers without a specific formula but through qualitative laws of induction. In , Sadi Carnot's Reflections on the Motive Power of Fire (1824) analyzed ideal heat engines, deriving the maximum efficiency as η=1−TcTh\eta = 1 - \frac{T_c}{T_h}, where TcT_c and ThT_h are the absolute temperatures of the cold and hot reservoirs, respectively, introducing the second law in caloric terms. James Prescott Joule's paddle-wheel experiments in the quantitatively established the mechanical equivalent of , measuring approximately 4.18 joules per calorie and affirming by showing work converts directly to . James Clerk Maxwell unified , , and in the 1860s, culminating in his 1865 paper "A Dynamical of the ," where he formulated equations showing that varying electric and magnetic fields propagate as waves at the , implying is an electromagnetic phenomenon and completing the classical synthesis.

20th Century and Beyond

The marked a profound transformation in physics, driven by two foundational theories that challenged classical notions of space, time, and matter. Albert Einstein's special , published in 1905, revolutionized understanding by positing that the laws of physics are invariant across inertial frames and that the is constant, leading to the famous mass-energy equivalence . This reconciled with , predicting phenomena like time and length . Complementing this, Einstein's general , finalized in 1915, extended these ideas to accelerated frames through the , which states that the effects of are indistinguishable from , describing as the curvature of caused by and . These frameworks provided a new geometric interpretation of the universe, enabling predictions such as the bending of by , confirmed during the 1919 . Parallel to relativity, the quantum revolution emerged to address atomic-scale phenomena, beginning with Max Planck's 1900 introduction of the constant hh (Planck's constant) to explain , where energy is emitted in discrete quanta rather than continuously. Niels Bohr's 1913 atomic model built on this by proposing quantized electron orbits in atoms, resolving the stability of matter and emissions without classical radiation losses. The theory matured with Erwin Schrödinger's 1926 , iℏ∂ψ∂t=H^ψi\hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi, which describes the time evolution of a quantum system's ψ\psi, incorporating probabilistic interpretations of particle behavior and superposition. These developments formed the basis of , revealing a probabilistic, non-deterministic at microscopic scales, fundamentally altering physics from classical . In , the mid-20th century saw the formulation of the , a unifying electromagnetic, weak, and strong nuclear forces, developed through seminal works by , , and in the late 1960s and 1970s. This model predicted the , a scalar particle responsible for electroweak and imparting to fundamental particles via the . Its discovery was confirmed in 2012 by the ATLAS and CMS experiments at CERN's , with the boson observed decaying into pairs of photons and other particles at a mass of approximately 125 GeV. The has since been rigorously tested, accounting for three of the four fundamental forces with extraordinary precision. Post-2000 milestones have further validated these theories while probing their limits. The first direct detection of in 2015 by the collaboration confirmed general relativity's predictions for merging black holes, opening multimessenger astronomy and revealing the abundance of stellar-mass black holes. In 2019, the Event Horizon Telescope imaged the shadow of the in M87, providing visual evidence of curvature around event horizons. experiments, culminating in loophole-free Bell tests and the 2022 awarded to , , and , have affirmed , ruling out local hidden variables and enabling applications in . Subsequent s highlighted further quantum advancements: in 2023, , , and Anne L’Huillier were recognized for experimental methods generating pulses of light, enabling the study of dynamics on ultrafast timescales; in 2024, John J. Hopfield and Geoffrey E. Hinton received the prize for foundational discoveries and inventions enabling with artificial neural networks, applying statistical physics to AI; and in 2025, John Clarke, Michel H. Devoret, and John M. Martinis were awarded for discoveries of in superconducting electrical circuits, foundational to . Efforts to unify quantum mechanics and general relativity persist as key pursuits beyond the 20th century. String theory posits that fundamental particles are one-dimensional vibrating strings in higher dimensions, potentially reconciling with quantum forces through and , with anomaly cancellation demonstrated in the 1984 Green-Schwarz mechanism. , an alternative approach, quantizes itself into discrete loops, emerging from Abhay Ashtekar's 1986 reformulation of variables, aiming to resolve singularities like those in black holes without additional dimensions. These ongoing theories represent high-impact attempts at a quantum theory of gravity, though experimental verification remains elusive.

Branches of Physics

Mechanics

Mechanics is the branch of physics that studies the motion of macroscopic bodies and their equilibrium under the action of forces, providing foundational principles for understanding physical systems from everyday objects to celestial bodies. , the core of this field, assumes non-relativistic speeds and deterministic trajectories. This discipline emphasizes forces as causes of motion changes, enabling predictions of system behavior through mathematical formulations. The cornerstone of classical mechanics is Isaac Newton's three laws of motion, articulated in his 1687 work Philosophiæ Naturalis Principia Mathematica. The first law, or law of inertia, states that an object at rest remains at rest, and an object in uniform rectilinear motion continues indefinitely unless compelled to change by external forces. The second law posits that the net force on an object equals the time rate of change of its linear momentum, directed along the force: F=dpdt,\mathbf{F} = \frac{d\mathbf{p}}{dt}, where momentum p\mathbf{p} for a body of constant mass mm and velocity v\mathbf{v} is derived as p=mv\mathbf{p} = m \mathbf{v}. This expression for momentum arises directly from Newton's definition of "quantity of motion" as a measure proportional to the product of mass and velocity, with the second law quantifying acceleration a=dv/dt\mathbf{a} = d\mathbf{v}/dt as F=ma\mathbf{F} = m \mathbf{a} when mass is invariant. The third law declares that every action has an equal and opposite reaction, meaning forces between interacting bodies are mutual and collinear. Lagrangian mechanics offers an alternative, coordinate-independent formulation of , developed by in his 1788 treatise Mécanique Analytique. Here, the Lagrangian function LL is defined as the difference between TT and VV: L=T−V.L = T - V. The dynamics follow from the Euler-Lagrange equations for qiq_i: ddt(∂L∂q˙i)−∂L∂qi=0,\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0, where q˙i=dqi/dt\dot{q}_i = dq_i/dt. These equations derive from the principle of stationary action, integrating the Lagrangian over time paths to yield equations of motion equivalent to Newton's laws but more versatile for complex constraints. For rigid bodies—systems where distances between points remain fixed—dynamics extend to rotational motion, as pioneered by Leonhard Euler in his 1765 Theoria motus corporum solidorum. The rotational analog of Newton's second law relates net τ\boldsymbol{\tau} to the rate of change of L\mathbf{L}, yielding τ=Iα\boldsymbol{\tau} = I \boldsymbol{\alpha} for rotation about a principal axis, where II is the and α\boldsymbol{\alpha} is . conservation, dLdt=0\frac{d\mathbf{L}}{dt} = 0 in the absence of external torques, follows as a direct consequence, with L=Iω\mathbf{L} = I \boldsymbol{\omega} for ω\boldsymbol{\omega}. These principles govern phenomena like spinning tops and planetary s. Applications of mechanics abound in macroscopic systems. , such as a cannonball's parabolic path under constant , emerges from Newton's second , yielding range R=v02sin⁡2θgR = \frac{v_0^2 \sin 2\theta}{g} for initial speed v0v_0 and angle θ\theta, with gg . In , Newton derived Johannes Kepler's three empirical laws from his universal gravitation in Principia Book III: the first (elliptical orbits with the sun at one focus), second (equal areas swept in equal times, implying conserved ), and third (T2∝a3T^2 \propto a^3, relating period TT to semi-major axis aa). These derivations confirm inverse-square 's role in stable planetary motion. Energy concepts in , like work-energy theorems, provide a bridge to thermodynamic analyses of and efficiency.

Thermodynamics and Statistical Mechanics

Thermodynamics is the branch of physics that describes the relationships between , work, and other forms of in macroscopic systems, particularly those at or near . It provides a framework for understanding processes involving energy transfer and transformation, without delving into the underlying microscopic details. Statistical mechanics, on the other hand, offers a microscopic interpretation of thermodynamic phenomena by applying probability and statistics to large ensembles of particles, bridging the gap between atomic-scale behaviors and observable macroscopic properties. Together, these fields explain why flows from hot to cold objects and how systems evolve toward equilibrium states. The establishes the concept of as a measurable that defines . It states that if two systems are each in with a third system, then they are in with each other, allowing for the construction of consistent temperature scales such as the or scales. This law, formalized by Ralph H. Fowler in 1931 based on earlier ideas from James Clerk Maxwell and others, underpins thermometry and the notion of as an intensive independent of system size. The first law of thermodynamics expresses the in thermal systems, stating that the change in ΔU of a system equals the added Q minus the work done by the system W, or ΔU = Q - W. This principle, rooted in the work of Julius Robert von Mayer, , and in the mid-19th century, implies that is neither created nor destroyed in thermodynamic processes, only converted between forms like and mechanical work. For an , this law combines with the PV = nRT—where P is , V is , n is the number of moles, R is the , and T is —to describe how depends solely on for such systems. The second law of thermodynamics introduces the concept of S, a measure of disorder or the number of microscopic configurations consistent with a macroscopic state. It asserts that the of an never decreases; instead, ΔS ≥ 0 for any , with equality holding only for reversible processes. Formulated by in 1850 and later interpreted statistically by , this law explains the directionality of natural processes, such as the irreversible mixing of gases or the dissipation of heat, and sets limits on the efficiency of heat engines. The third law of thermodynamics states that the entropy of a perfect approaches a minimum value (often taken as zero) as approaches , implying that it becomes increasingly difficult to remove from a at low temperatures. Proposed by in 1906–1912 and later refined by , this law has profound implications for absolute calculations and the unattainability of in finite steps. provides a foundational link to by deriving macroscopic laws from the probabilistic behavior of microscopic particles. In the , the probability P of a occupying a state with E at T follows the P ∝ e^{-E/kT}, where k is Boltzmann's constant. This distribution, developed by in the late 19th century, arises from maximizing subject to constraints and predicts that lower-energy states are more probable at any finite . The partition function Z = ∑ e^{-E_i/kT}, summing over all possible states i, encapsulates the statistical weight of the system and allows computation of thermodynamic quantities like average ⟨E⟩ = -∂(ln Z)/∂β, where β = 1/kT. Phase transitions occur when a undergoes a qualitative change in structure or properties, such as or , driven by variations in , , or composition. At a critical point, the distinction between phases vanishes, leading to phenomena like critical where fluctuations become long-range. The G = H - TS, with H as and S as , serves as the that minimizes at equilibrium under constant and ; phase stability is determined by comparing G values across phases. These concepts, advanced by in the 1870s, enable predictions of transition temperatures and the behavior of mixtures. Applications of thermodynamics and statistical mechanics are central to engineering and natural phenomena. Heat engines, such as the , convert into mechanical work with maximum efficiency η = 1 - T_c/T_h, where T_h and T_c are the hot and cold reservoir temperatures, illustrating the second law's constraints. Refrigerators and heat pumps reverse this to achieve cooling, albeit with work input exceeding the heat extracted by the Carnot limit. In radiative es, —modeled classically by the Rayleigh-Jeans law but requiring quantum corrections—exemplifies how statistical mechanics preludes quantum theory by revealing ultraviolet divergences resolved by Max Planck's 1900 hypothesis of energy quantization. Conservation laws, particularly , underlie these energy transformations in thermal contexts.

Electromagnetism and Optics

Electromagnetism encompasses the study of electric and magnetic fields and their interactions with matter, forming a cornerstone of classical physics. Electric fields arise from electric charges, exerting forces on other charges according to Coulomb's law, which states that the force FF between two point charges q1q_1 and q2q_2 separated by distance rr is F=14πϵ0q1q2r2F = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2}, where ϵ0\epsilon_0 is the vacuum permittivity. Magnetic fields, generated by moving charges or currents, produce forces on other currents as described by Ampère's law, which relates the magnetic field around a closed loop to the current enclosed: ∮B⋅dl=μ0I\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I, with μ0\mu_0 the vacuum permeability. These phenomena, initially treated separately, were unified in the 19th century, revealing that electricity and magnetism are aspects of a single electromagnetic force. The unification culminated in , a set of four differential equations that govern electromagnetic fields. In their modern vector form, formulated by , they are: ∇⋅E=ρϵ0\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} ∇⋅B=0\nabla \cdot \mathbf{B} = 0 ∇×E=−∂B∂t\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} ∇×B=μ0J+μ0ϵ0∂E∂t\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} These equations, derived from Maxwell's original 20 scalar equations in his 1865 paper, predict that changing electric fields generate magnetic fields and vice versa, leading to self-sustaining electromagnetic waves propagating at speed c=1μ0ϵ0≈3×108c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8
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