classical mechanics

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Classical mechanics is a physical theory describing the
motion In physics, motion is the phenomenon in which an object changes its Position (geometry), position with respect to time. Motion is mathematically described in terms of Displacement (geometry), displacement, distance, velocity, acceleration, speed ...

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 scale, microscopic. Overview When applied to ph ...
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 (physics), motion through space are projectiles, they are ...

s to parts of
machinery A machine is a physical system using Power (physics), power to apply Force, forces and control Motion, movement to perform an action. The term is commonly applied to artificial devices, such as those employing engines or motors, but also to na ...
, and astronomical objects, such as
spacecraft A spacecraft is a vehicle or machine designed to spaceflight, fly in outer space. A type of artificial satellite, spacecraft are used for a variety of purposes, including Telecommunications, communications, Earth observation satellite, Earth ...

,
planets A planet is a large, rounded Astronomical object, astronomical body that is neither a star nor its Stellar remnant, remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud colla ...
,
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 . 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 Scientific law, 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 re ...
. 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 mathema ...

,
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 ma ...

, 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 Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to accelerate. Force can ...

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-Loui ...
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) ''momenta ...
. 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 relativity, special theory of relativity, is ...
. 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: wikt:μηχανική#Ancient_Greek, μηχανική, ''mēkhanikḗ'', "of machine, machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among Ph ...

:
quantum mechanics Quantum mechanics is a fundamental Scientific theory, 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 qua ...
. 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 Spacetime, space and time. In Albert Einstein's original treatment, the theory is based on two Postulates of ...
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 differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...
becomes applicable. However, a number of modern sources do include relativistic mechanics 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 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 and extrinsic properties, intrinsic property of a body. It was traditionally believed to be related to the physical quantity, quantity of matter in a Physical object, physical body, until the discovery of the atom and par ...
, and the
force In physics, a force is an influence that can change the motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to accelerate. Force can ...

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 size. (The physics of ''very'' small particles, such as the
electron The electron ( or ) is a subatomic particle with a negative one elementary charge, elementary electric charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are generally thought t ...

, is more accurately described by
quantum mechanics Quantum mechanics is a fundamental Scientific theory, 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 qua ...
.) 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 games, bat-and-ball sport played between two team sport, teams of nine players each, taking turns batting (baseball), batting and Fielding (baseball), fielding. The game occurs over the course of several Pitch ...
can while it is moving. However, the results for point particles can be used to study such objects by treating them as composite 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 weight function, weighted relative position (vector), position of the distributed mass sums to zero. Thi ...
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 Kingdom of Great Britain, Great Britain to people in the Thirteen Colonies. Writing in clear and persuasive prose, Paine collected variou ...
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 Contact mechanics, mechanical contact) by another object. That is, it is the non-local interaction of objects that are separat ...
).

## Position and its derivatives

The ''position'' of a point particle 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 (geometry), position of the Point (geometry), points or other geometric elements on a manifold such as Euclidean space ...
centered on an arbitrary fixed reference point in
space Space is the boundless Three-dimensional space, three-dimensional extent in which Physical body, objects and events have relative position (geometry), position and direction (geometry), direction. In classical physics, physical space is often ...
called the origin ''O''. A simple coordinate system might describe the position of a
particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object which can be described by several physical property, physical or chemical property, chemical ...
''P'' with a
vector Vector most often refers to: *Euclidean vector In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities ...
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 Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial frames of reference In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, i ...
), 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 preferr ...
, classical mechanics assumes
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
for the structure of space.

### Velocity and speed

The ''
velocity Velocity is the directional derivative, directional speed of an physical object, object in motion as an indication of its time derivative, rate of change in position (vector), position as observed from a particular frame of reference and as m ...
'', or the rate of change of displacement with time, is defined as the
derivative In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
of the position with respect to time: :$\mathbf = \,\!$. 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 derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for ...
. 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_(mathematics_and_physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
s in the directions of motion of each object respectively, then the velocity of the first object as seen by the second object is: :$\mathbf\text{'} = \mathbf - \mathbf \, .$ Similarly, the first object sees the velocity of the second object as: :$\mathbf= \mathbf - \mathbf \, .$ When both objects are moving in the same direction, this equation can be simplified to: :$\mathbf\text{'} = \left( u - v \right) \mathbf \, .$ Or, by ignoring direction, the difference can be given in terms of speed only: :$u\text{'} = u - v \, .$

### Acceleration

The ''
acceleration In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Accelerations are Euclidean vector, vector quantities (in that they have Magnitude (mathematics), magnitude and Direction ...
'', or rate of change of velocity, is the
derivative In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
of the velocity with respect to time (the
second derivative In 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 stud ...
of the position with respect to time): :$\mathbf = = .$ 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 In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object which can be described by several physical property, physical or chemical property, chemical ...
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 con ...
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 (mathematics), origin, orientation (geometry), orientation, and scale (geometry), scale are specified by a set of reference point ...
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 frame A non-inertial reference frame is a frame of reference that undergoes acceleration with respect to an inertial frame. An accelerometer at rest in a non-inertial frame will, in general, detect a non-zero acceleration. While the laws of motion ...
s 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 reference frame, non-inertial frame of reference, such as a linearly accelerating or rotating reference frame. It is related to Newton's la ...
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 (mathematics), origin, orientation (geometry), orientation, and scale (geometry), scale are specified by a set of reference point ...
''S'' and 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: :$x\text{'} = x - u t \,$ :$y\text{'} = y \,$ :$z\text{'} = z \,$ :$t\text{'} = t \, .$ This set of formulas defines a group transformation known as the
Galilean transformation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science i ...
(informally, the ''Galilean transform''). This group is a limiting case of the
Poincaré group The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at University of Königsberg, Königsberg, Un ...
used in
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between Spacetime, space and time. In Albert Einstein's original treatment, the theory is based on two Postulates of ...
. 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 relativity, special theory of relativity, is ...
. 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 relativity, special theory of relativity, is ...
is not a constant in classical mechanics, nor does the special position given to the speed of light in relativistic mechanics 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 parallel ...
and
Coriolis force In physics, the Coriolis force is an fictitious force, inertial or fictitious force that acts on objects in motion within a rotating reference frame, frame of reference that rotates with respect to an Inertial frame of reference, inertial fram ...
.

## 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 In physics, a force is an influence that can change the motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to accelerate. Force can ...

and
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the Multiplication, product of the mass and velocity of an object. It is a Euclidean vector, vector quantity, possessing a magnitude and a dire ...
. Some physicists interpret
Newton's second law of motion Newton's laws of motion are three basic Scientific law, 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 re ...
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": :$\mathbf = = .$ The quantity ''m''v is called the (
canonical The adjective canonical is applied in many contexts to mean "according to the canon (basic principle), canon" the standard (metrology), standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in t ...
)
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the Multiplication, product of the mass and velocity of an object. It is a Euclidean vector, vector quantity, possessing a magnitude and a dire ...
. 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: :$\mathbf = m \mathbf \, .$ 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 (mathematics), variable and involves the derivatives of those functions. The term ''ordinary ...
, 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: :$\mathbf_ = - \lambda \mathbf \, ,$ 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 :$- \lambda \mathbf = m \mathbf = m \, .$ This can be integrated to obtain :$\mathbf = \mathbf_0 e^$ where v0 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 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 Conservation law, ''conserved'' over time. This law, first proposed and tested by Émilie du Chât ...
), 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 strong ...
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 elect ...
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 of a ...
Newton's third law Newton's laws of motion are three basic law Law is a set of rules that are created and are law enforcement, enforceable by social or governmental institutions to regulate behavior,Robertson, ''Crimes against humanity'', 90. with its ...
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 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: : $W = \mathbf \cdot \Delta \mathbf \, .$ More generally, if the force varies as a function of position as the particle moves from r1 to r2 along a path ''C'', the work done on the particle is given by the
line integral In mathematics, a line integral is an integral where the function (mathematics), function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integra ...
: $W = \int_C \mathbf\left(\mathbf\right) \cdot \mathrm\mathbf \, .$ If the work done in moving the particle from r1 to r2 is the same no matter what path is taken, the force is said to be
conservative Conservatism is a Philosophy of culture, cultural, Social philosophy, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in r ...
.
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 strong ...
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 (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that ...
. The force due to
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding (motion), sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative la ...
is non-conservative. The
kinetic energy In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical sci ...
''E''k of a particle of mass ''m'' travelling at speed ''v'' is given by : $E_\mathrm = \tfracmv^2 \, .$ 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 r1 to r2 is equal to the change in
kinetic energy In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical sci ...
''E''k of the particle: :$W = \Delta E_\mathrm = E_\mathrm - E_\mathrm = \tfrac m \left\left(v_2^ - v_1^\right\right) .$ Conservative forces can be expressed as the
gradient In vector calculus, the gradient of a scalar-valued function, scalar-valued differentiable function of Function of several variables, several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "d ...
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 potentia ...
and denoted ''E''p: : $\mathbf = - \mathbf E_\mathrm \, .$ 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 : $\mathbf \cdot \Delta \mathbf = - \mathbf E_\mathrm \cdot \Delta \mathbf = - \Delta E_\mathrm \, .$ The decrease in the potential energy is equal to the increase in the kinetic energy : $-\Delta E_\mathrm = \Delta E_\mathrm \Rightarrow \Delta \left(E_\mathrm + E_\mathrm\right) = 0 \, .$ This result is known as ''conservation of energy'' and states that the total
energy In physics, energy (from Ancient Greek: wikt:ἐνέργεια#Ancient_Greek, ἐνέργεια, ''enérgeia'', “activity”) is the physical quantity, quantitative physical property, property that is #Energy transfer, transferred to a phy ...
, : $\sum E = E_\mathrm + E_\mathrm \, ,$ 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. O ...
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 Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity—the total angular ...
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 Acceleration, accelerate without using the surrounding Atmosphere of Earth, air. A rocket engine produces thrust by Reaction (physics), reaction to exhaust ...
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-Loui ...
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) ''momenta ...
. 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 (physics), configuration space. These parameters must uniquely define the configuration of the system relati ...
. 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 Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scienc ...
divided by ''c''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 relativity, special theory of relativity, is ...
in free space.

# Limits of validity

Many branches of classical mechanics are simplifications or approximations of more accurate forms; two of the most accurate being
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...
and relativistic
statistical mechanics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scien ...
.
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 Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict List of natural phenomena, natural phenomena. This is in c ...
(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 Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scien ...
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 (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
for systems that lie outside the bounds of the assumptions of classical thermodynamics. In the case of high
velocity Velocity is the directional derivative, directional speed of an physical object, object in motion as an indication of its time derivative, rate of change in position (vector), position as observed from a particular frame of reference and as m ...
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 Spacetime, space and time. In Albert Einstein's original treatment, the theory is based on two Postulates of ...
. 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 characteristic ...
is not negligibly small for a given application), deviations from
Newtonian mechanics Newton's laws of motion are three basic Scientific law, 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 re ...
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 differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...
(GR) becomes applicable. However, until now there is no theory of
quantum gravity Quantum gravity (QG) is a field of theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict List of natural phenomena, ...
unifying GR and QFT in the sense that it could be used when objects become extremely small and heavy. /sup>

## The Newtonian approximation to special relativity

In special relativity, the momentum of a particle is given by :$\mathbf = \frac \, ,$ 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 :$\mathbf \approx m\mathbf \, .$ 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: Janu ...
,
gyrotron High-power 140 GHz gyrotron for plasma heating in the Wendelstein 7-X fusion experiment, Germany. A gyrotron is a class of high-power linear-beam vacuum tube A vacuum tube, electron tube, valve (British usage), or tube (North America), is ...
, or high voltage
magnetron The cavity magnetron is a high-power vacuum tube used in early radar systems and currently in microwave oven, microwave ovens and linear particle accelerators. It generates microwaves using the interaction of a stream of electrons with a magn ...
is given by :$f = f_\mathrm\frac \, ,$ 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 entertain ...
) 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 Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatom ...
is not much smaller than other dimensions of the system. For non-relativistic particles, this wavelength is :$\lambda=\frac$ 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 charge, elementary electric charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are generally thought t ...
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 Umbra, penumbra and antumbra, geometrical shadow of the obstacle/aperture. The diffracting object or ape ...
side lobe In antenna (radio), antenna engineering, sidelobes are the lobes (local maxima) of the Near and far field, far field radiation pattern of an antenna or other radiation source, that are not the ''main lobe''. The radiation pattern of most an ...
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, macrosc ...
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 A vacuum pump is a device that draws gas molecules from a sealed volume in order to leave behind a partial vacuum. The job of a vacuum pump is ...
, it would seem relatively easy to increase the
angular resolution Angular resolution describes the ability of any image-forming device such as an Optical telescope, optical or radio telescope, a microscope, a camera, or an Human eye, eye, to distinguish small details of an object, thereby making it a major det ...
from around a radian to a
milliradian A milliradian (International System of Units, 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 ...
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 ti ...
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 fundamental interaction, interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the ...
in
tunnel diode A tunnel diode or Esaki diode is a type of semiconductor diode that has effectively "negative resistance" due to the quantum mechanics, quantum mechanical effect called quantum tunneling, tunneling. It was invented in August 1957 by Leo Esa ...
s and very narrow
transistor file:MOSFET Structure.png, upright=1.4, Metal-oxide-semiconductor field-effect transistor (MOSFET), showing Metal gate, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an insulating layer (pink). A ...
gates 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 ti ...
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 A system is a group of interacting or interrelated elements that act according ...
. 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 Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earli ...
,
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 specializ ...
, and
technology Technology is the application of knowledge to reach practical goals in a specifiable and Reproducibility, reproducible way. The word ''technology'' may also mean the product of such an endeavor. The use of technology is widely prevalent in me ...
. 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 Greece, Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatet ...
, founder of
Aristotelian physics Aristotelian physics is the form of natural science described in the works of the Greek philosopher Aristotle (384–322 BC). In his work '' Physics'', Aristotle intended to establish general principles of change that govern all natural bodies, ...
, 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 wh ...
, 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 strong ...
" 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 (relativity), event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''eff ...
explanation of the motions of
planets A planet is a large, rounded Astronomical object, astronomical body that is neither a star nor its Stellar remnant, remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud colla ...
was Johannes Kepler's ''
Astronomia nova ''Astronomia nova'' (English language, 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. ...
,'' 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 k ...
'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 atmosph ...
, that the planet's orbits were
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), 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 ty ...
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 was ...
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 A mathematician is someone who uses an extensive knowledge of mathematics in their work, typica ...
described in his ''
Horologium Oscillatorium (English language, 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 horol ...
'' 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 mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is ...
. 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 language, 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 Newton's law of universal gravitation, law of universa ...
.'' 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 Multiplication, product of the mass and velocity of an object. It is a Euclidean vector, vector quantity, possessing a magnitude and a dire ...
and
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity—the total angular ...
. 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 strong ...
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 In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the Kepler orbit, orbits of planets around the Sun. The scientific law, laws modified the Copernican heliocentrism, heliocentric theor ...
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 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 mathema ...
who developed the notation of the
derivative In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
and
integral In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
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 visual perception, perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequency, fr ...
, 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 Newton's rings is a phenomenon in which an interference pattern is created by the reflection of light Light or visible light is electromagnetic radiation that can be visual perception, perceived by the human eye. Visible light is usua ...
(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 destructive ...
phenomenon) he maintained his own
corpuscular theory of light In optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually ...
. 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 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 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 ...
, 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 1887 ...
. 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 Spacetime, space and time. In Albert Einstein's original treatment, the theory is based on two Postulates of ...
, 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 (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
, classical mechanics leads to the
Gibbs paradox In statistical mechanics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy ...
of classical
statistical mechanics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scien ...
, 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 thermodynam ...
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 spe ...
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 mechanics, quantum mechanical system or particle that is Bound state, bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with Classical mechanics, classical pa ...
and sizes of
atoms Every atom is composed of a atomic nucleus, 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, l ...
and the photo-electric effect. The effort at resolving these problems led to the development of
quantum mechanics Quantum mechanics is a fundamental Scientific theory, 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 qua ...
. Since the end of the 20th century, classical mechanics in
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
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 Particle physics or high energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The fundamental particles ...
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 i
Asymptotics 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 mechanical equilibrium, st ...
, the study of equilibrium and its relation to
force In physics, a force is an influence that can change the motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to accelerate. Force can ...

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 (physics), motion of points, Physical object, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause ...
, 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 Scientific law, 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 re ...
*
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-Loui ...
*
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) ''momenta ...
Alternatively, a division can be made by region of application: *
Celestial mechanics Celestial mechanics is the branch of astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronol ...
, 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,
planet A planet is a large, rounded Astronomical object, astronomical body that is neither a star nor its Stellar remnant, remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud colla ...
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 point particle, discrete particles. The French mathematician Augustin-Louis Cauchy was the first to fo ...
, for materials modelled as a continuum, e.g.,
solid Solid is one of the State of matter#Four fundamental states, four fundamental states of matter (the others being liquid, gas, and Plasma (physics), plasma). The molecules in a solid are closely packed together and contain the least amount o ...
s and
fluid In physics, a fluid is a liquid, gas, or other material that continuously Deformation (physics), deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are Matter, substances wh ...
s (i.e.,
liquid A liquid is a nearly Compressibility, 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 State of matter#Four fundamental states, the four fund ...
s and
gas Gas is one of the four fundamental states of matter In physics, a state of matter is one of the distinct forms in which matter can exist. Four states of matter are observable in everyday life: solid, liquid, gas, and Plasma (physics), pl ...
es). *
Relativistic mechanics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science ...
(i.e. including the
special Special or specials may refer to: Policing * Specials, Ulster Special Constabulary The Ulster Special Constabulary (USC; commonly called the "B-Specials" or "B Men") was a quasi-military reserve special constable police force in what wou ...
and
general A general officer is an Officer (armed forces), officer of highest military ranks, high rank in the army, armies, and in some nations' air forces, space forces, and marines or naval infantry. In some usages the term "general officer" refers t ...
theories of relativity), for bodies whose speed is close to the speed of light. *
Statistical mechanics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scien ...
, 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 (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
properties of materials.

*
Dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
*
History of classical mechanics This article deals with the history of classical mechanics. Precursors to classical mechanics Antiquity The ancient Greek philosophy, Greek philosophers, Aristotle in particular, were among the first to propose that abstract principles go ...
* 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 Motion (physics), 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 dynamics (m ...
*
Newton's laws of motion Newton's laws of motion are three basic Scientific law, 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 re ...
*
Special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between Spacetime, space and time. In Albert Einstein's original treatment, the theory is based on two Postulates of ...
*
Quantum mechanics Quantum mechanics is a fundamental Scientific theory, 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 qua ...
*
Quantum field theory In theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict List of natural phenomena, natural phenomena. This is in c ...

# References

* * * * * * * * * * *

* Crowell, Benjamin
Light 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 a
e-book library
of classic texts on mechanical design and engineering.
MIT OpenCourseWare 8.01: Classical Mechanics
Free videos of actual course lectures with links to lecture notes, assignments and exams. * Alejandro A. Torassa

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