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In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, equations of motion are equations that describe the behavior of a physical system in terms of its
motion In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and m ...
as a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1 (VHC Inc.) 0-89573-752-3 More specifically, the equations of motion describe the behavior of a physical system as a set of mathematical functions in terms of dynamic variables. These variables are usually spatial coordinates and time, but may include momentum components. The most general choice are
generalized coordinates In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 39 ...
which can be any convenient variables characteristic of the physical system.''Analytical Mechanics'', L.N. Hand, J.D. Finch, Cambridge University Press, 2008, The functions are defined in a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
in
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics.


Types

There are two main descriptions of motion: dynamics and kinematics. Dynamics is general, since the momenta, forces and
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
of the
particles In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from s ...
are taken into account. In this instance, sometimes the term ''dynamics'' refers to the differential equations that the system satisfies (e.g.,
Newton's second law 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 moti ...
or Euler–Lagrange equations), and sometimes to the solutions to those equations. However, kinematics is simpler. It concerns only variables derived from the positions of objects and time. In circumstances of constant acceleration, these simpler equations of motion are usually referred to as the SUVAT equations, arising from the definitions of kinematic quantities: displacement (), initial velocity (), final velocity (), acceleration (), and time (). A differential equation of motion, usually identified as some
physical law Scientific laws or laws of science are statements, based on repeated experiments or observations, that describe or predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, accurate, broad, or narrow) ...
and applying
definitions A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definiti ...
of physical quantities, is used to set up an equation for the problem. Solving the differential equation will lead to a general solution with arbitrary constants, the arbitrariness corresponding to a family of solutions. A particular solution can be obtained by setting the
initial value In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or ot ...
s, which fixes the values of the constants. To state this formally, in general an equation of motion is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
of the position of the object, its
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
(the first time
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of , ), and its acceleration (the second
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of , ), and time .
Euclidean vector In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ...
s in 3D are denoted throughout in bold. This is equivalent to saying an equation of motion in is a second-order
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
(ODE) in , :M\left mathbf(t),\mathbf(t),\mathbf(t),t\right0\,, where is time, and each overdot denotes one
time derivative A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t. Notation A variety of notations are used to denote th ...
. The
initial conditions In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). For ...
are given by the ''constant'' values at , : \mathbf(0) \,, \quad \mathbf(0) \,. The solution to the equation of motion, with specified initial values, describes the system for all times after . Other dynamical variables like the momentum of the object, or quantities derived from and like
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
, can be used in place of as the quantity to solve for from some equation of motion, although the position of the object at time is by far the most sought-after quantity. Sometimes, the equation will be
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
and is more likely to be exactly solvable. In general, the equation will be non-linear, and cannot be solved exactly so a variety of approximations must be used. The solutions to nonlinear equations may show chaotic behavior depending on how ''sensitive'' the system is to the initial conditions.


History

Kinematics, dynamics and the mathematical models of the universe developed incrementally over three millennia, thanks to many thinkers, only some of whose names we know. In antiquity,
priest A priest is a religious leader authorized to perform the sacred rituals of a religion, especially as a mediatory agent between humans and one or more deities. They also have the authority or power to administer religious rites; in partic ...
s,
astrologers Astrology is a range of divinatory practices, recognized as pseudoscientific since the 18th century, that claim to discern information about human affairs and terrestrial events by studying the apparent positions of celestial objects. Di ...
and
astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, moons, comets and galaxies – in either ...
s predicted solar and lunar
eclipses An eclipse is an astronomical event that occurs when an astronomical object or spacecraft is temporarily obscured, by passing into the shadow of another body or by having another body pass between it and the viewer. This alignment of three ce ...
, the solstices and the equinoxes of the
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
and the period of the
Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
. But they had nothing other than a set of algorithms to guide them. Equations of motion were not written down for another thousand years. Medieval scholars in the thirteenth century — for example at the relatively new universities in Oxford and Paris — drew on ancient mathematicians (Euclid and Archimedes) and philosophers (Aristotle) to develop a new body of knowledge, now called physics. At Oxford, Merton College sheltered a group of scholars devoted to natural science, mainly physics, astronomy and mathematics, who were of similar stature to the intellectuals at the University of Paris. Thomas Bradwardine extended Aristotelian quantities such as distance and velocity, and assigned intensity and extension to them. Bradwardine suggested an exponential law involving force, resistance, distance, velocity and time.
Nicholas Oresme Nicole Oresme (; c. 1320–1325 – 11 July 1382), also known as Nicolas Oresme, Nicholas Oresme, or Nicolas d'Oresme, was a French philosopher of the later Middle Ages. He wrote influential works on economics, mathematics, physics, astrology ...
further extended Bradwardine's arguments. The Merton school proved that the quantity of motion of a body undergoing a uniformly accelerated motion is equal to the quantity of a uniform motion at the speed achieved halfway through the accelerated motion. For writers on kinematics before Galileo, since small time intervals could not be measured, the affinity between time and motion was obscure. They used time as a function of distance, and in free fall, greater velocity as a result of greater elevation. Only Domingo de Soto, a Spanish theologian, in his commentary on
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ph ...
's ''
Physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
'' published in 1545, after defining "uniform difform" motion (which is uniformly accelerated motion) – the word velocity wasn't used – as proportional to time, declared correctly that this kind of motion was identifiable with freely falling bodies and projectiles, without his proving these propositions or suggesting a formula relating time, velocity and distance. De Soto's comments are remarkably correct regarding the definitions of acceleration (acceleration was a rate of change of motion (velocity) in time) and the observation that acceleration would be negative during ascent. Discourses such as these spread throughout Europe, shaping the work of
Galileo Galilei 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 wa ...
and others, and helped in laying the foundation of kinematics.The Britannica Guide to History of Mathematics, ed. Erik Gregersen Galileo deduced the equation in his work geometrically, using the Merton rule, now known as a special case of one of the equations of kinematics. Galileo was the first to show that the path of a projectile is a
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...
. Galileo had an understanding of
centrifugal force In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is parall ...
and gave a correct definition of momentum. This emphasis of momentum as a fundamental quantity in dynamics is of prime importance. He measured momentum by the product of velocity and weight; mass is a later concept, developed by Huygens and Newton. In the swinging of a simple pendulum, Galileo says in '' Discourses'' that "every momentum acquired in the descent along an arc is equal to that which causes the same moving body to ascend through the same arc." His analysis on projectiles indicates that Galileo had grasped the first law and the second law of motion. He did not generalize and make them applicable to bodies not subject to the earth's gravitation. That step was Newton's contribution. The term "inertia" was used by Kepler who applied it to bodies at rest. (The first law of motion is now often called the law of inertia.) Galileo did not fully grasp the third law of motion, the law of the equality of action and reaction, though he corrected some errors of Aristotle. With Stevin and others Galileo also wrote on statics. He formulated the principle of the parallelogram of forces, but he did not fully recognize its scope. Galileo also was interested by the laws of the pendulum, his first observations of which were as a young man. In 1583, while he was praying in the cathedral at Pisa, his attention was arrested by the motion of the great lamp lighted and left swinging, referencing his own pulse for time keeping. To him the period appeared the same, even after the motion had greatly diminished, discovering the isochronism of the pendulum. More careful experiments carried out by him later, and described in his Discourses, revealed the period of oscillation varies with the square root of length but is independent of the mass the pendulum. Thus we arrive at
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Ma ...
,
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...
,
Gottfried Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...
, et al.; and the evolved forms of the equations of motion that begin to be recognized as the modern ones. Later the equations of motion also appeared in
electrodynamics In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
, when describing the motion of charged particles in electric and magnetic fields, the Lorentz force is the general equation which serves as the definition of what is meant by an electric field and magnetic field. With the advent of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws ...
and
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, the theoretical modifications to
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
meant the classical equations of motion were also modified to account for the finite
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
, and curvature of spacetime. In all these cases the differential equations were in terms of a function describing the particle's trajectory in terms of space and time coordinates, as influenced by forces or energy transformations. However, the equations of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
can also be considered "equations of motion", since they are differential equations of the
wavefunction A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
, which describes how a quantum state behaves analogously using the space and time coordinates of the particles. There are analogs of equations of motion in other areas of physics, for collections of physical phenomena that can be considered waves, fluids, or fields.


Kinematic equations for one particle


Kinematic quantities

From the instantaneous position , instantaneous meaning at an instant value of time , the instantaneous velocity and acceleration have the general, coordinate-independent definitions;Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, : \mathbf = \frac \,, \quad \mathbf = \frac = \frac \,\! Notice that velocity always points in the direction of motion, in other words for a curved path it is the
tangent vector In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are e ...
. Loosely speaking, first order derivatives are related to tangents of curves. Still for curved paths, the acceleration is directed towards the center of curvature of the path. Again, loosely speaking, second order derivatives are related to curvature. The rotational analogues are the "angular vector" (angle the particle rotates about some axis) , angular velocity , and angular acceleration : :\boldsymbol = \theta \hat \,,\quad \boldsymbol = \frac \,, \quad \boldsymbol= \frac \,, where is a
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
in the direction of the axis of rotation, and is the angle the object turns through about the axis. The following relation holds for a point-like particle, orbiting about some axis with angular velocity : : \mathbf = \boldsymbol\times \mathbf \,\! where is the position vector of the particle (radial from the rotation axis) and the tangential velocity of the particle. For a rotating continuum
rigid body In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...
, these relations hold for each point in the rigid body.


Uniform acceleration

The differential equation of motion for a particle of constant or uniform acceleration in a straight line is simple: the acceleration is constant, so the second derivative of the position of the object is constant. The results of this case are summarized below.


Constant translational acceleration in a straight line

These equations apply to a particle moving linearly, in three dimensions in a straight line with constant
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...
.Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, second Edition, 1978, John Murray, Since the position, velocity, and acceleration are collinear (parallel, and lie on the same line) – only the magnitudes of these vectors are necessary, and because the motion is along a straight line, the problem effectively reduces from three dimensions to one. :\begin v & = at+v_0 & \ r & = r_0 + v_0 t + \tfrac12 t^2 & \ r & = r_0 + \tfrac12 \left( v+v_0 \right )t & \ v^2 & = v_0^2 + 2a\left( r - r_0 \right) & \ r & = r_0 + vt - \tfrac12 t^2 & \ \end where: * is the particle's initial position * is the particle's final position * is the particle's initial
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
* is the particle's final velocity * is the particle's
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...
* is the
time interval Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to c ...
Equations and are from integrating the definitions of velocity and acceleration, subject to the initial conditions and ; :\begin \mathbf & = \int \mathbf dt = \mathbft+\mathbf_0 \,, & \\ \mathbf & = \int (\mathbft+\mathbf_0) dt = \frac+\mathbf_0t +\mathbf_0 \,, & \\ \end in magnitudes, :\begin v & = at+v_0 \,, & \\ r & = \frac+v_0t +r_0 \,. & \\ \end Equation involves the average velocity . Intuitively, the velocity increases linearly, so the average velocity multiplied by time is the distance traveled while increasing the velocity from to , as can be illustrated graphically by plotting velocity against time as a straight line graph. Algebraically, it follows from solving for : \mathbf = \frac and substituting into : \mathbf = \mathbf_0 + \mathbf_0 t + \frac(\mathbf - \mathbf_0) \,, then simplifying to get : \mathbf = \mathbf_0 + \frac(\mathbf + \mathbf_0) or in magnitudes : r = r_0 + \left( \frac \right )t \quad From :t = \left( r - r_0 \right)\left( \frac \right ) substituting for in :\begin v & = a\left( r - r_0 \right)\left( \frac \right )+v_0 \\ v\left( v+v_0 \right ) & = 2a\left( r - r_0 \right)+v_0\left( v+v_0 \right ) \\ v^2+vv_0 & = 2a\left( r - r_0 \right)+v_0v+v_0^2 \\ v^2 & = v_0^2 + 2a\left( r - r_0 \right) & \\ \end From : 2\left(r - r_0\right) - vt = v_0 t substituting into : \begin r & = \frac + 2r - 2r_0 - vt + r_0 \\ 0 & = \frac+r - r_0 - vt \\ r & = r_0 + vt - \frac & \end Usually only the first 4 are needed, the fifth is optional. Here is ''constant'' acceleration, or in the case of bodies moving under the influence of
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
, the standard gravity is used. Note that each of the equations contains four of the five variables, so in this situation it is sufficient to know three out of the five variables to calculate the remaining two. In elementary physics the same formulae are frequently written in different notation as: :\begin v & = u + at & \\ s & = ut + \tfrac12 at^2 & \\ s & = \tfrac(u + v)t & \\ v^2 & = u^2 + 2as & \\ s & = vt - \tfrac12 at^2 & \\ \end where has replaced , replaces . They are often referred to as the SUVAT equations, where "SUVAT" is an
acronym An acronym is a word or name formed from the initial components of a longer name or phrase. Acronyms are usually formed from the initial letters of words, as in ''NATO'' (''North Atlantic Treaty Organization''), but sometimes use syllables, as ...
from the variables: = displacement, = initial velocity, = final velocity, = acceleration, = time.


Constant linear acceleration in any direction

The initial position, initial velocity, and acceleration vectors need not be collinear, and take an almost identical form. The only difference is that the square magnitudes of the velocities require the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
. The derivations are essentially the same as in the collinear case, :\begin \mathbf & = \mathbft+\mathbf_0 & \ \mathbf & = \mathbf_0 + \mathbf_0 t + \tfrac12\mathbft^2 & \ \mathbf & = \mathbf_0 + \tfrac12 \left(\mathbf+\mathbf_0\right) t & \ v^2 & = v_0^2 + 2\mathbf\cdot\left( \mathbf - \mathbf_0 \right) & \ \mathbf & = \mathbf_0 + \mathbft - \tfrac12\mathbft^2 & \ \end although the Torricelli equation can be derived using the
distributive property In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmet ...
of the dot product as follows: :v^ = \mathbf\cdot\mathbf = (\mathbf_0+\mathbft)\cdot(\mathbf_0+\mathbft) = v_0^+2t(\mathbf\cdot\mathbf_0)+a^t^ :(2\mathbf)\cdot(\mathbf-\mathbf_0) = (2\mathbf)\cdot\left(\mathbf_0t+\tfrac\mathbft^\right)=2t(\mathbf\cdot\mathbf_0)+a^t^ = v^ - v_0^ :\therefore v^ = v_0^ + 2(\mathbf\cdot(\mathbf-\mathbf_0))


Applications

Elementary and frequent examples in kinematics involve projectiles, for example a ball thrown upwards into the air. Given initial speed , one can calculate how high the ball will travel before it begins to fall. The acceleration is local acceleration of gravity . While these quantities appear to be scalars, the direction of displacement, speed and acceleration is important. They could in fact be considered as unidirectional vectors. Choosing to measure up from the ground, the acceleration must be in fact , since the force of
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
acts downwards and therefore also the acceleration on the ball due to it. At the highest point, the ball will be at rest: therefore . Using equation in the set above, we have: :s= \frac. Substituting and cancelling minus signs gives: :s = \frac.


Constant circular acceleration

The analogues of the above equations can be written for rotation. Again these axial vectors must all be parallel to the axis of rotation, so only the magnitudes of the vectors are necessary, :\begin \omega & = \omega_0 + \alpha t \\ \theta &= \theta_0 + \omega_0t + \tfrac12\alpha t^2 \\ \theta & = \theta_0 + \tfrac12(\omega_0 + \omega)t \\ \omega^2 & = \omega_0^2 + 2\alpha(\theta - \theta_0) \\ \theta & = \theta_0 + \omega t - \tfrac12\alpha t^2 \\ \end where is the constant angular acceleration, is the angular velocity, is the initial angular velocity, is the angle turned through (
angular displacement Angular displacement of a body is the angle (in radians, degrees or revolutions) through which a point revolves around a centre or a specified axis in a specified sense. When a body rotates about its axis, the motion cannot simply be analyzed ...
), is the initial angle, and is the time taken to rotate from the initial state to the final state.


General planar motion

These are the kinematic equations for a particle traversing a path in a plane, described by position . They are simply the time derivatives of the position vector in plane
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
using the definitions of physical quantities above for angular velocity and angular acceleration . These are instantaneous quantities which change with time. The position of the particle is : \mathbf =\mathbf\left ( r(t),\theta(t) \right ) = r \mathbf_r where and are the polar unit vectors. Differentiating with respect to time gives the velocity :\mathbf = \mathbf_r \frac + r \omega \mathbf_\theta with radial component and an additional component due to the rotation. Differentiating with respect to time again obtains the acceleration :\mathbf =\left ( \frac - r\omega^2\right )\mathbf_r + \left ( r \alpha + 2 \omega \frac \right )\mathbf_\theta which breaks into the radial acceleration ,
centripetal acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by th ...
,
Coriolis acceleration 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 ...
, and angular acceleration . Special cases of motion described by these equations are summarized qualitatively in the table below. Two have already been discussed above, in the cases that either the radial components or the angular components are zero, and the non-zero component of motion describes uniform acceleration.


General 3D motions

In 3D space, the equations in spherical coordinates with corresponding unit vectors , and , the position, velocity, and acceleration generalize respectively to : \begin \mathbf & =\mathbf\left ( t \right ) = r \mathbf_r\\ \mathbf & = v \mathbf_r + r\,\frac\mathbf_\theta + r\,\frac\,\sin\theta \mathbf_\varphi \\ \mathbf & = \left( a - r\left(\frac\right)^2 - r\left(\frac\right)^2\sin^2\theta \right)\mathbf_r \\ & + \left( r \frac + 2v\frac - r\left(\frac\right)^2\sin\theta\cos\theta \right) \mathbf_\theta \\ & + \left( r\frac\,\sin\theta + 2v\,\frac\,\sin\theta + 2 r\,\frac\,\frac\,\cos\theta \right) \mathbf_\varphi \end \,\! In the case of a constant this reduces to the planar equations above.


Dynamic equations of motion


Newtonian mechanics

The first general equation of motion developed was
Newton's second law 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 moti ...
of motion. In its most general form it states the rate of change of momentum of an object equals the force acting on it,An Introduction to Mechanics, D. Kleppner, R.J. Kolenkow, Cambridge University Press, 2010, p. 112, : \mathbf = \frac The force in the equation is ''not'' the force the object exerts. Replacing momentum by mass times velocity, the law is also written more famously as : \mathbf = m\mathbf since is a constant in Newtonian mechanics. Newton's second law applies to point-like particles, and to all points in a
rigid body In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...
. They also apply to each point in a mass continuum, like deformable solids or fluids, but the motion of the system must be accounted for; see
material derivative In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material der ...
. In the case the mass is not constant, it is not sufficient to use the
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
for the time derivative on the mass and velocity, and Newton's second law requires some modification consistent with
conservation of 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 ...
; see variable-mass system. It may be simple to write down the equations of motion in vector form using Newton's laws of motion, but the components may vary in complicated ways with spatial coordinates and time, and solving them is not easy. Often there is an excess of variables to solve for the problem completely, so Newton's laws are not always the most efficient way to determine the motion of a system. In simple cases of rectangular geometry, Newton's laws work fine in Cartesian coordinates, but in other coordinate systems can become dramatically complex. The momentum form is preferable since this is readily generalized to more complex systems, such as
special Special or specials may refer to: Policing * Specials, Ulster Special Constabulary, the Northern Ireland police force * Specials, Special Constable, an auxiliary, volunteer, or temporary; police worker or police officer Literature * ''Specia ...
and
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
(see
four-momentum In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is ...
). It can also be used with the momentum conservation. However, Newton's laws are not more fundamental than momentum conservation, because Newton's laws are merely consistent with the fact that zero resultant force acting on an object implies constant momentum, while a resultant force implies the momentum is not constant. Momentum conservation is always true for an isolated system not subject to resultant forces. For a number of particles (see
many body problem The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provid ...
), the equation of motion for one particle influenced by other particles is : \frac = \mathbf_ + \sum_ \mathbf_ \,\! where is the momentum of particle , is the force on particle by particle , and is the resultant external force due to any agent not part of system. Particle does not exert a force on itself. Euler's laws of motion are similar to Newton's laws, but they are applied specifically to the motion of
rigid bodies In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...
. The
Newton–Euler equations In classical mechanics, the Newton–Euler equations describe the combined translational and rotational dynamics of a rigid body. Traditionally the Newton–Euler equations is the grouping together of Euler's two laws of motion for a rig ...
combine the forces and torques acting on a rigid body into a single equation. Newton's second law for rotation takes a similar form to the translational case, :\boldsymbol = \frac \,, by equating the
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
acting on the body to the rate of change of its
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
. Analogous to mass times acceleration, the moment of inertia
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
depends on the distribution of mass about the axis of rotation, and the angular acceleration is the rate of change of angular velocity, : \boldsymbol = \mathbf \cdot \boldsymbol. Again, these equations apply to point like particles, or at each point of a rigid body. Likewise, for a number of particles, the equation of motion for one particle is : \frac = \boldsymbol_E + \sum_ \boldsymbol_ \,, where is the angular momentum of particle , the torque on particle by particle , and is resultant external torque (due to any agent not part of system). Particle does not exert a torque on itself.


Applications

Some examplesThe Physics of Vibrations and Waves (3rd edition), H.J. Pain, John Wiley & Sons, 1983, of Newton's law include describing the motion of a
simple pendulum A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward th ...
, : - mg\sin\theta = m\frac \quad \Rightarrow \quad \frac = - \frac\sin\theta \,, and a damped, sinusoidally driven harmonic oscillator, : F_0 \sin(\omega t) = m\left(\frac + 2\zeta\omega_0\frac + \omega_0^2 x \right)\,. For describing the motion of masses due to gravity, Newton's law of gravity can be combined with Newton's second law. For two examples, a ball of mass thrown in the air, in air currents (such as wind) described by a vector field of resistive forces , : - \frac \mathbf_r + \mathbf = m\frac + 0 \quad \Rightarrow \quad \frac = - \frac \mathbf_r + \mathbf where is the gravitational constant, the mass of the Earth, and is the acceleration of the projectile due to the air currents at position and time . The classical -body problem for particles each interacting with each other due to gravity is a set of nonlinear coupled second order ODEs, :\frac = G\sum_\frac (\mathbf_j - \mathbf_i) where labels the quantities (mass, position, etc.) associated with each particle.


Analytical mechanics

Using all three coordinates of 3D space is unnecessary if there are constraints on the system. If the system has degrees of freedom, then one can use a set of
generalized coordinates In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 39 ...
, to define the configuration of the system. They can be in the form of
arc length ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...
s or
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...
s. They are a considerable simplification to describe motion, since they take advantage of the intrinsic constraints that limit the system's motion, and the number of coordinates is reduced to a minimum. The
time derivative A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t. Notation A variety of notations are used to denote th ...
s of the generalized coordinates are the ''generalized velocities'' :\mathbf = \frac \,. The Euler–Lagrange equations areClassical Mechanics (second edition), T.W.B. Kibble, European Physics Series, 1973, : \frac \left ( \frac \right ) = \frac \,, where the ''Lagrangian'' is a function of the configuration and its time rate of change (and possibly time ) :L = L\left \mathbf(t), \mathbf(t), t \right \,. Setting up the Lagrangian of the system, then substituting into the equations and evaluating the partial derivatives and simplifying, a set of coupled second order
ODE An ode (from grc, ᾠδή, ōdḗ) is a type of lyric poetry. Odes are elaborately structured poems praising or glorifying an event or individual, describing nature intellectually as well as emotionally. A classic ode is structured in three majo ...
s in the coordinates are obtained.
Hamilton's equations Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
are :\mathbf = -\frac \,, \quad \mathbf = + \frac \,, where the Hamiltonian :H = H\left \mathbf(t), \mathbf(t), t \right \,, is a function of the configuration and conjugate '' "generalized" momenta'' :\mathbf = \frac \,, in which is a shorthand notation for a vector of partial derivatives with respect to the indicated variables (see for example
matrix calculus In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a ...
for this denominator notation), and possibly time , Setting up the Hamiltonian of the system, then substituting into the equations and evaluating the partial derivatives and simplifying, a set of coupled first order ODEs in the coordinates and momenta are obtained. The
Hamilton–Jacobi equation In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mecha ...
is : - \frac = H\left(\mathbf, \mathbf, t \right) \,. where :S mathbf,t= \int_^L(\mathbf, \mathbf, t)\,dt \,, is ''Hamilton's principal function'', also called the '' classical action'' is a functional of . In this case, the momenta are given by :\mathbf = \frac\,. Although the equation has a simple general form, for a given Hamiltonian it is actually a single first order '' non-linear'' PDE, in variables. The action allows identification of conserved quantities for mechanical systems, even when the mechanical problem itself cannot be solved fully, because any
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
symmetry of the
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
of a physical system has a corresponding conservation law, a theorem due to
Emmy Noether Amalie Emmy NoetherEmmy is the '' Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noeth ...
. All classical equations of motion can be derived from the
variational principle In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations, which concerns finding functions that optimize the values of quantities that depend on those funct ...
known as Hamilton's principle of least action :\delta S = 0 \,, stating the path the system takes through the configuration space is the one with the least action .


Electrodynamics

In electrodynamics, the force on a charged particle of charge is the Lorentz force: :\mathbf = q\left(\mathbf + \mathbf \times \mathbf\right) \,\! Combining with Newton's second law gives a first order differential equation of motion, in terms of position of the particle: :m\frac = q\left(\mathbf + \frac \times \mathbf\right) \,\! or its momentum: :\frac = q\left(\mathbf + \frac\right) \,\! The same equation can be obtained using the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
(and applying Lagrange's equations above) for a charged particle of mass and charge : :L=\tfrac12 m \mathbf\cdot\mathbf+q\mathbf\cdot\dot-q\phi where and are the electromagnetic scalar and
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
potential fields. The Lagrangian indicates an additional detail: the
canonical momentum In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of cla ...
in Lagrangian mechanics is given by: : \mathbf = \frac = m \dot + q \mathbf instead of just , implying the motion of a charged particle is fundamentally determined by the mass and charge of the particle. The Lagrangian expression was first used to derive the force equation. Alternatively the Hamiltonian (and substituting into the equations): : H = \frac + q\phi \,\! can derive the Lorentz force equation.


General relativity


Geodesic equation of motion

The above equations are valid in flat spacetime. In
curved In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that a ...
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
, things become mathematically more complicated since there is no straight line; this is generalized and replaced by a '' geodesic'' of the curved spacetime (the shortest length of curve between two points). For curved manifolds with a metric tensor , the metric provides the notion of arc length (see
line element In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc ...
for details). The differential arc length is given by: :ds = \sqrt and the geodesic equation is a second-order differential equation in the coordinates. The general solution is a family of geodesics: :\frac = - \Gamma^\mu_\frac\frac where is a Christoffel symbol of the second kind, which contains the metric (with respect to the coordinate system). Given the mass-energy distribution provided by the stress–energy tensor , the
Einstein field equations In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form ...
are a set of non-linear second-order partial differential equations in the metric, and imply the curvature of spacetime is equivalent to a gravitational field (see equivalence principle). Mass falling in curved spacetime is equivalent to a mass falling in a gravitational field - because gravity is a fictitious force. The ''relative acceleration'' of one geodesic to another in curved spacetime is given by the ''
geodesic deviation equation In general relativity, if two objects are set in motion along two initially parallel trajectories, the presence of a tidal gravitational force will cause the trajectories to bend towards or away from each other, producing a relative acceleration be ...
'': :\frac = -R^\alpha_\frac\xi^\gamma\frac where is the separation vector between two geodesics, (''not'' just ) is the covariant derivative, and is the
Riemann curvature tensor In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
, containing the Christoffel symbols. In other words, the geodesic deviation equation is the equation of motion for masses in curved spacetime, analogous to the Lorentz force equation for charges in an electromagnetic field. For flat spacetime, the metric is a constant tensor so the Christoffel symbols vanish, and the geodesic equation has the solutions of straight lines. This is also the limiting case when masses move according to Newton's law of gravity.


Spinning objects

In general relativity, rotational motion is described by the
relativistic angular momentum In physics, relativistic angular momentum refers to the mathematical formalisms and physical concepts that define angular momentum in special relativity (SR) and general relativity (GR). The relativistic quantity is subtly different from the thr ...
tensor, including the
spin tensor In mathematics, mathematical physics, and theoretical physics, the spin tensor is a quantity used to describe the rotational motion of particles in spacetime. The tensor has application in general relativity and special relativity, as well as qu ...
, which enter the equations of motion under covariant derivatives with respect to
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval ...
. The
Mathisson–Papapetrou–Dixon equations In physics, specifically general relativity, the Mathisson–Papapetrou–Dixon equations describe the motion of a massive spinning body moving in a gravitational field. Other equations with similar names and mathematical forms are the Mathisson ...
describe the motion of spinning objects moving in a gravitational field.


Analogues for waves and fields

Unlike the equations of motion for describing particle mechanics, which are systems of coupled ordinary differential equations, the analogous equations governing the dynamics of
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
s and
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
s are always partial differential equations, since the waves or fields are functions of space and time. For a particular solution, boundary conditions along with initial conditions need to be specified. Sometimes in the following contexts, the wave or field equations are also called "equations of motion".


Field equations

Equations that describe the spatial dependence and time evolution of fields are called '' field equations''. These include *
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
for the electromagnetic field, *
Poisson's equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...
for
Newtonian gravitation Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distanc ...
al or
electrostatic Electrostatics is a branch of physics that studies electric charges at rest ( static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amb ...
field potentials, * the
Einstein field equation In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form ...
for gravitation ( Newton's law of gravity is a special case for weak gravitational fields and low velocities of particles). This terminology is not universal: for example although the
Navier–Stokes equations In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
govern the
velocity field In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
of a fluid, they are not usually called "field equations", since in this context they represent the momentum of the fluid and are called the "momentum equations" instead.


Wave equations

Equations of wave motion are called ''
wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seism ...
s''. The solutions to a wave equation give the time-evolution and spatial dependence of the
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
. Boundary conditions determine if the solutions describe
traveling wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
s or
standing waves In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
. From classical equations of motion and field equations; mechanical, gravitational wave, and
electromagnetic wave In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
equations can be derived. The general linear wave equation in 3D is: :\frac\frac = \nabla^2 X where is any mechanical or electromagnetic field amplitude, say: * the transverse or longitudinal displacement of a vibrating rod, wire, cable, membrane etc., * the fluctuating
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
of a medium, sound pressure, * the electric fields or , or the magnetic fields or , * the
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
or
current Currents, Current or The Current may refer to: Science and technology * Current (fluid), the flow of a liquid or a gas ** Air current, a flow of air ** Ocean current, a current in the ocean *** Rip current, a kind of water current ** Current (stre ...
in an
alternating current Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in whic ...
circuit, and is the phase velocity. Nonlinear equations model the dependence of phase velocity on amplitude, replacing by . There are other linear and nonlinear wave equations for very specific applications, see for example the Korteweg–de Vries equation.


Quantum theory

In quantum theory, the wave and field concepts both appear. In
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
, in which particles also have wave-like properties according to
wave–particle duality Wave–particle duality is the concept in quantum mechanics that every particle or quantum entity may be described as either a particle or a wave. It expresses the inability of the classical concepts "particle" or "wave" to fully describe the b ...
, the analogue of the classical equations of motion (Newton's law, Euler–Lagrange equation, Hamilton–Jacobi equation, etc.) is the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
in its most general form: :i\hbar\frac = \hat\Psi \,, where is the
wavefunction A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
of the system, is the quantum
Hamiltonian operator Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonia ...
, rather than a function as in classical mechanics, and is the
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
divided by 2. Setting up the Hamiltonian and inserting it into the equation results in a wave equation, the solution is the wavefunction as a function of space and time. The Schrödinger equation itself reduces to the Hamilton–Jacobi equation when one considers the correspondence principle, in the limit that becomes zero. Throughout all aspects of quantum theory, relativistic or non-relativistic, there are various formulations alternative to the Schrödinger equation that govern the time evolution and behavior of a quantum system, for instance: *the Heisenberg equation of motion resembles the time evolution of classical observables as functions of position, momentum, and time, if one replaces dynamical observables by their quantum operators and the classical Poisson bracket by the commutator, *the
phase space formulation The phase-space formulation of quantum mechanics places the position ''and'' momentum variables on equal footing in phase space. In contrast, the Schrödinger picture uses the position ''or'' momentum representations (see also position and mome ...
closely follows classical Hamiltonian mechanics, placing position and momentum on equal footing, *the Feynman
path integral formulation The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional i ...
extends the
principle of least action The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the '' action'' of a mechanical system, yields the equations of motion for that system. The principle states tha ...
to quantum mechanics and field theory, placing emphasis on the use of a Lagrangians rather than Hamiltonians.


See also

* Scalar (physics) *
Vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
*
Distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
* Displacement *
Speed In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (ma ...
*
Velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
*
Acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...
*
Angular displacement Angular displacement of a body is the angle (in radians, degrees or revolutions) through which a point revolves around a centre or a specified axis in a specified sense. When a body rotates about its axis, the motion cannot simply be analyzed ...
* Angular speed * Angular velocity * Angular acceleration * Equations for a falling body * Parabolic trajectory *
Curvilinear coordinates In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally inve ...
*
Orthogonal coordinates In mathematics, orthogonal coordinates are defined as a set of ''d'' coordinates q = (''q''1, ''q''2, ..., ''q'd'') in which the coordinate hypersurfaces all meet at right angles (note: superscripts are indices, not exponents). A coordinate su ...
*
Newton's laws of motion Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in moti ...
*
Projectile motion Projectile motion is a form of motion experienced by an object or particle (a projectile) that is projected in a gravitational field, such as from Earth's surface, and moves along a curved path under the action of gravity only. In the particul ...
*
Torricelli's equation In physics, Torricelli's equation, or Torricelli's formula, is an equation created by Evangelista Torricelli to find the final velocity of an object moving with a constant acceleration along an axis (for example, the x axis) without having a kno ...
*
Euler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
*
Generalized forces Generalized forces find use in Lagrangian mechanics, where they play a role conjugate to generalized coordinates. They are obtained from the applied forces, Fi, i=1,..., n, acting on a system that has its configuration defined in terms of generaliz ...
*
Defining equation (physics) In physics, defining equations are equations that define new quantities in terms of base quantities. This article uses the current SI system of units, not natural or characteristic units. Description of units and physical quantities Physical ...
* Newton–Euler laws of motion for a rigid body


References

{{reflist Classical mechanics Equations of physics