Diagram of orbital motion of a satellite around the Earth, showing perpendicular velocity and acceleration (force) vectors, represented through a classical interpretation
If the present state of an object that obeys the laws of classical mechanics is known, it is possible todetermine 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 thespeed of light. With objects about the size of an atom's diameter, it becomes necessary to usequantum mechanics. To describe velocities approaching the speed of light,special relativity is needed. In cases where objects become extremely massive,general relativity becomes applicable.
Classical mechanics was traditionally divided into three main branches.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 inequilibrium with its environment.[5]Kinematics describes themotion of points,bodies (objects), and systems of bodies (groups of objects) without considering theforces that cause them to move.[6][7][5] Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch ofmathematics.[8][9][10]Dynamics goes beyond merely describing objects' behavior and also considers the forces which explain it.Some authors (for example, Taylor (2005)[11] and Greenwood (1997)[12]) includespecial relativity within classical dynamics.
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 onNewton's laws of motion. Newtonian mechanics in this sense emphasizes force as avector quantity.[13]
Statistical mechanics, which provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulkthermodynamic properties of materials.
Classical mechanics models real-world objects aspoint particles, that is, objects with negligible size. Classical mechanics describes the motions of extended non-pointlike objects by considering these objects to be aggregates of rigidly connected particles.[14]: 1
In the Newtonian form of classical mechanics the motion of a point particle is determined by a small number ofparameters: its position,mass, and theforces applied to it.[14]: 3 Euler's laws provide extensions to Newton's laws in this area. The concepts ofangular momentum rely on the samecalculus used to describe one-dimensional motion. Therocket 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 anon-zero size. (The behavior ofvery small particles, such as theelectron, is more accurately described byquantum mechanics.) Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the additionaldegrees of freedom, e.g., abaseball canspin while it is moving. However, the results for point particles can be used to study such objects by treating them ascomposite objects, made of a large number of collectively acting point particles. Thecenter of mass 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 alsoAction at a distance).
Theposition of apoint particle is defined in relation to acoordinate system centered on an arbitrary fixed reference point inspace called the originO. A simple coordinate system might describe the position of aparticleP with avector notated by an arrow labeledr that points from the originO to pointP. In general, the point particle does not need to be stationary relative toO. In cases whereP is moving relative toO,r is defined as a function oft,time. In pre-Einstein relativity (known asGalilean relativity), time is considered an absolute, i.e., thetime interval that is observed to elapse between any given pair of events is the same for all observers.[15] In addition to relying onabsolute time, classical mechanics assumesEuclidean geometry for the structure of space.[16]
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 at60 − 50 = 10 km/h. 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 asvector quantities; they must be dealt with usingvector analysis.
Mathematically, if the velocity of the first object in the previous discussion is denoted by the vectoru =ud and the velocity of the second object by the vectorv =ve, whereu is the speed of the first object,v is the speed of the second object, andd ande areunit vectors in the directions of motion of each object respectively, then the velocity of the first object as seen by the second object is:
Similarly, the first object sees the velocity of the second object as:
When both objects are moving in the same direction, this equation can be simplified to:
Or, by ignoring direction, the difference can be given in terms of speed only:
Theacceleration, or rate of change of velocity, is thederivative of the velocity with respect to time (thesecond derivative of the position with respect to time):
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 asdeceleration, but generally any change in the velocity over time, including deceleration, is referred to as acceleration.
While the position, velocity and acceleration of aparticle can be described with respect to anyobserver in any state of motion, classical mechanics assumes the existence of a special family ofreference frames in which the mechanical laws of nature take a comparatively simple form. These special reference frames are calledinertial 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,, is valid.[3]: 4
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.[3]: 105 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 asfictitious forces, inertia forces, or pseudo-forces.
Consider tworeference framesS andS'. For observers in each of the reference frames an event has space-time coordinates of (x,y,z,t) in frameS and (x',y',z',t') in frameS'. Assuming time is measured the same in all reference frames, if we requirex =x' whent = 0, then the relation between the space-time coordinates of the same event observed from the reference framesS' andS, which are moving at a relative velocityu in thex direction, is:
The transformations have the following consequences:
v′ =v −u (the velocityv′ of a particle from the perspective ofS′ is slower byu than its velocityv from the perspective ofS)
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)
thespeed of light is not a constant in classical mechanics, nor does the special position given to the speed of light inrelativistic 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 fictitiouscentrifugal force andCoriolis force.
A force in physics is any action that causes an object's velocity to change; that is, to accelerate. A force originates from within afield, such as an electro-static field (caused by static electrical charges), electro-magnetic field (caused by moving charges), or gravitational field (caused by mass), among others.
Newton was the first to mathematically express the relationship betweenforce andmomentum. Some physicists interpretNewton's second law of motion as a definition of force and mass, while others consider it a fundamental postulate, a law of nature.[17] Either interpretation has the same mathematical consequences, historically known as "Newton's second law":
The quantitymv 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 isa = dv/dt, the second law can be written in the simplified and more familiar form:
So long as the force acting on a particle is known, Newton's second law is sufficient to describe the motion of a particle. Once independent relations for each force acting on a particle are available, they can be substituted into Newton's second law to obtain anordinary differential equation, which is called theequation 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:
whereλ is a positive constant, the negative sign states that the force is opposite the sense of the velocity. Then the equation of motion is
wherev0 is the initial velocity. This means that the velocity of this particledecays 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 theconservation of energy), and the particle is slowing down. This expression can be further integrated to obtain the positionr of the particle as a function of time.
Illustration of how the non-central Lorentz force needs to take into account the direction and relative reference frames to calculate the cross product of vectors.
Newton's third law can be used to deduce the forces acting on a particle when in a closed system. 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. Forconservative forces, this means that the line integral around a closed loop is zero. The strong form of Newton's third law requires that F and −F act along the line connecting A and B, and these forces are defined ascentral forces. However, central forces are an approximation since objects that are at rest are only at rest with respect to one another.[18] This limitation to Newton's third law can be shown using theCoulomb force, where charges must remain stationary with respect to a nonaccelerating frame of reference.[19] When dealing with non-central forces like theLorentz force, the weak form of Newton's third law is used by identifyingconservation of momentum. Illustrations of the weak form of Newton's third law can be found for magnetic forces like the Lorentz force while discussing the curl or cross product of vectors. Thus, the forces acting on objects cannot be identified without accounting for relative acceleration and direction by utilizing reference frames.[20]
If a constant forceF applied to a particle displaces it from positionrinitial torfinal, then thework done, by the force is defined as thescalar product of that force and the displacement vectorΔr =rfinal −rinitial:
More generally, if the force varies as a function of position as the particle moves fromr1 tor2 along a pathC, the work done on the particle is given by theline integral
If the work done in moving the particle fromr1 tor2 is the same no matter what path is taken, the force is said to beconservative.Gravity is a conservative force, as is the force due to an idealizedspring, as given byHooke's law. The force due tofriction is non-conservative.
Thekinetic energyEk of a particle of massm travelling at speedv is given by
For extended objects composed of many particles, the kinetic energy of the composite body is the sum of the kinetic energies of the particles.
Thework–energy theorem states that for a particle of constant massm, the total workW done on the particle as it moves from positionr1 tor2 is equal to the change inkinetic energyEk of the particle:
Conservative forces can be expressed as thegradient of a scalar function, known as thepotential energy and denotedEp:
If all the forces acting on a particle are conservative, andEp 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
The decrease in the potential energy is equal to the increase in the kinetic energy
This result is known asconservation of energy and states that the totalenergy,
is constant in time. It is often useful, because many commonly encountered forces are conservative.
Lagrangian mechanics is a formulation of classical mechanics founded on thestationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomerJoseph-Louis Lagrange in his presentation to the Turin Academy of Science in 1760[21] culminating in his 1788 grand opus,Mécanique analytique. Lagrangian mechanics describes a mechanical system as a pair consisting of aconfiguration space and a smooth function within that space called a Lagrangian. For many systems, where and are thekinetic andpotential energy of the system, respectively. The stationary action principle requires that theaction functional of the system derived from must remain at a stationary point (amaximum,minimum, orsaddle) throughout the time evolution of the system. This constraint allows the calculation of the equations of motion of the system using Lagrange's equations.[22]
Hamiltonian mechanics emerged in 1833 as a reformulation ofLagrangian mechanics. Introduced bySir William Rowan Hamilton,[23] Hamiltonian mechanics replaces (generalized) velocities 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 andPoisson structures) and serves as a link between classical andquantum mechanics.
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 ofpartial derivatives of a function called the Hamiltonian:The Hamiltonian is theLegendre transform of the Lagrangian, and in many situations of physical interest it is equal to the total energy of the system.
Classical mechanics has a wide range of application but its impact on physics is not limited to its practical applications. The techniques and point of view in classical mechanics is a critical foundation for modern physics.[3]: viii The mathematical techniques of classical mechanics have been adapted far beyond their original source of inspiration.[24]: 2
When both quantum mechanics and classical mechanics cannot apply, such as at the quantum level with many degrees of freedom,quantum field theory (QFT) is of use. QFT deals with small distances, and large speeds with many degrees of freedom as well as the possibility of any change in the number of particles throughout the interaction. When treating large degrees of freedom at the macroscopic level,statistical mechanics becomes useful. Statistical mechanics describes the behavior of large (but countable) numbers of particles and their interactions as a whole at the macroscopic level. Statistical mechanics is mainly used inthermodynamics for systems that lie outside the bounds of the assumptions of classical thermodynamics. In the case of highvelocity objects approaching the speed of light, classical mechanics is enhanced byspecial relativity. In case that objects become extremely heavy (i.e., theirSchwarzschild radius is not negligibly small for a given application), deviations fromNewtonian mechanics become apparent and can be quantified by using theparameterized post-Newtonian formalism. In that case,general relativity (GR) becomes applicable. However, until now there is no theory ofquantum gravity unifying GR and QFT in the sense that it could be used when objects become extremely small and heavy.[4][5]
In special relativity, the momentum of a particle is given by
wherem is the particle's rest mass,v its velocity,v is the modulus ofv, andc is the speed of light.
Ifv is very small compared toc,v2/c2 is approximately zero, and so
Thus the Newtonian equationp =mv 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 acyclotron,gyrotron, or high voltagemagnetron is given by
wherefc is the classical frequency of an electron (or other charged particle) with kinetic energyT and (rest) massm0 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 ray approximation of classical mechanics breaks down when thede Broglie wavelength is not much smaller than other dimensions of the system. For non-relativistic particles, this wavelength is
Again, this happens withelectrons before it happens with heavier particles. For example, the electrons used byClinton Davisson andLester Germer in 1927, accelerated by 54 V, had a wavelength of 0.167 nm, which was long enough to exhibit a singlediffractionside lobe when reflecting from the face of a nickelcrystal with atomic spacing of 0.215 nm. With a largervacuum chamber, it would seem relatively easy to increase theangular resolution from around a radian to amilliradian and see quantum diffraction from the periodic patterns ofintegrated circuit computer memory.
Classical mechanics is the same extremehigh frequency approximation asgeometric optics. It is more often accurate because it describes particles and bodies withrest mass. These have more momentum and therefore shorter De Broglie wavelengths than massless particles, such as light, with the same kinetic energies.
The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects inscience,engineering, andtechnology. The development of classical mechanics lead to the development of many areas of mathematics.[25]: 54
SomeGreek philosophers of antiquity, among themAristotle, founder ofAristotelian 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 mathematicaltheory and controlledexperiment, 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 hisElementa super demonstrationem ponderum, medieval mathematicianJordanus de Nemore introduced the concept of "positionalgravity" and the use of componentforces.
The first publishedcausal explanation of the motions ofplanets was Johannes Kepler'sAstronomia nova, published in 1609. He concluded, based onTycho Brahe's observations on the orbit ofMars, that the planet's orbits wereellipses. This break withancient thought was happening around the same time thatGalileo 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 thetower 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 aninclined plane. His theory of accelerated motion was derived from the results of such experiments and forms a cornerstone of classical mechanics. In 1673Christiaan Huygens described in hisHorologium Oscillatorium the first twolaws of motion.[26] The work is also the first modern treatise in which a physical problem (theaccelerated motion of a falling body) isidealized by a set of parameters then analyzed mathematically and constitutes one of the seminal works ofapplied mathematics.[27]
SirIsaac Newton (1643–1727), an influential figure in the history of physics and whosethree laws of motion form the basis of classical mechanics
Newton founded his principles of natural philosophy on three proposedlaws of motion: thelaw of inertia, his second law of acceleration (mentioned above), and the law ofaction 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'sPhilosophiæ 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 ofconservation of momentum andangular momentum. In mechanics, Newton was also the first to provide the first correct scientific and mathematical formulation ofgravity inNewton's law of universal gravitation. 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 ofKepler's laws of motion of the planets.
Newton had previously invented thecalculus; however, thePrincipia was formulated entirely in terms of long-established geometric methods in emulation ofEuclid. Newton, and most of his contemporaries, with the notable exception ofHuygens, worked on the assumption that classical mechanics would be able to explain all phenomena, includinglight, in the form ofgeometric optics. Even when discovering the so-calledNewton's rings (awave interference phenomenon) he maintained his owncorpuscular theory of light.
Lagrange's contribution was realising Newton's ideas in the language of modern mathematics, now calledLagrangian mechanics.
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 byJoseph Louis Lagrange. Lagrangian mechanics was in turn re-formulated in 1833 byWilliam 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 withelectromagnetic theory, and the famousMichelson–Morley experiment. The resolution of these problems led to thespecial theory of relativity, often still considered a part of classical mechanics.
A second set of difficulties were related to thermodynamics. When combined withthermodynamics, classical mechanics leads to theGibbs paradox of classicalstatistical mechanics, in whichentropy is not a well-defined quantity.Black-body radiation was not explained without the introduction ofquanta. As experiments reached the atomic level, classical mechanics failed to explain, even approximately, such basic things as theenergy levels and sizes ofatoms and thephoto-electric effect. The effort at resolving these problems led to the development ofquantum mechanics.
Since the end of the 20th century, classical mechanics inphysics 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 theStandard Model and its more modern extensions into a unifiedtheory 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.
^Ben-Chaim, Michael (2004),Experimental Philosophy and the Birth of Empirical Science: Boyle, Locke and Newton, Aldershot: Ashgate,ISBN0-7546-4091-4,OCLC53887772.
^Robert Rynasiewicz; Edward N. Zalta (2022).Newton's Views on Space, Time, and Motion (Spring 2022 ed.). Metaphysics Research Lab, Stanford University: The Stanford Encyclopedia of Philosophy.
^Taylor, John (2005).Classical Mechanics. University Science Books. pp. 133–138.ISBN1-891389-22-X.
^Griffiths, David (2023).Introduction to Electrodynamics (4th ed.). Cambridge: Cambridge University Press. pp. 316–318.
^Fraser, Craig (1983). "J. L. Lagrange's Early Contributions to the Principles and Methods of Mechanics".Archive for History of Exact Sciences.28 (3):197–241.doi:10.1007/BF00328268.JSTOR41133689.
^Hamilton, William Rowan (1833).On a general method of expressing the paths of light, & of the planets, by the coefficients of a characteristic function. Printed by P.D. Hardy.OCLC68159539.
