Classical mechanics is a physical theory describing the motion of macroscopic objects, from

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

s to parts of

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

, and astronomical objects, such as spacecraft,

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

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

s, and

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

. For objects governed by classical mechanics, if the present state is known, it is possible to predict how it will move in the future (determinism), and how it has moved in the past (reversibility). The earliest development of classical mechanics is often referred to as

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

. It consists of the physical concepts based on foundational works of Sir Isaac Newton, and the mathematical methods invented by Gottfried Wilhelm Leibniz,

Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaLeonhard Euler, and other contemporaries, in the 17th century to describe the motion of

bodies Bodies may refer to: * The plural of body * ''Bodies'' (2004 TV series), BBC television programme * Bodies (upcoming TV series), an upcoming British crime thriller limited series * "Bodies" (''Law & Order''), 2003 episode of ''Law & Order'' * ...

under the influence of a system of

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

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

analytical mechanics 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. 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: quantum mechanics. To describe velocities that are not small compared to the speed of light, special relativity is needed. In cases where objects become extremely massive, general relativity 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 parameters: its position, mass, and the

s applied to it. Each of these parameters is discussed in turn. In reality, the kind of objects that classical mechanics can describe always have a non-zero size. (The physics of ''very'' small particles, such as the electron, is more accurately described by quantum mechanics.) Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the additional

degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...

, e.g., a baseball can

spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...

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 weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...

of a composite object behaves like a point particle. Classical mechanics uses common sense notions of how matter and forces exist and interact. It assumes that matter and energy have definite, knowable attributes such as location in space and speed. Non-relativistic mechanics also assumes that forces act instantaneously (see also Action at a distance).

Position and its derivatives

The ''position'' of a point particle is defined in relation to a

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

centered on an arbitrary fixed reference point in space 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. 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 for the structure of space.

Velocity and speed

The '' velocity'', or the rate of change of displacement with time, is defined as the derivative 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 vectors 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'', or rate of change of velocity, is the derivative of the velocity with respect to time (the second derivative 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 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. 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 are ...

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 frame of reference, such as a linearly accelerating or rotating reference frame. It is related to Newton's second law of motion, which trea ...

s, 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: :

x' = x - u t \,

y' = y \,

z' = z \,

t' = t \, .

This set of formulas defines a

group transformation In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...

known as the

Galilean transformation In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotatio ...

(informally, the ''Galilean transform''). This group is a limiting case of the Poincaré group used in special relativity. The limiting case applies when the velocity ''u'' is very small compared to ''c'', the speed of light. 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 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 In physics, the Coriolis force is an inertial or fictitious force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the ...

Forces and Newton's second law

A force in physics is any action that causes an object's velocity to change; that is, to accelerate. A force originates from within a field, such as an electro-static field (caused by static electrical charges), electro-magnetic field (caused by moving charges), or gravitational field (caused by mass), among others.

Newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * ''Newton'' (film), a 2017 Indian film * Newton ( ...

was the first to mathematically express the relationship between

and

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

. Some physicists interpret

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

as a definition of force and mass, while others consider it a fundamental postulate, a law of nature. Either interpretation has the same mathematical consequences, historically known as "Newton's Second Law": :

\mathbf =  = .

The quantity ''m''v is called the ( canonical)

. 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, 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 A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and ( lambda) is a positive rat ...

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 ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means th ...

), 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 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 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 :

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. Gravity 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 ''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 In physics, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force stre ...

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 ''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 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 (E_\mathrm + E_\mathrm) = 0 \, .

This result is known as ''conservation of energy'' and states that the total energy, :

\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 provide extensions to Newton's laws in this area. The concepts of angular momentum rely on the same calculus 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.) There are two important alternative formulations of classical mechanics: Lagrangian mechanics and Hamiltonian mechanics. These, and other modern formulations, usually bypass the concept of "force", instead referring to other physical quantities, such as energy, speed and momentum, for describing mechanical systems in generalized coordinates. These are basically mathematical rewriting of Newton's laws, but complicated mechanical problems are much easier to solve in these forms. Also, analogy with quantum mechanics is more explicit in Hamiltonian formalism. The expressions given above for momentum and kinetic energy are only valid when there is no significant electromagnetic contribution. In electromagnetism, Newton's second law for current-carrying wires breaks down unless one includes the electromagnetic field contribution to the momentum of the system as expressed by the

Poynting vector In physics, the Poynting vector (or Umov–Poynting vector) represents the directional energy flux (the energy transfer per unit area per unit time) or '' power flow'' of an electromagnetic field. The SI unit of the Poynting vector is the watt ...

divided by ''c''², where ''c'' is the speed of light 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 and relativistic

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

. Geometric optics is an approximation to the quantum theory of light, and does not have a superior "classical" form. When both quantum mechanics and classical mechanics cannot apply, such as at the quantum level with many degrees of freedom,

quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...

(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,

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 for systems that lie outside the bounds of the assumptions of classical thermodynamics. In the case of high velocity objects approaching the speed of light, classical mechanics is enhanced by special relativity. In case that objects become extremely heavy (i.e., their Schwarzschild radius is not negligibly small for a given application), deviations from

become apparent and can be quantified by using the parameterized post-Newtonian formalism. In that case, general relativity (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 vi ...

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, gyrotron, or high voltage magnetron 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.

The classical approximation to quantum mechanics

The ray approximation of classical mechanics breaks down when the de Broglie wavelength is not much smaller than other dimensions of the system. For non-relativistic particles, this wavelength is
: $\lambda=\frac$

where ''h'' is Planck's constant and ''p'' is the momentum.

Again, this happens with electrons before it happens with heavier particles. For example, the electrons used by Clinton Davisson and Lester Germer Lester Halbert Germer (October 10, 1896 – October 3, 1971) was an American physicist. With Clinton Davisson, he proved the wave-particle duality of matter in the Davisson–Germer experiment, which was important to the development of the elec ...
in 1927, accelerated by 54 V, had a wavelength of 0.167 nm, which was long enough to exhibit a single diffraction

Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...
side lobe when reflecting from the face of a nickel crystal with atomic spacing of 0.215 nm. With a larger vacuum chamber, it would seem relatively easy to increase the angular resolution

Angular resolution describes the ability of any image-forming device such as an optical or radio telescope, a microscope, a camera, or an eye, to distinguish small details of an object, thereby making it a major determinant of image resolution. ...
from around a radian to a milliradian 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 tiny ...
computer memory.

More practical examples of the failure of classical mechanics on an engineering scale are conduction by quantum tunneling in tunnel diodes and very narrow transistor gates in integrated 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 tiny ...
s.

Classical mechanics is the same extreme high frequency approximation
A high-frequency approximation (or "high energy approximation") for scattering or other wave propagation problems, in physics or engineering, is an approximation whose accuracy increases with the size of features on the scatterer or medium relativ ...
as geometric optics. It is more often accurate because it describes particles and bodies with rest mass. These have more momentum and therefore shorter De Broglie wavelengths than massless particles, such as light, with the same kinetic energies.

History

The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in science, engineering, and technology.

Some Greek philosophers of antiquity, among them Aristotle, 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 and controlled experiment, 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" and the use of component forces.

The first published causal explanation of the motions of planets

A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a young ...
was Johannes Kepler's '' Astronomia nova,'' published in 1609. He concluded, based on Tycho Brahe's observations on the orbit of Mars, 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

The Leaning Tower of Pisa ( it, torre pendente di Pisa), or simply, the Tower of Pisa (''torre di Pisa'' ), is the '' campanile'', or freestanding bell tower, of Pisa Cathedral. It is known for its nearly four-degree lean, the result of an unst ...
, showing that they both hit the ground at the same time. The reality of that particular experiment is disputed, but he did carry out quantitative experiments by rolling balls on an inclined plane. His theory of accelerated motion was derived from the results of such experiments and forms a cornerstone of classical mechanics. In 1673 Christiaan Huygens

Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists of ...
described in his '' Horologium Oscillatorium'' the first two laws of motion. The work is also the first modern treatise in which a physical problem (the accelerated motion

In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
of a falling body) is idealized by a set of parameters then analyzed mathematically and constitutes one of the seminal works of applied mathematics.

Newton founded his principles of natural philosophy on three proposed laws of motion: the law of inertia

Newton's laws of motion are three basic 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 ...
, his second law of acceleration (mentioned above), and the law of action and reaction; and hence laid the foundations for classical mechanics. Both Newton's second and third laws were given the proper scientific and mathematical treatment in Newton's ''Philosophiæ Naturalis Principia Mathematica

(English: ''Mathematical Principles of Natural Philosophy'') often referred to as simply the (), is a book by Isaac Newton that expounds Newton's laws of motion and his law of universal gravitation. The ''Principia'' is written in Latin and ...
.'' Here they are distinguished from earlier attempts at explaining similar phenomena, which were either incomplete, incorrect, or given little accurate mathematical expression. Newton also enunciated the principles of conservation of momentum and angular momentum. In mechanics, Newton was also the first to provide the first correct scientific and mathematical formulation of gravity in Newton's law of universal gravitation. The combination of Newton's laws of motion and gravitation provide the fullest and most accurate description of classical mechanics. He demonstrated that these laws apply to everyday objects as well as to celestial objects. In particular, he obtained a theoretical explanation of Kepler's laws of motion of the planets.

Newton had previously invented the calculus, of mathematics, and used it to perform the mathematical calculations. For acceptability, his book, the ''Principia'', was formulated entirely in terms of the long-established geometric methods, which were soon eclipsed by his calculus. However, it was Leibniz who developed the notation of the derivative and integral preferred today. Newton, and most of his contemporaries, with the notable exception of Huygens
Huygens (also Huijgens, Huigens, Huijgen/Huygen, or Huigen) is a Dutch patronymic surname, meaning "son of Hugo". Most references to "Huygens" are to the polymath Christiaan Huygens. Notable people with the surname include:

* Jan Huygen (1563– ...
, worked on the assumption that classical mechanics would be able to explain all phenomena, including light, in the form of geometric optics. 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. Lagrangian mechanics was in turn re-formulated in 1833 by William 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, classical mechanics leads to the Gibbs paradox of classical statistical mechanics

In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
, in which entropy 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 spect ...
was not explained without the introduction of quanta
Quanta is the plural of quantum.

Quanta may also refer to:

Organisations

* Quanta Computer, a Taiwan-based manufacturer of electronic and computer equipment
* Quanta Display Inc., a Taiwanese TFT-LCD panel manufacturer acquired by AU Optronic ...
. As experiments reached the atomic level, classical mechanics failed to explain, even approximately, such basic things as the energy levels and sizes of atoms

Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.

Every solid, liquid, gas, an ...
and the photo-electric effect. The effort at resolving these problems led to the development of quantum mechanics.

Since the end of the 20th century, classical mechanics in physics has no longer been an independent theory. Instead, classical mechanics is now considered an approximate theory to the more general quantum mechanics. Emphasis has shifted to understanding the fundamental forces of nature as in the Standard model

The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...
and its more modern extensions into a unified theory of everything. Classical mechanics is a theory useful for the study of the motion of non-quantum mechanical, low-energy particles in weak gravitational fields. Also, it has been extended into the complex domain

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
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
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Branches

Classical mechanics was traditionally divided into three main branches:
* Statics

Statics is the branch of classical mechanics that is concerned with the analysis of force and torque (also called moment) acting on physical systems that do not experience an acceleration (''a''=0), but rather, are in static equilibrium with ...
, the study of equilibrium and its relation to force

In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
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 laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:

# A body remains at rest, or in motion ...

* Lagrangian mechanics
* Hamiltonian mechanics

Alternatively, a division can be made by region of application:
* Celestial mechanics, relating to star

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

Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
, for materials modelled as a continuum, e.g., solids and fluid

In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
s (i.e., liquid

A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, a ...
s and gases).
* Relativistic mechanics (i.e. including the special and general theories of relativity), for bodies whose speed is close to the speed of light.
* Statistical mechanics

In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
, which provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk thermodynamic properties of materials.

See also

* Dynamical system
* History of classical mechanics
* List of equations in classical mechanics
* List of publications in classical mechanics
* List of textbooks on classical mechanics and quantum mechanics
* Molecular dynamics
* Newton's laws of motion
* Special relativity
* Quantum mechanics
* Quantum field theory

In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...

Notes

References

Further reading

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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. 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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