Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as

`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 conservation of energy), 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 and the Lorentz force for

_{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 conservative. _{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 fo ...

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, gyrotron, or high voltage magnetron 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) 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 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 n ... 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 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, macro ... with atomic spacing of 0.215 nm. With a larger vacuum chamber, it would seem relatively easy to increase the angular resolution from around a radian to a milliradian and see quantum diffraction from the periodic patterns of integrated circuit computer memory.
More practical examples of the failure of classical mechanics on an engineering scale are conduction by quantum tunneling in tunnel diodes and very narrow transistor gates in integrated circuits.
Classical mechanics is the same extreme high frequency approximation as geometric optics. It is more often accurate because it describes particles and bodies with rest mass. 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 evidence ..., 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, scien ....
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 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 stro ..." and the use of component forces.
The first published causal explanation of the motions of planets was Johannes Kepler's '' Astronomia nova,'' published in 1609. He concluded, based on Tycho Brahe'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 ellipses. This break with ancient thought was happening around the same time that Galileo 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
The Leaning Tower of Pisa ( it, torre pendente di Pisa), or simply, the Tower of Pisa (''torre di Pisa'' ), is the ''campanile'', or freestanding bell tower, of Pisa Cathedral. It is known for its nearly four-degree lean, the result of an unst ..., 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 cla .... His theory of accelerated motion was derived from the results of such experiments and forms a cornerstone of classical mechanics. In 1673 Christiaan Huygens described in his '' Horologium Oscillatorium'' 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 mathemati ....
Newton founded his principles of natural philosophy on three proposed laws of motion: the law of inertia
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 motion ..., 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.'' 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 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 syst .... 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 stro ... in Newton's law of universal gravitation. 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 mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ..., 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 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 t ..., in the form of geometric optics. Even when discovering the so-called Newton's rings
Newton's rings is a phenomenon in which an interference pattern is created by the reflection of light between two surfaces, typically a spherical surface and an adjacent touching flat surface. It is named after Isaac Newton, who investigated ... (a wave interference 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. Lagrangian mechanics was in turn re-formulated in 1833 by William Rowan Hamilton
Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ....
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 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 ..., classical mechanics leads to the Gibbs paradox of classical statistical mechanics, 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 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 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 and its more modern extensions into a unified theory of everything. 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
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ... 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 forces
* Dynamics, the study of motion and its relation to forces
* Kinematics, 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
* Lagrangian mechanics
* Hamiltonian mechanics
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, ..., relating to star
A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...s, planets and other celestial bodies
* Continuum mechanics, 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 structur ...s and fluids (i.e., liquids and gases).
* 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.". O ... theories of relativity), for bodies whose speed is close to the speed of light.
* Statistical mechanics, 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 ... 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 i ...
* History of classical mechanics
* 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 th ...
* Newton's laws 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 mo ...
* 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 law ...
* 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
Notes
References
Further reading
*
*
*
*
*
*
*
*
*
*
*
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. 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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}

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 colonization, ...

, planets, star
A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...

s, and galaxies. 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. 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, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...

, 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, Leonhard 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 ...

, and other contemporaries, in the 17th century to describe the motion of bodies
Bodies may refer to:
* The plural of body
* ''Bodies'' (2004 TV series), BBC television programme
* Bodies (upcoming TV series), an upcoming British crime thriller limited series
* "Bodies" (''Law & Order''), 2003 episode of ''Law & Order''
* B ...

under the influence of a system of forces. Later, more abstract methods were developed, leading to the reformulations of classical mechanics known as Lagrangian mechanics and Hamiltonian mechanics. These advances, made predominantly in the 18th and 19th centuries, extend substantially beyond earlier works, particularly through their use of analytical mechanics. 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 fo ...

. 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 law ...

is needed. In cases where objects become extremely massive, general relativity 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 as point particles (objects with negligible size). The motion of a point particle is characterized by a small number ofparameter
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 forces 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, 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 t ...

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 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 arg ...

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).
Position and its derivatives

The ''position'' of a point particle is defined in relation to a coordinate system centered on an arbitrary fixed reference point inspace
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 consi ...

called the origin ''O''. A simple coordinate system might describe the position of a particle ''P'' with a vector 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 event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

. In pre-Einstein relativity (known as Galilean relativity), 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
Absolute space and time is a concept in physics and philosophy about the properties of the universe. In physics, absolute space and time may be a preferred frame.
Before Newton
A version of the concept of absolute space (in the sense of a pre ...

, classical mechanics assumes Euclidean geometry 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 i ...

'', 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.
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 vectors 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 ...

'', 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 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 a particle can be described with respect to any observer in any state of motion, classical mechanics assumes the existence of a special family of reference frames in which the mechanical laws of nature take a comparatively simple form. These special reference frames are calledinertial frames
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...

. 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 forces, inertia forces, or pseudo-forces.
Consider two reference frames ''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 law ...

. 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 fo ...

.
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 fo ...

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.
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 between force and momentum. Some physicists interpret Newton's second law of motion 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) momentum. 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 anordinary 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 velectromagnetism
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 the scalar product 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, ...

: $W\; =\; \backslash int\_C\; \backslash mathbf(\backslash mathbf)\; \backslash cdot\; \backslash mathrm\backslash mathbf\; \backslash ,\; .$
If the work done in moving the particle from rGravity
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 stro ...

is a conservative force, as is the force due to an idealized spring
Spring(s) may refer to:
Common uses
* Spring (season), a season of the year
* Spring (device), a mechanical device that stores energy
* Spring (hydrology), a natural source of water
* Spring (mathematics), a geometric surface in the shape of a h ...

, 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 t ...

. 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 ...

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 acce ...

''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 acce ...

''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 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 ...

,
: $\backslash sum\; E\; =\; E\_\backslash mathrm\; +\; E\_\backslash mathrm\; \backslash ,\; ,$
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 provide extensions to Newton's laws in this area. The concepts ofangular 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 syst ...

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

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 entirely ...

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 and Hamiltonian mechanics. 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. 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 divided by ''c''Limits of validity

Many branches of classical mechanics are simplifications or approximations of more accurate forms; two of the most accurate being general relativity and relativistic statistical mechanics. Geometric optics 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 (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 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 inthermodynamics
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 ...

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 i ...

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 law ...

. 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 characteri ...

is not negligibly small for a given application), deviations from Newtonian mechanics become apparent and can be quantified by using the parameterized post-Newtonian formalism
In physics, precisely in the study of the theory of general relativity and many alternatives to it, the post-Newtonian formalism is a calculational tool that expresses Einstein's (nonlinear) equations of gravity in terms of the lowest-order devi ...

. In that case, general relativity (GR) becomes applicable. However, until now there is no theory of quantum gravity unifying GR and QFT in the sense that it could be used when objects become extremely small and heavy.