Classical mechanics is a

`S'`. For observers in each of the reference frames an event has space-time coordinates of (''x'',''y'',''z'',''t'') in frame ''S'' and (`x'`,`y'`,`z'`,`t'`) in frame `S'`. Assuming time is measured the same in all reference frames, if we require when , then the relation between the space-time coordinates of the same event observed from the reference frames `S'` and ''S'', which are moving at a relative velocity ''u'' in the ''x'' direction, is:
:$x\text{'}\; =\; x\; -\; u\; t\; \backslash ,$
:$y\text{'}\; =\; y\; \backslash ,$
:$z\text{'}\; =\; z\; \backslash ,$
:$t\text{'}\; =\; t\; \backslash ,\; .$
This set of formulas defines a group transformation known as the Galilean transformation (informally, the ''Galilean transform''). This group is a limiting case of the Poincaré group used in

_{0} is the initial velocity. This means that the velocity of this particle decays exponentially to zero as time progresses. In this case, an equivalent viewpoint is that the kinetic energy of the particle is absorbed by friction (which converts it to heat energy in accordance with the

_{1} to r_{2} along a path ''C'', the work done on the particle is given by the _{1} to r_{2} is the same no matter what path is taken, the force is said to be _{k} of a particle of mass ''m'' travelling at speed ''v'' is given by
: $E\_\backslash mathrm\; =\; \backslash tfracmv^2\; \backslash ,\; .$
For extended objects composed of many particles, the kinetic energy of the composite body is the sum of the kinetic energies of the particles.
The work–energy theorem states that for a particle of constant mass ''m'', the total work ''W'' done on the particle as it moves from position r_{1} to r_{2} is equal to the change in _{k} of the particle:
:$W\; =\; \backslash Delta\; E\_\backslash mathrm\; =\; E\_\backslash mathrm\; -\; E\_\backslash mathrm\; =\; \backslash tfrac\; m\; \backslash left(v\_2^\; -\; v\_1^\backslash right)\; .$
Conservative forces can be expressed as the _{p}:
: $\backslash mathbf\; =\; -\; \backslash mathbf\; E\_\backslash mathrm\; \backslash ,\; .$
If all the forces acting on a particle are conservative, and ''E''_{p} is the total potential energy (which is defined as a work of involved forces to rearrange mutual positions of bodies), obtained by summing the potential energies corresponding to each force
: $\backslash mathbf\; \backslash cdot\; \backslash Delta\; \backslash mathbf\; =\; -\; \backslash mathbf\; E\_\backslash mathrm\; \backslash cdot\; \backslash Delta\; \backslash mathbf\; =\; -\; \backslash Delta\; E\_\backslash mathrm\; \backslash ,\; .$
The decrease in the potential energy is equal to the increase in the kinetic energy
: $-\backslash Delta\; E\_\backslash mathrm\; =\; \backslash Delta\; E\_\backslash mathrm\; \backslash Rightarrow\; \backslash Delta\; (E\_\backslash mathrm\; +\; E\_\backslash mathrm)\; =\; 0\; \backslash ,\; .$
This result is known as ''conservation of energy'' and states that the total

^{2}, where ''c'' is the speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...

in free space.

^{ /sup>
The Newtonian approximation to special relativity
In special relativity, the momentum of a particle is given by
:$\backslash mathbf\; =\; \backslash frac\; \backslash ,\; ,$
where ''m'' is the particle's rest mass, v its velocity, ''v'' is the modulus of v, and ''c'' is the speed of light.
If ''v'' is very small compared to ''c'', ''v''2/''c''2 is approximately zero, and so
:$\backslash mathbf\; \backslash approx\; m\backslash mathbf\; \backslash ,\; .$
Thus the Newtonian equation is an approximation of the relativistic equation for bodies moving with low speeds compared to the speed of light.
For example, the relativistic cyclotron frequency of a cyclotron
A cyclotron is a type of particle accelerator invented by Ernest O. Lawrence in 1929–1930 at the University of California, Berkeley, and patented in 1932. Lawrence, Ernest O. ''Method and apparatus for the acceleration of ions'', filed: Ja ..., gyrotron, or high voltage magnetron
The cavity magnetron is a high-power vacuum tube used in early radar systems and currently in microwave ovens and linear particle accelerators. It generates microwaves using the interaction of a stream of electrons with a magnetic field whil ... is given by
:$f\; =\; f\_\backslash mathrm\backslash frac\; \backslash ,\; ,$
where ''f''c is the classical frequency of an electron (or other charged particle) with kinetic energy ''T'' and (rest
Rest or REST may refer to:
Relief from activity
* Sleep
** Bed rest
* Kneeling
* Lying (position)
* Sitting
* Squatting position
Structural support
* Structural support
** Rest (cue sports)
** Armrest
** Headrest
** Footrest
Arts and ...) mass ''m''0 circling in a magnetic field. The (rest) mass of an electron is 511 keV. So the frequency correction is 1% for a magnetic vacuum tube with a 5.11 kV direct current accelerating voltage.
The classical approximation to quantum mechanics
The ray approximation of classical mechanics breaks down when the de Broglie wavelength
Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wa ... is not much smaller than other dimensions of the system. For non-relativistic particles, this wavelength is
:$\backslash lambda=\backslash frac$
where ''h'' is Planck's constant and ''p'' is the momentum.
Again, this happens with electrons
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ... before it happens with heavier particles. For example, the electrons used by Clinton Davisson and Lester Germer in 1927, accelerated by 54 V, had a wavelength of 0.167 nm, which was long enough to exhibit a single diffraction
Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ... side lobe when reflecting from the face of a nickel crystal
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, ma ... with atomic spacing of 0.215 nm. With a larger vacuum chamber
A vacuum chamber is a rigid enclosure from which air and other gases are removed by a vacuum pump. This results in a low-pressure environment within the chamber, commonly referred to as a vacuum. A vacuum environment allows researchers to co ..., it would seem relatively easy to increase the angular resolution
Angular resolution describes the ability of any image-forming device such as an optical or radio telescope, a microscope, a camera, or an eye, to distinguish small details of an object, thereby making it a major determinant of image resolu ... from around a radian to a milliradian
A milliradian ( SI-symbol mrad, sometimes also abbreviated mil) is an SI derived unit for angular measurement which is defined as a thousandth of a radian (0.001 radian). Milliradians are used in adjustment of firearm sights by adjustin ... and see quantum diffraction from the periodic patterns of integrated circuit
An integrated circuit or monolithic integrated circuit (also referred to as an IC, a chip, or a microchip) is a set of electronic circuits on one small flat piece (or "chip") of semiconductor material, usually silicon. Large numbers of tin ... computer memory.
More practical examples of the failure of classical mechanics on an engineering scale are conduction by quantum tunneling
In physics, a quantum (plural quanta) is the minimum amount of any physical entity ( physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantiza ... in tunnel diode
A tunnel diode or Esaki diode is a type of semiconductor diode that has effectively " negative resistance" due to the quantum mechanical effect called tunneling. It was invented in August 1957 by Leo Esaki, Yuriko Kurose, and Takashi Su ...s and very narrow transistor
upright=1.4, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an insulating layer (pink).
A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch e ... gates
Gates is the plural of gate, a point of entry to a space which is enclosed by walls. It may also refer to:
People
* Gates (surname), various people with the last name
* Gates Brown (1939-2013), American Major League Baseball player
* Gates McFa ... in integrated circuit
An integrated circuit or monolithic integrated circuit (also referred to as an IC, a chip, or a microchip) is a set of electronic circuits on one small flat piece (or "chip") of semiconductor material, usually silicon. Large numbers of tin ...s.
Classical mechanics is the same extreme high frequency approximation as geometric optics
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca .... It is more often accurate because it describes particles and bodies with rest mass
The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, .... These have more momentum and therefore shorter De Broglie wavelengths than massless particles, such as light, with the same kinetic energies.
History
The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in science
Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earliest archeological evidenc ..., engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ..., and technology
Technology is the application of knowledge to reach practical goals in a specifiable and reproducible way. The word ''technology'' may also mean the product of such an endeavor. The use of technology is widely prevalent in medicine, science ....
Some Greek philosophers of antiquity, among them Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ..., founder of Aristotelian physics, may have been the first to maintain the idea that "everything happens for a reason" and that theoretical principles can assist in the understanding of nature. While to a modern reader, many of these preserved ideas come forth as eminently reasonable, there is a conspicuous lack of both mathematical theory
A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may ... and controlled experiment
An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs whe ..., as we know it. These later became decisive factors in forming modern science, and their early application came to be known as classical mechanics. In his ''Elementa super demonstrationem ponderum'', medieval mathematician Jordanus de Nemore
Jordanus de Nemore (fl. 13th century), also known as Jordanus Nemorarius and Giordano of Nemi, was a thirteenth-century European mathematician and scientist. The literal translation of Jordanus de Nemore (Giordano of Nemi) would indicate that he w ... introduced the concept of "positional gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the str ..." and the use of component forces.
The first published causal
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the ca ... explanation of the motions of planets
A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a y ... was Johannes Kepler's ''Astronomia nova
''Astronomia nova'' (English: ''New Astronomy'', full title in original Latin: ) is a book, published in 1609, that contains the results of the astronomer Johannes Kepler's ten-year-long investigation of the motion of Mars.
One of the most si ...,'' published in 1609. He concluded, based on Tycho Brahe
Tycho Brahe ( ; born Tyge Ottesen Brahe; generally called Tycho (14 December 154624 October 1601) was a Danish astronomer, known for his comprehensive astronomical observations, generally considered to be the most accurate of his time. He was ...'s observations on the orbit of Mars
Mars is the fourth planet from the Sun and the second-smallest planet in the Solar System, only being larger than Mercury. In the English language, Mars is named for the Roman god of war. Mars is a terrestrial planet with a thin at ..., that the planet's orbits were ellipse
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse ...s. This break with ancient thought was happening around the same time that Galileo
Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He w ... was proposing abstract mathematical laws for the motion of objects. He may (or may not) have performed the famous experiment of dropping two cannonballs of different weights from the tower of Pisa, showing that they both hit the ground at the same time. The reality of that particular experiment is disputed, but he did carry out quantitative experiments by rolling balls on an inclined plane
An inclined plane, also known as a ramp, is a flat supporting surface tilted at an angle from the vertical direction, with one end higher than the other, used as an aid for raising or lowering a load. The inclined plane is one of the six clas .... His theory of accelerated motion was derived from the results of such experiments and forms a cornerstone of classical mechanics. In 1673 Christiaan Huygens
Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists ... described in his ''Horologium Oscillatorium
( English: ''The Pendulum Clock: or Geometrical Demonstrations Concerning the Motion of Pendula as Applied to Clocks'') is a book published by Dutch physicist Christiaan Huygens in 1673 and his major work on pendulums and horology. It is regard ...'' the first two laws of motion. The work is also the first modern treatise in which a physical problem (the accelerated motion of a falling body) is idealized by a set of parameters then analyzed mathematically and constitutes one of the seminal works of applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematica ....
Newton founded his principles of natural philosophy on three proposed laws of motion: the law of inertia, his second law of acceleration (mentioned above), and the law of action and reaction; and hence laid the foundations for classical mechanics. Both Newton's second and third laws were given the proper scientific and mathematical treatment in Newton's ''Philosophiæ Naturalis Principia Mathematica
( English: ''Mathematical Principles of Natural Philosophy'') often referred to as simply the (), is a book by Isaac Newton that expounds Newton's laws of motion and his law of universal gravitation. The ''Principia'' is written in Latin ....'' Here they are distinguished from earlier attempts at explaining similar phenomena, which were either incomplete, incorrect, or given little accurate mathematical expression. Newton also enunciated the principles of conservation of momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ... and angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed sy .... In mechanics, Newton was also the first to provide the first correct scientific and mathematical formulation of gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the str ... in Newton's law of universal gravitation
Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the dist .... The combination of Newton's laws of motion and gravitation provide the fullest and most accurate description of classical mechanics. He demonstrated that these laws apply to everyday objects as well as to celestial objects. In particular, he obtained a theoretical explanation of Kepler's laws of motion of the planets.
Newton had previously invented the calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of ari ..., of mathematics, and used it to perform the mathematical calculations. For acceptability, his book, the ''Principia'', was formulated entirely in terms of the long-established geometric methods, which were soon eclipsed by his calculus. However, it was Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathe ... who developed the notation of the derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ... and integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ... preferred today. Newton, and most of his contemporaries, with the notable exception of Huygens, worked on the assumption that classical mechanics would be able to explain all phenomena, including light
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ..., in the form of geometric optics
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca .... Even when discovering the so-called Newton's rings (a wave interference
In physics, interference is a phenomenon in which two waves combine by adding their displacement together at every single point in space and time, to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructi ... phenomenon) he maintained his own corpuscular theory of light.
After Newton, classical mechanics became a principal field of study in mathematics as well as physics. Mathematical formulations progressively allowed finding solutions to a far greater number of problems. The first notable mathematical treatment was in 1788 by Joseph Louis Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaWilliam Rowan Hamilton
Sir William Rowan Hamilton LL.D, DCL, MRIA, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Irel ....
Some difficulties were discovered in the late 19th century that could only be resolved by more modern physics. Some of these difficulties related to compatibility with electromagnetic theory, and the famous Michelson–Morley experiment
The Michelson–Morley experiment was an attempt to detect the existence of the luminiferous aether, a supposed medium permeating space that was thought to be the carrier of light waves. The experiment was performed between April and July 188 .... The resolution of these problems led to the special theory of relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The law ..., often still considered a part of classical mechanics.
A second set of difficulties were related to thermodynamics. When combined with thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of th ..., classical mechanics leads to the Gibbs paradox of classical statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ..., in which entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ... is not a well-defined quantity. Black-body radiation
Black-body radiation is the thermal electromagnetic radiation within, or surrounding, a body in thermodynamic equilibrium with its environment, emitted by a black body (an idealized opaque, non-reflective body). It has a specific, continuous sp ... was not explained without the introduction of quanta. As experiments reached the atomic level, classical mechanics failed to explain, even approximately, such basic things as the energy levels
A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The ... and sizes of atoms
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, an ... and the photo-electric effect. The effort at resolving these problems led to the development of quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry ....
Since the end of the 20th century, classical mechanics in physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ... has no longer been an independent theory. Instead, classical mechanics is now considered an approximate theory to the more general quantum mechanics. Emphasis has shifted to understanding the fundamental forces of nature as in the Standard model
The Standard Model of particle physics is the theory describing three of the four known fundamental forces ( electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. It ... and its more modern extensions into a unified theory of everything
A theory of everything (TOE or TOE/ToE), final theory, ultimate theory, unified field theory or master theory is a hypothetical, singular, all-encompassing, coherent theoretical framework of physics that fully explains and links together all asp .... Classical mechanics is a theory useful for the study of the motion of non-quantum mechanical, low-energy particles in weak gravitational fields. Also, it has been extended into the complex domain where complex classical mechanics exhibits behaviors very similar to quantum mechanics.Complex Elliptic Pendulum Carl M. Bender, Daniel W. Hook, Karta Kooner iAsymptotics in Dynamics, Geometry and PDEs; Generalized Borel Summation vol. I/ref>
Branches
Classical mechanics was traditionally divided into three main branches:
* Statics
Statics is the branch of classical mechanics that is concerned with the analysis of force and torque (also called moment) acting on physical systems that do not experience an acceleration (''a''=0), but rather, are in static equilibrium with ..., the study of equilibrium and its relation to force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as ...s
* Dynamics, the study of motion and its relation to forces
* Kinematics
Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fie ..., dealing with the implications of observed motions without regard for circumstances causing them
Another division is based on the choice of mathematical formalism:
* Newtonian mechanics
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motio ...
* Lagrangian mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lo ...
* Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momen ...
Alternatively, a division can be made by region of application:
* Celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ..., relating to star
A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Eart ...s, planet
A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a ...s and other celestial bodies
* Continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such ..., for materials modelled as a continuum, e.g., solid
Solid is one of the four fundamental states of matter (the others being liquid, gas, and plasma). The molecules in a solid are closely packed together and contain the least amount of kinetic energy. A solid is characterized by structura ...s and fluid
In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any she ...s (i.e., liquid
A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas ...s and gas
Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma).
A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...es).
* Relativistic mechanics
In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities of ... (i.e. including the special and general
A general officer is an officer of high rank in the armies, and in some nations' air forces, space forces, and marines or naval infantry.
In some usages the term "general officer" refers to a rank above colonel."general, adj. and n.". OED ... theories of relativity), for bodies whose speed is close to the speed of light.
* Statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ..., which provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk thermodynamic
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ... properties of materials.
See also
* Dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a ...
* History of classical mechanics
This article deals with the history of classical mechanics.
Precursors to classical mechanics
Antiquity
The ancient Greek philosophers, Aristotle in particular, were among the first to propose that abstract principles govern nature. Aris ...
* List of equations in classical mechanics
* List of publications in classical mechanics
* List of textbooks on classical mechanics and quantum mechanics
* Molecular dynamics
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the ...
* Newton's laws of motion
Newton most commonly refers to:
* Isaac Newton (1642–1726/1727), English scientist
* Newton (unit), SI unit of force named after Isaac Newton
Newton may also refer to:
Arts and entertainment
* ''Newton'' (film), a 2017 Indian film
* Newton ...
* Special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The la ...
* Quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry ...
* Quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particl ...
Notes
References
Further reading
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External links
* Crowell, BenjaminLight and Matter(an introductory text, uses algebra with optional sections involving calculus)
* Fitzpatrick, Richard(uses calculus)
* Hoiland, Paul (2004)Preferred Frames of Reference & Relativity* Horbatsch, Marko, "'".
* Rosu, Haret C., "Classical Mechanics'". Physics Education. 1999. rxiv.org : physics/9909035* Shapiro, Joel A. (2003)Classical Mechanics* Sussman, Gerald Jay & Wisdom, Jack & Mayer, Meinhard E. (2001)Structure and Interpretation of Classical Mechanics* Tong, David(Cambridge lecture notes on Lagrangian and Hamiltonian formalism)
Kinematic Models for Design Digital Library (KMODDL)br /> Movies and photos of hundreds of working mechanical-systems models at Cornell University
Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to .... Also includes ae-book libraryof classic texts on mechanical design and engineering.
MIT OpenCourseWare 8.01: Classical MechanicsFree videos of actual course lectures with links to lecture notes, assignments and exams.
* Alejandro A. Torassa
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physical theory
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...

describing the motion
In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer a ...

of macroscopic
The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying optical instruments. It is the opposite of microscopic.
Overview
When applied to physical phenomena ...

objects, from projectile
A projectile is an object that is propelled by the application of an external force and then moves freely under the influence of gravity and air resistance. Although any objects in motion through space are projectiles, they are commonly found in ...

s to parts of machinery
A machine is a physical system using power to apply forces and control movement to perform an action. The term is commonly applied to artificial devices, such as those employing engines or motors, but also to natural biological macromolecul ...

, and astronomical objects, such as spacecraft
A spacecraft is a vehicle or machine designed to fly in outer space. A type of artificial satellite, spacecraft are used for a variety of purposes, including communications, Earth observation, meteorology, navigation, space colonizati ...

, planets
A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a y ...

, star
A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Eart ...

s, and galaxies
A galaxy is a system of stars, stellar remnants, interstellar gas, dust, dark matter, bound together by gravity. The word is derived from the Greek ' (), literally 'milky', a reference to the Milky Way galaxy that contains the Solar Syste ...

. For objects governed by classical mechanics, if the present state is known, it is possible to predict how it will move in the future (determinism), and how it has moved in the past (reversibility).
The earliest development of classical mechanics is often referred to as Newtonian mechanics
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motio ...

. It consists of the physical concepts based on foundational works of Sir Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the ...

, and the mathematical methods invented by Gottfried Wilhelm Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ...

, Joseph-Louis Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaLeonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...

, and other contemporaries, in the 17th century to describe the motion of bodies under the influence of a system of force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as ...

s. Later, more abstract methods were developed, leading to the reformulations of classical mechanics known as Lagrangian mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lo ...

and Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momen ...

. These advances, made predominantly in the 18th and 19th centuries, extend substantially beyond earlier works, particularly through their use of analytical mechanics
In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the ...

. They are, with some modification, also used in all areas of modern physics.
Classical mechanics provides extremely accurate results when studying large objects that are not extremely massive and speeds not approaching the speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...

. When the objects being examined have about the size of an atom diameter, it becomes necessary to introduce the other major sub-field of mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objec ...

: quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry ...

. To describe velocities that are not small compared to the speed of light, special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The la ...

is needed. In cases where objects become extremely massive, general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...

becomes applicable. However, a number of modern sources do include relativistic mechanics
In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities of ...

in classical physics, which in their view represents classical mechanics in its most developed and accurate form.
Description of the theory

The following introduces the basic concepts of classical mechanics. For simplicity, it often models real-world objects aspoint particle
A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take u ...

s (objects with negligible size). The motion of a point particle is characterized by a small number of parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...

s: its position, mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...

, and the force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as ...

s applied to it. Each of these parameters is discussed in turn.
In reality, the kind of objects that classical mechanics can describe always have a non-zero size. (The physics of ''very'' small particles, such as the electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have n ...

, is more accurately described by quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry ...

.) Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the additional degrees of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...

, e.g., a baseball
Baseball is a bat-and-ball sport played between two teams of nine players each, taking turns batting and fielding. The game occurs over the course of several plays, with each play generally beginning when a player on the fielding ...

can spin while it is moving. However, the results for point particles can be used to study such objects by treating them as composite
Composite or compositing may refer to:
Materials
* Composite material, a material that is made from several different substances
** Metal matrix composite, composed of metal and other parts
** Cermet, a composite of ceramic and metallic materials
...

objects, made of a large number of collectively acting point particles. The center of mass
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force m ...

of a composite object behaves like a point particle.
Classical mechanics uses common sense
''Common Sense'' is a 47-page pamphlet written by Thomas Paine in 1775–1776 advocating independence from Great Britain to people in the Thirteen Colonies. Writing in clear and persuasive prose, Paine collected various moral and political argu ...

notions of how matter and forces exist and interact. It assumes that matter and energy have definite, knowable attributes such as location in space and speed. Non-relativistic mechanics also assumes that forces act instantaneously (see also Action at a distance
In physics, action at a distance is the concept that an object can be affected without being physically touched (as in mechanical contact) by another object. That is, it is the non-local interaction of objects that are separated in space.
Non- ...

).
Position and its derivatives

The ''position'' of apoint particle
A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take u ...

is defined in relation to a coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...

centered on an arbitrary fixed reference point in space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually con ...

called the origin ''O''. A simple coordinate system might describe the position of a particle
In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, from ...

''P'' with a vector
Vector most often refers to:
* Euclidean vector, a quantity with a magnitude and a direction
* Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathemat ...

notated by an arrow labeled r that points from the origin ''O'' to point ''P''. In general, the point particle does not need to be stationary relative to ''O''. In cases where ''P'' is moving relative to ''O'', r is defined as a function of ''t'', time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, ...

. In pre-Einstein relativity (known as Galilean relativity
Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial frames of reference. Galileo Galilei first described this principle in 1632 in his '' Dialogue Concerning the Two Chief World Systems'' using ...

), time is considered an absolute, i.e., the time interval that is observed to elapse between any given pair of events is the same for all observers. In addition to relying on absolute time, classical mechanics assumes Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axio ...

for the structure of space.
Velocity and speed

The ''velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity ...

'', or the rate of change of displacement with time, is defined as the derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

of the position with respect to time:
:$\backslash mathbf\; =\; \backslash ,\backslash !$.
In classical mechanics, velocities are directly additive and subtractive. For example, if one car travels east at 60 km/h and passes another car traveling in the same direction at 50 km/h, the slower car perceives the faster car as traveling east at . However, from the perspective of the faster car, the slower car is moving 10 km/h to the west, often denoted as −10 km/h where the sign implies opposite direction. Velocities are directly additive as vector quantities; they must be dealt with using vector analysis
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...

.
Mathematically, if the velocity of the first object in the previous discussion is denoted by the vector and the velocity of the second object by the vector , where ''u'' is the speed of the first object, ''v'' is the speed of the second object, and d and e are unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vec ...

s in the directions of motion of each object respectively, then the velocity of the first object as seen by the second object is:
:$\backslash mathbf\text{'}\; =\; \backslash mathbf\; -\; \backslash mathbf\; \backslash ,\; .$
Similarly, the first object sees the velocity of the second object as:
:$\backslash mathbf=\; \backslash mathbf\; -\; \backslash mathbf\; \backslash ,\; .$
When both objects are moving in the same direction, this equation can be simplified to:
:$\backslash mathbf\text{'}\; =\; (\; u\; -\; v\; )\; \backslash mathbf\; \backslash ,\; .$
Or, by ignoring direction, the difference can be given in terms of speed only:
:$u\text{'}\; =\; u\; -\; v\; \backslash ,\; .$
Acceleration

The ''acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by th ...

'', or rate of change of velocity, is the derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

of the velocity with respect to time (the second derivative
In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, ...

of the position with respect to time):
:$\backslash mathbf\; =\; =\; .$
Acceleration represents the velocity's change over time. Velocity can change in either magnitude or direction, or both. Occasionally, a decrease in the magnitude of velocity "''v''" is referred to as ''deceleration'', but generally any change in the velocity over time, including deceleration, is referred to as acceleration.
Frames of reference

While the position, velocity and acceleration of aparticle
In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, from ...

can be described with respect to any observer
An observer is one who engages in observation or in watching an experiment.
Observer may also refer to:
Computer science and information theory
* In information theory, any system which receives information from an object
* State observer in co ...

in any state of motion, classical mechanics assumes the existence of a special family of reference frames
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both mathe ...

in which the mechanical laws of nature take a comparatively simple form. These special reference frames are called inertial frames. An inertial frame is an idealized frame of reference within which an object with zero net force acting upon it moves with a constant velocity; that is, it is either at rest or moving uniformly in a straight line.
A key concept of inertial frames is the method for identifying them. For practical purposes, reference frames that do not accelerate with respect to distant stars (an extremely distant point) are regarded as good approximations to inertial frames. Non-inertial reference frames accelerate in relation to an existing inertial frame. They form the basis for Einstein's relativity. Due to the relative motion, particles in the non-inertial frame appear to move in ways not explained by forces from existing fields in the reference frame. Hence, it appears that there are other forces that enter the equations of motion solely as a result of the relative acceleration. These forces are referred to as fictitious force
A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as a linearly accelerating or rotating reference frame.
It is related to Newton's second law of motion, which t ...

s, inertia forces, or pseudo-forces.
Consider two reference frames
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both mathe ...

''S'' and special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The la ...

. The limiting case applies when the velocity ''u'' is very small compared to ''c'', the speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...

.
The transformations have the following consequences:
* v′ = v − u (the velocity v′ of a particle from the perspective of ''S''′ is slower by u than its velocity v from the perspective of ''S'')
* a′ = a (the acceleration of a particle is the same in any inertial reference frame)
* F′ = F (the force on a particle is the same in any inertial reference frame)
* the speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...

is not a constant in classical mechanics, nor does the special position given to the speed of light in relativistic mechanics
In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities of ...

have a counterpart in classical mechanics.
For some problems, it is convenient to use rotating coordinates (reference frames). Thereby one can either keep a mapping to a convenient inertial frame, or introduce additionally a fictitious centrifugal force
In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is paralle ...

and Coriolis force
In physics, the Coriolis force is an inertial or fictitious force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the ...

.
Forces and Newton's second law

A force in physics is any action that causes an object's velocity to change; that is, to accelerate. A force originates from within a field, such as an electro-static field (caused by static electrical charges), electro-magnetic field (caused by moving charges), or gravitational field (caused by mass), among others. Newton was the first to mathematically express the relationship betweenforce
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as ...

and momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass a ...

. Some physicists interpret Newton's second law of motion
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motio ...

as a definition of force and mass, while others consider it a fundamental postulate, a law of nature. Either interpretation has the same mathematical consequences, historically known as "Newton's Second Law":
:$\backslash mathbf\; =\; =\; .$
The quantity ''m''v is called the (canonical
The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical examp ...

) momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass a ...

. The net force on a particle is thus equal to the rate of change of the momentum of the particle with time. Since the definition of acceleration is , the second law can be written in the simplified and more familiar form:
:$\backslash mathbf\; =\; m\; \backslash mathbf\; \backslash ,\; .$
So long as the force acting on a particle is known, Newton's second law is sufficient to describe the motion of a particle. Once independent relations for each force acting on a particle are available, they can be substituted into Newton's second law to obtain an ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...

, which is called the ''equation of motion''.
As an example, assume that friction is the only force acting on the particle, and that it may be modeled as a function of the velocity of the particle, for example:
:$\backslash mathbf\_\; =\; -\; \backslash lambda\; \backslash mathbf\; \backslash ,\; ,$
where ''λ'' is a positive constant, the negative sign states that the force is opposite the sense of the velocity. Then the equation of motion is
:$-\; \backslash lambda\; \backslash mathbf\; =\; m\; \backslash mathbf\; =\; m\; \backslash ,\; .$
This can be integrated to obtain
:$\backslash mathbf\; =\; \backslash mathbf\_0\; e^$
where vconservation of energy
In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means tha ...

), and the particle is slowing down. This expression can be further integrated to obtain the position r of the particle as a function of time.
Important forces include the gravitational force
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...

and the Lorentz force
In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an el ...

for electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...

. In addition, Newton's third law can sometimes be used to deduce the forces acting on a particle: if it is known that particle ''A'' exerts a force F on another particle ''B'', it follows that ''B'' must exert an equal and opposite ''reaction force'', −F, on ''A''. The strong form of Newton's third law requires that F and −F act along the line connecting ''A'' and ''B'', while the weak form does not. Illustrations of the weak form of Newton's third law are often found for magnetic forces.
Work and energy

If a constant force F is applied to a particle that makes a displacement Δr, the ''work done'' by the force is defined as thescalar product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar (mathematics), scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidea ...

of the force and displacement vectors:
: $W\; =\; \backslash mathbf\; \backslash cdot\; \backslash Delta\; \backslash mathbf\; \backslash ,\; .$
More generally, if the force varies as a function of position as the particle moves from rline integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; '' contour integral'' is used as well, a ...

: $W\; =\; \backslash int\_C\; \backslash mathbf(\backslash mathbf)\; \backslash cdot\; \backslash mathrm\backslash mathbf\; \backslash ,\; .$
If the work done in moving the particle from rconservative
Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization ...

. Gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the str ...

is a conservative force, as is the force due to an idealized spring, as given by Hooke's law
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...

. The force due to friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of t ...

is non-conservative.
The kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its ac ...

''E''kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its ac ...

''E''gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...

of a scalar function, known as the potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potenti ...

and denoted ''E''energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of he ...

,
: $\backslash sum\; E\; =\; E\_\backslash mathrm\; +\; E\_\backslash mathrm\; \backslash ,\; ,$
is constant in time. It is often useful, because many commonly encountered forces are conservative.
Beyond Newton's laws

Classical mechanics also describes the more complex motions of extended non-pointlike objects.Euler's laws
In classical mechanics, Euler's laws of motion are equations of motion which extend Newton's laws of motion for point particle to rigid body motion. They were formulated by Leonhard Euler about 50 years after Isaac Newton formulated his laws. ...

provide extensions to Newton's laws in this area. The concepts of angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed sy ...

rely on the same calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of ari ...

used to describe one-dimensional motion. The rocket equation
A rocket (from it, rocchetto, , bobbin/spool) is a vehicle that uses jet propulsion to accelerate without using the surrounding air. A rocket engine produces thrust by reaction to exhaust expelled at high speed. Rocket engines work entirel ...

extends the notion of rate of change of an object's momentum to include the effects of an object "losing mass".
(These generalizations/extensions are derived from Newton's laws, say, by decomposing a solid body into a collection of points.)
There are two important alternative formulations of classical mechanics: Lagrangian mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lo ...

and Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momen ...

. These, and other modern formulations, usually bypass the concept of "force", instead referring to other physical quantities, such as energy, speed and momentum, for describing mechanical systems in generalized coordinates
In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 3 ...

. These are basically mathematical rewriting of Newton's laws, but complicated mechanical problems are much easier to solve in these forms. Also, analogy with quantum mechanics is more explicit in Hamiltonian formalism.
The expressions given above for momentum and kinetic energy are only valid when there is no significant electromagnetic contribution. In electromagnetism, Newton's second law for current-carrying wires breaks down unless one includes the electromagnetic field contribution to the momentum of the system as expressed by the Poynting vector
In physics, the Poynting vector (or Umov–Poynting vector) represents the directional energy flux (the energy transfer per unit area per unit time) or '' power flow'' of an electromagnetic field. The SI unit of the Poynting vector is the watt ...

divided by ''c''Limits of validity

Many branches of classical mechanics are simplifications or approximations of more accurate forms; two of the most accurate beinggeneral relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...

and relativistic statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...

. Geometric optics
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...

is an approximation to the quantum theory of light, and does not have a superior "classical" form.
When both quantum mechanics and classical mechanics cannot apply, such as at the quantum level with many degrees of freedom, quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particl ...

(QFT) is of use. QFT deals with small distances, and large speeds with many degrees of freedom as well as the possibility of any change in the number of particles throughout the interaction. When treating large degrees of freedom at the macroscopic level, statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...

becomes useful. Statistical mechanics describes the behavior of large (but countable) numbers of particles and their interactions as a whole at the macroscopic level. Statistical mechanics is mainly used in thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of th ...

for systems that lie outside the bounds of the assumptions of classical thermodynamics. In the case of high velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity ...

objects approaching the speed of light, classical mechanics is enhanced by special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The la ...

. In case that objects become extremely heavy (i.e., their Schwarzschild radius
The Schwarzschild radius or the gravitational radius is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius defining the event horizon of a Schwarzschild black hole. It is a characteris ...

is not negligibly small for a given application), deviations from Newtonian mechanics
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motio ...

become apparent and can be quantified by using the parameterized post-Newtonian formalism. In that case, general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...

(GR) becomes applicable. However, until now there is no theory of quantum gravity
Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the v ...

unifying GR and QFT in the sense that it could be used when objects become extremely small and heavy.