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Classical mechanics is a physical theory describing the
motion In physics, motion is when an object changes its position with respect to a reference point in a given time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed, and frame of reference to an o ...
of objects such as
projectile A projectile is an object that is propelled by the application of an external force and then moves freely under the influence of gravity and air resistance. Although any objects in motion through space are projectiles, they are commonly found ...
s, parts of machinery,
spacecraft A spacecraft is a vehicle that is designed spaceflight, to fly and operate in outer space. Spacecraft are used for a variety of purposes, including Telecommunications, communications, Earth observation satellite, Earth observation, Weather s ...
, planets,
star A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
s, and galaxies. The development of classical mechanics involved substantial change in the methods and philosophy of physics. The qualifier ''classical'' distinguishes this type of mechanics from physics developed after the revolutions in physics of the early 20th century, all of which revealed limitations in classical mechanics. The earliest formulation of classical mechanics is often referred to as Newtonian mechanics. It consists of the physical concepts based on the 17th century foundational works of Sir
Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
, and the mathematical methods invented by Newton,
Gottfried Wilhelm Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to ...
,
Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
and others to describe the motion of bodies under the influence of
force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
s. Later, methods based on
energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
were developed by Euler,
Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaWilliam Rowan Hamilton and others, leading to the development of analytical mechanics (which includes
Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the d'Alembert principle of virtual work. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the ...
and
Hamiltonian mechanics In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (gener ...
). These advances, made predominantly in the 18th and 19th centuries, extended beyond earlier works; they are, with some modification, used in all areas of modern physics. If the present state of an object that obeys the laws of classical mechanics is known, it is possible to determine how it will move in the future, and how it has moved in the past. Chaos theory shows that the long term predictions of classical mechanics are not reliable. Classical mechanics provides accurate results when studying objects that are not extremely massive and have speeds not approaching the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
. With objects about the size of an atom's diameter, it becomes necessary to use
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
. To describe velocities approaching the speed of light,
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
is needed. In cases where objects become extremely massive,
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
becomes applicable. Some modern sources include relativistic mechanics in classical physics, as representing the field in its most developed and accurate form.


Branches


Traditional division

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 acting on a physical system that does not experience an acceleration, but rather is in mechanical equilibrium, equilibrium with its environment ...
'' is the branch of classical mechanics that is concerned with the analysis of force and torque acting on a physical system that does not experience an acceleration, but rather is in equilibrium with its environment. ''
Kinematics In physics, kinematics studies the geometrical aspects of motion of physical objects independent of forces that set them in motion. Constrained motion such as linked machine parts are also described as kinematics. Kinematics is concerned with s ...
'' describes the
motion In physics, motion is when an object changes its position with respect to a reference point in a given time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed, and frame of reference to an o ...
of points, bodies (objects), and systems of bodies (groups of objects) without considering the
force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
s that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
. ''Dynamics'' goes beyond merely describing objects' behavior and also considers the forces which explain it. Some authors (for example, Taylor (2005) and Greenwood (1997) ) include
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
within classical dynamics.


Forces vs. energy

Another division is based on the choice of mathematical formalism. Classical mechanics can be mathematically presented in multiple different ways. The physical content of these different formulations is the same, but they provide different insights and facilitate different types of calculations. While the term "Newtonian mechanics" is sometimes used as a synonym for non-relativistic classical physics, it can also refer to a particular formalism based on
Newton's laws of motion Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows: # A body re ...
. Newtonian mechanics in this sense emphasizes force as a vector quantity. In contrast, analytical mechanics uses '' scalar'' properties of motion representing the system as a whole—usually its
kinetic energy In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...
and
potential energy In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity ...
. The equations of motion are derived from the scalar quantity by some underlying principle about the scalar's variation. Two dominant branches of analytical mechanics are
Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the d'Alembert principle of virtual work. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the ...
, which uses generalized coordinates and corresponding generalized velocities in
tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
space (the tangent bundle of the configuration space and sometimes called "state space"), and
Hamiltonian mechanics In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (gener ...
, which uses coordinates and corresponding momenta in phase space (the cotangent bundle of the configuration space). Both formulations are equivalent by a Legendre transformation on the generalized coordinates, velocities and momenta; therefore, both contain the same information for describing the dynamics of a system. There are other formulations such as Hamilton–Jacobi theory, Routhian mechanics, and Appell's equation of motion. All equations of motion for particles and fields, in any formalism, can be derived from the widely applicable result called the principle of least action. One result is Noether's theorem, a statement which connects conservation laws to their associated symmetries.


By region of application

Alternatively, a division can be made by region of application: *
Celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
, relating to
star A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
s,
planet A planet is a large, Hydrostatic equilibrium, rounded Astronomical object, astronomical body that is generally required to be in orbit around a star, stellar remnant, or brown dwarf, and is not one itself. The Solar System has eight planets b ...
s and other celestial bodies *
Continuum mechanics Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a ''continuous medium'' (also called a ''continuum'') rather than as discrete particles. Continuum mec ...
, for materials modelled as a continuum, e.g.,
solid Solid is a state of matter where molecules are closely packed and can not slide past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree to which they are resisted dependent upon the ...
s and
fluid In physics, a fluid is a liquid, gas, or other material that may continuously motion, move and Deformation (physics), deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are M ...
s (i.e.,
liquid Liquid is a state of matter with a definite volume but no fixed shape. Liquids adapt to the shape of their container and are nearly incompressible, maintaining their volume even under pressure. The density of a liquid is usually close to th ...
s and gases). * Relativistic mechanics (i.e. including the special and
general A general officer is an Officer (armed forces), officer of high rank in the army, armies, and in some nations' air force, air and space forces, marines or naval infantry. In some usages, the term "general officer" refers to a rank above colone ...
theories of relativity), for bodies whose speed is close to the speed of light. *
Statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
, which provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk thermodynamic properties of materials.


Description of objects and their motion

For simplicity, classical mechanics often models real-world objects as point particles, that is, objects with negligible size. The motion of a point particle is determined 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 physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
, and the
force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
s applied to it. 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 of
angular momentum Angular momentum (sometimes called 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 ang ...
rely on the same
calculus Calculus 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 generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...
used to describe one-dimensional motion. The rocket equation 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.) In reality, the kind of objects that classical mechanics can describe always have a non-zero size. (The behavior of ''very'' small particles, such as the
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
, is more accurately described by
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
.) Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the additional
degrees of freedom In many scientific fields, the degrees of freedom of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation: its two coordinates; a non-infinite ...
, 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 spin 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 barycenter or balance point) is the unique point at any given time where the weight function, weighted relative position (vector), position of the d ...
of a composite object behaves like a point particle. Classical mechanics 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 Action at a distance is the concept in physics that an object's motion (physics), motion can be affected by another object without the two being in Contact mechanics, physical contact; that is, it is the concept of the non-local interaction of ob ...
).


Kinematics

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 and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...
centered on an arbitrary fixed reference point in
space Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...
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 continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...
. 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, 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, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
for the structure of space.


Velocity and speed

The ''
velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
'', or the rate of change of displacement with time, is defined as the
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
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. 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' = \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' = ( u - v ) \mathbf \, . Or, by ignoring direction, the difference can be given in terms of speed only: : u' = 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. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector, vector ...
'', or rate of change of velocity, is the
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
of the velocity with respect to time (the
second derivative In calculus, the second derivative, or the second-order derivative, of a function is the derivative of the derivative of . Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the secon ...
of the position with respect to time): : \mathbf = = . Acceleration represents the velocity's change over time. Velocity can change in magnitude, 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 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. In an inertial frame Newton's law of motion, F=ma, is valid.Goldstein, Herbert (1950). Classical Mechanics (1st ed.). Addison-Wesley. Non-inertial reference frames accelerate in relation to another inertial frame. A body rotating with respect to an inertial frame is not an inertial frame. When viewed from an inertial frame, 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 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: : \begin x' &= x - tu,\\ y' &= y, \\ z' &= z, \\ t' &= t. \end 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
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
. 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 exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
. 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 exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
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 and Coriolis force.


Newtonian mechanics

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 cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
and momentum. Some physicists interpret
Newton's second law of motion Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows: # A body 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) 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: : \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 (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
, 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), 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
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
. 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 = \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 integr ...
: W = \int_C \mathbf(\mathbf) \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 cultural, social, and political philosophy and ideology that seeks to promote and preserve traditional institutions, customs, and values. The central tenets of conservatism may vary in relation to the culture and civiliza ...
.
Gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
is a conservative force, as is the force due to an idealized spring, as given by Hooke's law. The force due to friction is non-conservative. The
kinetic energy In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...
''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, the kinetic energy of an object is the form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...
''E''k of the particle: : W = \Delta E_\mathrm = E_\mathrm - E_\mathrm = \tfrac m \left(v_2^ - v_1^\right) . Conservative forces can be expressed as the gradient of a scalar function, known as the
potential energy In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity ...
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 (E_\mathrm + E_\mathrm) = 0 \, . This result is known as ''conservation of energy'' and states that the total
energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
, : \sum E = E_\mathrm + E_\mathrm \, , is constant in time. It is often useful, because many commonly encountered forces are conservative.


Lagrangian mechanics

Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the d'Alembert principle of virtual work. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the ...
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-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaMécanique analytique''. Lagrangian mechanics describes a mechanical system as a pair (M,L) consisting of a configuration space M and a smooth function L within that space called a Lagrangian. For many systems, L = T - V, where T and V are the kinetic and potential energy of the system, respectively. The stationary action principle requires that the action functional of the system derived from L must remain at a stationary point (a maximum, minimum, or saddle) throughout the time evolution of the system. This constraint allows the calculation of the equations of motion of the system using Lagrange's equations.


Hamiltonian mechanics

Hamiltonian mechanics In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (gener ...
emerged in 1833 as a reformulation of
Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the d'Alembert principle of virtual work. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the ...
. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta''. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
. In this formalism, the dynamics of a system are governed by Hamilton's equations, which express the time derivatives of position and momentum variables in terms of
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...
s of a function called the Hamiltonian: \frac = \frac,\quad \frac = -\frac. The Hamiltonian is the Legendre transform of the Lagrangian, and in many situations of physical interest it is equal to the total energy of the system.


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 as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
and relativistic
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
.
Geometric optics Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician ...
is an approximation to the quantum theory of light, and does not have a superior "classical" form. When both quantum mechanics and classical mechanics cannot apply, such as at the quantum level with many degrees of freedom,
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
(QFT) is of use. QFT deals with small distances, and large speeds with many degrees of freedom as well as the possibility of any change in the number of particles throughout the interaction. When treating large degrees of freedom at the macroscopic level,
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
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 a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
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 of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
. In case that objects become extremely heavy (i.e., their Schwarzschild radius 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 that case,
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
(GR) becomes applicable. However, until now there is no theory of
quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics. It deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the v ...
unifying GR and QFT in the sense that it could be used when objects become extremely small and heavy. /sup>


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 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: Januar ...
, gyrotron, or high voltage
magnetron The cavity magnetron is a high-power vacuum tube used in early radar systems and subsequently in microwave oven, microwave ovens and in linear particle accelerators. A cavity magnetron generates microwaves using the interaction of a stream of ...
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) 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.


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 : \lambda=\frac where ''h'' is the
Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
and ''p'' is the momentum. Again, this happens with
electrons The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
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 the deviation of waves from straight-line propagation without any change in their energy due to an obstacle or through an aperture. The diffracting object or aperture effectively becomes a secondary source of the Wave propagation ...
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, macros ...
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 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 (IC), also known as a microchip or simply chip, is a set of electronic circuits, consisting of various electronic components (such as transistors, resistors, and capacitors) and their interconnections. These components a ...
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 A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch electrical signals and electric power, power. It is one of the basic building blocks of modern electronics. It is composed of semicondu ...
gates in
integrated circuit An integrated circuit (IC), also known as a microchip or simply chip, is a set of electronic circuits, consisting of various electronic components (such as transistors, resistors, and capacitors) and their interconnections. These components a ...
s. Classical mechanics is the same extreme high frequency approximation as
geometric optics Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician ...
. 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 discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural sciences, which stu ...
,
engineering Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...
, and
technology Technology is the application of Conceptual model, conceptual knowledge to achieve practical goals, especially in a reproducible way. The word ''technology'' can also mean the products resulting from such efforts, including both tangible too ...
. The development of classical mechanics lead to the development of many areas of mathematics. Some
Greek philosophers Ancient Greek philosophy arose in the 6th century BC. Philosophy was used to make sense of the world using reason. It dealt with a wide variety of subjects, including astronomy, epistemology, mathematics, political philosophy, ethics, metaphysics ...
of antiquity, among them
Aristotle Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...
, 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 systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...
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 (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
" 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. It is also known as the "Red Planet", because of its orange-red appearance. Mars is a desert-like rocky planet with a tenuous carbon dioxide () atmosphere. At the average surface level the 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), commonly referred to as Galileo Galilei ( , , ) or mononymously as Galileo, was an Italian astronomer, physicist and engineer, sometimes described as a poly ...
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. 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, Halen, Lord of Zeelhem, ( , ; ; also spelled Huyghens; ; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor who is regarded as a key figure in the Scientific Revolution ...
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 mathematician and physicist Christiaan Huygens in 1673 and his major work on p ...
'' 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 a ...
. 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''. 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 (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...
and
angular momentum Angular momentum (sometimes called 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 ang ...
. In mechanics, Newton was also the first to provide the first correct scientific and mathematical formulation of
gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
in
Newton's law of universal gravitation Newton's law of universal gravitation describes gravity as a force by stating that every particle attracts every other particle in the universe with a force that is Proportionality (mathematics)#Direct proportionality, proportional to the product ...
. The combination of Newton's laws of motion and gravitation provides 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 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 generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...
; however, the ''Principia'' was formulated entirely in terms of long-established geometric methods in emulation of
Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...
. 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, visible light, or visible radiation is electromagnetic radiation that can be visual perception, perceived by the human eye. Visible light spans the visible spectrum and is usually defined as having wavelengths in the range of 400– ...
, in the form of
geometric optics Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician ...
. Even when discovering the so-called Newton's rings (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 Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaWilliam Rowan Hamilton. 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, 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 of classical
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
, in which
entropy Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
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 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 Atoms are the basic particles of the chemical elements. An atom consists of a nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished from each other ...
and the photo-electric effect. The effort at resolving these problems led to the development of
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
. Since the end of the 20th century, classical mechanics in
physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...
has no longer been an independent theory. Instead, classical mechanics is now considered an approximate theory to the more general quantum mechanics. Emphasis has shifted to understanding the fundamental forces of nature as in the
Standard Model The Standard Model of particle physics is the Scientific theory, theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions – excluding gravity) in the unive ...
and its more modern extensions into a unified
theory of everything A theory of everything (TOE), final theory, ultimate theory, unified field theory, or master theory is a hypothetical singular, all-encompassing, coherent theoretical physics, theoretical framework of physics that fully explains and links togeth ...
. Classical mechanics is a theory useful for the study of the motion of non-quantum mechanical, low-energy particles in weak gravitational fields.


See also

*
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, such as in a parametric curve. Examples include the mathematical models ...
* 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 ( ...
*
Newton's laws of motion Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows: # A body re ...
*
Special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
*
Quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
*
Quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...


Notes


References


Further reading

* * * * * * * * * * *


External links

* 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 university, private Ivy League research university based in Ithaca, New York, United States. The university was co-founded by American philanthropist Ezra Cornell and historian and educator Andrew Dickson W ...
. 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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