Talk:Motion

Intro - Motion, momentum, etc.
There were a few flaws in the introduction section.
 * "motion is the result of applied force" is 99% correct, but it could be misunderstood as that sustained motion required sustained applied force (as was wrongly believed in ancient times). Basically, applied force results in change in motion. Linear motion is the result of absent (resulting) force).


 * "Constant motion is the natural state of anything in the universe"‽ Give me a break! That's something you can tell your kids when they are 3. But it's just too wrong for an encyclopedia. Please don't believe everything that some kind of professor said... There is no notion of a "natural state" in physics. And most motion is definitely not constant!


 * Laws in physics (or in any other science) are not responsible for anything, they do not cause anything. They are describe the nature. (Think about that!)

Tang Wenlong (talk) 20:34, 18 March 2008 (UTC)


 * Nice job in improving clarity and brevity of the intro. Earthdirt (talk) 02:42, 19 March 2008 (UTC)

MOTION is 1st cause, and last effect possible. Motion is creator/maker of all using energy potential states it changes. MOTION is teacher that magic and supernatural do not exist by it being the example of no mind no brain yet able to do all. MOTION in form teach the key to everything is based in concentric circles and spheres.

My name is Peter...i am teaching you this. — Preceding unsigned comment added by 64.229.70.228 (talk) 11:33, 15 March 2022 (UTC)

Removing ERIKA
hi, i'm removing some stuff is hard to make sense of in terms of modern physics, and i'm pasting it here in case someone can rescue it. Boud 15:50, 1 Jun 2005 (UTC)


 * relative space and relative time result in relative motion, which means that the unit values of space and time can change for observers moving at high speeds relative to each other.

Kinds of motion
all kinds of motion are- 1.rotatory motion, 2.rectilinear motion, 3.translatory motion, 4.curvelinear motion, 5.oscilatory motion, 6.vibratory motion, 7.periodic motion, 8.non-periodic motion, 9.unoform motion, 10.non-uniform motion.

According to rate
1) uniform motion

2) (uniformly) accelerated motion 3) (uniformly) deceleretated

--Ionn-Korr 21:30, 27 October 2005 (UTC)

Motion defined by time = speed, yes?
I think we have the definition of motion wrong. If you put time into the definition, doesn't that become a measurement of speed? Hmmm?

Ye mi too (am JAZ 4rm Nig)no at all.


 * I could argue otherwise. If time equals speed, then speed wouldn't neccessarily exist. You see, all of time, every impossibly small fraction of a second, contains an impossibly large amount of moments. To measure speed, you would need some kind of motion, even 0.0(repeating)1x10-99999(repeating) fm. However, an instant, a nothingth of a second, can have no movement. Therefore, even if multiplied infinitely, the 0 speed of an object in an instant would be 0. Now some would argue that 0 time has 0 speed, and x time would have y speed, so it isn't of any effect. However, if this moment in question doesn't matter, neither would any other. What would happen if you didn't have a single instant ever? There would be no time. There just isn't motion in a moment, so that means that if you don't have any of something in an increment, it wouldn't be in any larger incriments. By using time=speed, I could argue that motion is an illusion and all motion is stroboscopic. ChristopherEdwards 17:49, 5 October 2007 (UTC)

motion
There are four main types of motion.There's rotational, linear, reciprocating and also oscillating. —The preceding unsigned comment was added by 74.117.131.130 (talk) 22:15, 25 February 2007 (UTC).

that is a good point, but there is also another way of looking at it, namely continuous and discrete or alternating motion, with starts and stops. Why lok at the one half or one quarter of cycle only? And what is reciprocating motion? What about spiral motion?

Don 't you find that you perceive motion through the trace that a moving object leaves behind?

Genezistan (talk) 09:33, 2 December 2011 (UTC)

WikiProject class rating
This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 04:18, 10 November 2007 (UTC)

huh?
i do not understand any of this info!!!!YOLO!!! —Preceding unsigned comment added by 71.194.212.28 (talk) 11:23, 9 May 2009 (UTC)

You just need too look at it the right way and use other articles, and you can understand. :) BlueKing360 (talk) 22:30, 16 May 2018 (UTC)

Newton and Motion
Those aren't Newton's Laws of Motion. Those are a modern rephrasing of Newton's Laws of Motion (and not all that accurate, defining force to be mass times acceleration!). Also, as far as defining motion is concerned, if this article has anything to do with Newton, you should look to how Newton defines "quantity of motion." It isn't merely an object's change in position with respect to time, which would be speed.

Further, the article needs a history section. There was a long line of thoughts about motion before Newton, going back to the ancient Greeks: pre-Socratics, Plato, Aristotle. JKeck (talk) 19:35, 25 April 2011 (UTC)

Its wrong Shagun sindhia (talk) 14:00, 8 April 2019 (UTC)

Absolute Reference Frame
The article says, "As there is no absolute frame of reference, absolute motion cannot be determined; this is emphasized by the term relative motion." As written, this is wrong.

There is an absolute standard of rest: it's that of the co-moving observers in the Friedmann–Lemaître–Robertson–Walker metric, and we can tell how fast we're moving with respect to it by looking at the cosmic microwave background radiation. Perhaps what the article means to say is that there are no locally detectable reference frames. In other words, if we were sealed in a (small enough) box that allowed no access to the outside world, we couldn't detect our absolute motion. But we can know our absolute motion with respect to the cosmos's origin.

In any event, the article should be corrected. JKeck (talk) 01:36, 9 July 2011 (UTC

Poor definition of motion
Motion is the change of place of an object - but not with respect to time, but with respect to its former place. Hence motion is the distance that a object covers in a span of space which at the same time is a span of time. If you take that distance as one unit, and the corresponding unit of time also as one, then the question is what is speed? Since we know that both the unit of time and the unit of length is an arbitrary measure, I wonder how this is sorted out?

Section on kinematics
The brief section on kinematics seems to state only that there are two types of motion uniform and non-uniform. This seems obvious and not very helpful. I think revisions are needed. Prof McCarthy (talk) 06:40, 2 January 2012 (UTC)
 * I replaced this section with a discussion of planar, spherical and spatial motion of machine systems.Prof McCarthy (talk) 12:10, 2 January 2012 (UTC)

Edits to the lead
The lead to this article talks about "change in action," momentum and forces more than it introduces motion. Prof McCarthy (talk) 12:32, 2 January 2012 (UTC)

Temporal change
I am not sure that motion is best understood to mean: "More generally, the term motion signifies any temporal change in a physical system." An increase in temperature, elongation under load, or a chemical change could correctly be viewed as motion at an atomic level, but "motion" is not the first thing that comes to mind in these physical processes. So this seems an awkward way of describing motion. In addition, I realize there is a desire to include motions of quantum particles, but I do not believe it is correct to say that "the concept of position does not apply." The specific position that a sub-atomic particle can occupy with a prescribed probability is well-defined in quantum mechanics. Prof McCarthy (talk) 06:15, 10 January 2012 (UTC)

Motion
motion is the changing in distance of an object to a reference point. you other people made that concept way to complicated. — Preceding unsigned comment added by 68.50.153.61 (talk) 17:35, 4 May 2013 (UTC)

Light??? And the definition of motion seems to be wrong...
The Light section in this article seems irrelevant. I really don't get how light came into this subject on motion.

Motion is not the change in position of an object with reference to time, but just with its initial reference point.... Motion is just the change in position of an object...

A_Random_Person (talk) 18:53, 7 June 2013 (UTC)

Good point BlueKing360 (talk) 22:32, 16 May 2018 (UTC)

Degrees of freedom
The article does not mention degrees of freedom (physics). It is not good, because the motion degrees of freedom pertain to any particle and have the same description (save for peculiarities of massless particles), whereas internal degrees of freedom do not have a uniform description, both classical and quantum. Also, the article should explicitly name and distinguish translational motion and its standard descriptions (velocity and wave function respectively), and rotational motion and its descriptions (angular velocity and angular momentum operator respectively). Incnis Mrsi (talk) 08:51, 17 January 2014 (UTC)

Newton's laws in motion
1: A is steady, when 2: B acts on A, then 3: A reacts to B, end.

--KYPark (talk) 16:39, 24 January 2016 (UTC)

Good faith addition by Varshit1234 cut&pasted here
In mathematical physics, equations of motion are equations that describe the behaviour of a physical system in terms of its motion as afunction of time.[1] More specifically, the equations of motion describe the behaviour of a physical system as a set of mathematical functions in terms of dynamic variables: normally spatial coordinates and time are used, but others are also possible, such as momentum components and time. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system.[2] The functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions to the differential equations describing the motion of the dynamics.

There are two main descriptions of motion: dynamics and kinematics. Dynamics is general, since momenta, forces and energy of the particlesare taken into account. In this instance, sometimes the term refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.

However, kinematics is simpler as 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 (s), initial velocity (u), final velocity (v), acceleration (a), and time (t). (see below).

Equations of motion can therefore be grouped under these main classifiers of motion. In all cases, the main types of motion are translations,rotations, oscillations, or any combinations of these.

A differential equation of motion, usually identified as some physical law and applying definitions 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 values, which fixes the values of the constants.

To state this formally, in general an equation of motion M is a function of the position r of the object, its velocity (the first time derivative of r, v = dr/dt), and its acceleration (the secondderivative of r, a = d2r/dt2), and time t. Euclidean vectors in 3D are denoted throughout in bold. This is equivalent to saying an equation of motion in r is a second order ordinary differential equation (ODE) in r, where t is time, and each overdot denotes one time derivative. The initial conditions are given by the constant values at t = 0, The solution r(t) to the equation of motion, with specified initial values, describes the system for all times t after t = 0. Other dynamical variables like the momentum p of the object, or quantities derived from r and p like angular momentum, can be used in place of r as the quantity to solve for from some equation of motion, although the position of the object at time t is by far the most sought-after quantity.

Sometimes, the equation will be 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.

Contents
[hide]
 * 1History
 * 2Kinematic equations for one particle
 * 2.1Kinematic quantities
 * 2.2Uniform acceleration
 * 2.2.1Constant translational acceleration in a straight line
 * 2.2.2Constant linear acceleration in any direction
 * 2.2.3Applications
 * 2.2.4Constant circular acceleration
 * 2.3General planar motion
 * 2.4General 3D motion
 * 3Dynamic equations of motion
 * 3.1Newtonian mechanics
 * 3.2Applications
 * 4Analytical mechanics
 * 5Electrodynamics
 * 6General relativity
 * 6.1Geodesic equation of motion
 * 6.2Spinning objects
 * 7Analogues for waves and fields
 * 7.1Field equations
 * 7.2Wave equations
 * 7.3Quantum theory
 * 8See also
 * 9References

History[edit source | edit]
Historically, equations of motion first appeared in classical mechanics to describe the motion of massive objects, a notable application was to celestial mechanics to predict the motion of the planets as if they orbit like clockwork (this was how Neptune was predicted before its discovery), and also investigate the stability of the solar system.

It is important to observe that the huge body of work involving kinematics, dynamics and the mathematical models of the universe developed in baby steps – faltering, getting up and correcting itself – over three millennia and included contributions of both known names and others who have since faded from the annals of history.

In antiquity, notwithstanding the success of priests, astrologers and astronomers in predicting solar and lunar eclipses, the solstices and the equinoxes of the Sun and the period of theMoon, there was nothing other than a set of algorithms to help them. Despite the great strides made in the development of geometry in the Ancient Greece and surveys in Rome, we were to wait for another thousand years before the first equations of motion arrive.

The exposure of Europe to the collected works by the Muslims of the Greeks, the Indians and the Islamic scholars, such as Euclid’s Elements, the works of Archimedes, and Al-Khwārizmī's treatises [3] began in Spain, and scholars from all over Europe went to Spain, read, copied and translated the learning into Latin. The exposure of Europe to Arabic numerals and their ease in computations encouraged first the scholars to learn them and then the merchants and invigorated the spread of knowledge throughout Europe.

By the 13th century the universities of Oxford and Paris had come up, and the scholars were now studying mathematics and philosophy with lesser worries about mundane chores of life—the fields were not as clearly demarcated as they are in the modern times. Of these, compendia and redactions, such as those of Johannes Campanus, of Euclid and Aristotle, confronted scholars with ideas about infinity and the ratio theory of elements as a means of expressing relations between various quantities involved with moving bodies. These studies led to a new body of knowledge that is now known as physics.

Of these institutes Merton College sheltered a group of scholars devoted to natural science, mainly physics, astronomy and mathematics, of similar in stature to the intellectuals at the University of Paris. Thomas Bradwardine, one of those scholars, 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 further extended Bradwardine's arguments. The Merton school proved the 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's Physics 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 shockingly correct regarding the definitions of acceleration (acceleration was a rate of change of motion (velocity) in time) and the observation that during the violent motion of ascent acceleration would be negative.

Discourses such as these spread throughout the Europe and definitely influenced Galileo and others, and helped in laying the foundation of kinematics.[4] Galileo deduced the equations = 1/2gt2 in his work geometrically,[5] using Merton's rule, now known as a special case of one of the equations of kinematics. He couldn't use the now-familiar mathematical reasoning. The relationships between speed, distance, time and acceleration was not known at the time.

Galileo was the first to show that the path of a projectile is a parabola. Galileo had an understanding of centrifugal force 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[6] 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 was when he was 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 to be independent of the mass and material of the pendulum and as the square root of its length.

Thus we arrive at René Descartes, Isaac Newton, Gottfried Leibniz, 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, 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 and general relativity, the theoretical modifications to spacetimemeant the classical equations of motion were also modified to account for the finite speed of light, 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.[7]

However, the equations of quantum mechanics can also be considered "equations of motion", since they are differential equations of the wavefunction, 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 quantities[edit source | edit]
Kinematic quantities of a classical particle of massm: position r, velocity v, acceleration a.

From the instantaneous position r = r(t), instantaneous meaning at an instant value of time t, the instantaneous velocity v = v(t) and acceleration a = a(t) have the general, coordinate-independent definitions;[8] Notice that velocity always points in the direction of motion, in other words for a curved path it is the tangent vector. Loosely speaking, first order derivatives are related to tangents of curves. Still for curved paths, the acceleration is directed towards the center of curvatureof 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) θ = θ(t), angular velocity ω = ω(t), and angular acceleration α = α(t): where n̂ is a unit vector in the direction of the axis of rotation, and θ 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 ω:[9] where r is the position vector of the particle (radial from the rotation axis) and v the tangential velocity of the particle. For a rotating continuum rigid body, these relations hold for each point in the rigid body.

Uniform acceleration[edit source | edit]
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[edit source | edit]
These equations apply to a particle moving linearly, in three dimensions in a straight line with constant acceleration.[10] 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. where: [show]
 * r0 is the particle's initial position
 * r is the particle's final position
 * v0 is the particle's initial velocity
 * v is the particle's final velocity
 * a is the particle's acceleration
 * t is the time interval

Derivation

Here a is constant acceleration, or in the case of bodies moving under the influence of gravity, the standard gravity g 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: where u has replaced v0, s replaces r, and s0 = 0. They are often referred to as the "SUVAT" equations, where "SUVAT" is an acronym from the variables: s = displacement (s0 = initial displacement), u = initial velocity, v = final velocity, a = acceleration, t = time.[11][12]

Constant linear acceleration in any direction[edit source | edit]
Trajectory of a particle with initial position vector r0 and velocity v0, subject to constant acceleration a, all three quantities in any direction, and the position r(t) and velocity v(t) after time t.

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. The derivations are essentially the same as in the collinear case, although the Torricelli equation [4] can be derived using the distributive property of the dot product as follows:

Applications[edit source | edit]
Elementary and frequent examples in kinematics involve projectiles, for example a ball thrown upwards into the air. Given initial speed u, one can calculate how high the ball will travel before it begins to fall. The acceleration is local acceleration of gravity g. At this point one must remember that 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 s to measure up from the ground, the acceleration a must be in fact −g, since the force ofgravity acts downwards and therefore also the acceleration on the ball due to it.

At the highest point, the ball will be at rest: therefore v = 0. Using equation [4] in the set above, we have: Substituting and cancelling minus signs gives:

Constant circular acceleration[edit source | edit]
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, where α is the constant angular acceleration, ω is the angular velocity, ω0 is the initial angular velocity, θ is the angle turned through (angular displacement), θ0 is the initial angle, and t is the time taken to rotate from the initial state to the final state.

General planar motion[edit source | edit]
Main article: General planar motion

Position vector r, always points radially from the origin.

Velocity vector v, always tangent to the path of motion.

Acceleration vector a, not parallel to the radial motion but offset by the angular and Coriolis accelerations, nor tangent to the path but offset by the centripetal and radial accelerations.

Kinematic vectors in plane polar coordinates. Notice the setup is not restricted to 2D space, but a plane in any higher dimension.

These are the kinematic equations for a particle traversing a path in a plane, described by position r = r(t).[13]They are simply the time derivatives of the position vector in plane polar coordinates using the definitions of physical quantities above for angular velocity ω and angular acceleration α.

The position, velocity and acceleration of the particle are respectively: where êr and êθ are the polar unit vectors. For the velocity v, dr/dt is the component of velocity in the radial direction, and rω is the additional component due to the rotation. For the acceleration a, –rω2 is the centripetal acceleration and 2ωdr/dt the Coriolis acceleration, in addition to the radial acceleration d2r/dt2 and angular acceleration rα.

Special cases of motion described be 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 motion[edit source | edit]
Main article: Spherical coordinate system

In 3D space, the equations in spherical coordinates (r, θ, φ) with corresponding unit vectors êr, êθ and êφ, the position, velocity, and acceleration generalize respectively to In the case of a constant φ this reduces to the planar equations above.

Newtonian mechanics[edit source | edit]
Main article: Newtonian mechanics

The first general equation of motion developed was Newton's second law of motion, in its most general form states the rate of change of momentum p = p(t) = mv(t) of an object equals the force F = F(x(t), v(t), t) acting on it,[14] 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 since m is a constant in Newtonian mechanics.

Newton's second law applies to point-like particles, and to all points in a rigid body. They also apply to each point in a mass continua, like deformable solids or fluids, but the motion of the system must be accounted for, see material derivative. In the case the mass is not constant, it is not sufficient to use the product rule for the time derivative on the mass and velocity, and Newton's second law requires some modification consistent with conservation of momentum, 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, generalizes to special and general relativity (see four-momentum).[14] 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 equation of motion for one particle i influenced by other particles is[8][15] where pi is the momentum of particle i, Fij is the force on particle i by particle j, and FE is the resultant external force due to any agent not part of system. Particle i 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. The Newton–Euler equations 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,[16] by equating the torque acting on the body to the rate of change of its angular momentum L. Analogous to mass times acceleration, the moment of inertia tensor I depends on the distribution of mass about the axis of rotation, and the angular acceleration is the rate of change of angular velocity, 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 i is[17] where Li is the angular momentum of particle i, τij the torque on particle i by particle j, and τE is resultant external torque (due to any agent not part of system). Particle i does not exert a torque on itself.

Applications[edit source | edit]
Some examples[18] of Newton's law include describing the motion of a simple pendulum, and a damped, sinusoidally driven harmonic oscillator, 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 m thrown in the air, in air currents (such as wind) described by a vector field of resistive forces R = R(r, t), where G is the gravitational constant, M the mass of the Earth, and A = R/m is the acceleration of the projectile due to the air currents at position r and time t.

The classical N-body problem for N particles each interacting with each other due to gravity is a set of N nonlinear coupled second order ODEs, where i = 1, 2, …, N labels the quantities (mass, position, etc.) associated with each particle.

Analytical mechanics[edit source | edit]
Main articles: Analytical mechanics, Lagrangian mechanics and Hamiltonian mechanics

As the system evolves, q traces a path through configuration space (only some are shown). The path taken by the system (red) has a stationary action (δS = 0) under small changes in the configuration of the system (δq).[19]

Using all three coordinates of 3D space is unnecessary if there are constraints on the system. If the system has N degrees of freedom, then one can use a set of N generalized coordinates q(t) = [q1(t), q2(t) ... qN(t)], to define the configuration of the system. They can be in the form of arc lengths or angles. 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 derivatives of the generalized coordinates are thegeneralized velocities The Euler–Lagrange equations are[2][20] where the Lagrangian is a function of the configuration q and its time rate of change dq/dt (and possibly time t) Setting up the Lagrangian of the system, then substituting into the equations and evaluating the partial derivatives and simplifying, a set of coupled N second order ODEs in the coordinates are obtained.

Hamilton's equations are[2][20] where the Hamiltonian is a function of the configuration q and conjugate "generalized" momenta in which ∂/∂q = (  ∂/∂q1, ∂/∂q2, …, ∂/∂qN  )  is a shorthand notation for a vector of partial derivatives with respect to the indicated variables (see for example matrix calculus for this denominator notation), and possibly time t,

Setting up the Hamiltonian of the system, then substituting into the equations and evaluating the partial derivatives and simplifying, a set of coupled 2N first order ODEs in the coordinates qiand momenta pi are obtained.

The Hamilton–Jacobi equation is[2] where is Hamilton's principal function, also called the classical action is a functional of L. In this case, the momenta are given by Although the equation has a simple general form, for a given Hamiltonian it is actually a single first order non-linear PDE, in N + 1 variables. The action S allows identification of conserved quantities for mechanical systems, even when the mechanical problem itself cannot be solved fully, because any differentiable symmetry of the action of a physical system has a corresponding conservation law, a theorem due to Emmy Noether.

All classical equations of motion can be derived from the variational principle known as Hamilton's principle of least action stating the path the system takes through the configuration space is the one with the least action S.

Electrodynamics[edit source | edit]
Lorentz force f on a charged particle (of charge q) in motion (instantaneous velocity v). The Efield and B field vary in space and time.

In electrodynamics, the force on a charged particle of charge q is the Lorentz force:[21] Combining with Newton's second law gives a first order differential equation of motion, in terms of position of the particle: or its momentum: The same equation can be obtained using the Lagrangian (and applying Lagrange's equations above) for a charged particle of mass m and charge q:[22] where A and ϕ are the electromagnetic scalar and vector potential fields. The Lagrangian indicates an additional detail: the canonical momentum in Lagrangian mechanics is given by: instead of just mv, 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):[20] can derive the Lorentz force equation.

Geodesic equation of motion[edit source | edit]
Geodesics on a sphere are arcs of great circles (yellow curve). On a2D–manifold (such as the sphere shown), the direction of the accelerating geodesic is uniquely fixed if the separation vector ξ isorthogonal to the "fiducial geodesic" (green curve). As the separation vector ξ0 changes to ξ after a distance s, the geodesics are not parallel (geodesic deviation).[23]

Main articles: Geodesics in general relativity and Geodesic equation

The above equations are valid in flat spacetime. In curved space spacetime, 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 g, the metric provides the notion of arc length (see line element for details), the differential arc length is given by:[24] and the geodesic equation is a second-order differential equation in the coordinates, the general solution is a family of geodesics:[25] where Γ μαβ 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 T αβ, the Einstein field equations are a set of non-linear second-order partial differential equations in the metric, and imply the curvature of space time is equivalent to a gravitational field (see principle of equivalence). 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: where ξα = x2α − x1α is the separation vector between two geodesics, D/ds (not just d/ds) is the covariant derivative, and Rαβγδ is the Riemann curvature tensor, 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.[26]

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[edit source | edit]
In general relativity, rotational motion is described by the relativistic angular momentum tensor, including the spin tensor, which enter the equations of motion under covariant derivatives with respect to proper time. The Mathisson–Papapetrou–Dixon equations describe the motion of spinning objects moving in a gravitational field.

Analogues for waves and fields[edit source | edit]
Unlike the equations of motion for describing particle mechanics, which are systems of coupled ordinary differential equations, the analogous equations governing the dynamics of waves andfields 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[edit source | edit]
Equations that describe the spatial dependence and time evolution of fields are called field equations. These include This terminology is not universal: for example although the Navier–Stokes equations govern the velocity field 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.
 * Maxwell's equations for the electromagnetic field,
 * Poisson's equation for Newtonian gravitational or electrostatic field potentials,
 * the Einstein field equation for gravitation (Newton's law of gravity is a special case for weak gravitational fields and low velocities of particles).

Wave equations[edit source | edit]
Equations of wave motion are called wave equations. The solutions to a wave equation give the time-evolution and spatial dependence of the amplitude. Boundary conditions determine if the solutions describe traveling waves or standing waves.

From classical equations of motion and field equations; mechanical, gravitational wave, and electromagnetic wave equations can be derived. The general linear wave equation in 3D is: where X = X(r, t) is any mechanical or electromagnetic field amplitude, say:[27] and v is the phase velocity. Nonlinear equations model the dependence of phase velocity on amplitude, replacing v by v(X). There are other linear and nonlinear wave equations for very specific applications, see for example the Korteweg–de Vries equation.
 * the transverse or longitudinal displacement of a vibrating rod, wire, cable, membrane etc.,
 * the fluctuating pressure of a medium, sound pressure,
 * the electric fields E or D, or the magnetic fields B or H,
 * the voltage V or current I in an alternating current circuit,

Quantum theory[edit source | edit]
In quantum theory, the wave and field concepts both appear.

In quantum mechanics, in which particles also have wave-like properties according to wave–particle duality, the analogue of the classical equations of motion (Newton's law, Euler–Lagrange equation, Hamilton–Jacobi equation, etc.) is the Schrödinger equation in its most general form: where Ψ is the wavefunction of the system, Ĥ is the quantum Hamiltonian operator, rather than a function as in classical mechanics, and ħ is the Planck constant divided by 2π. 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 in when one considers the correspondence principle, in the limit that ħ 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 theirquantum operators and the classical Poisson bracket by the commutator,
 * the phase space formulation closely follows classical Hamiltonian mechanics, placing position and momentum on equal footing,
 * the Feynman path integral formulation extends the principle of least action

The material above was added 13 March 2016 at 14:25 (UTC) by User:Varshit1234. Some of it may be useful in the article. Just plain Bill (talk) 15:21, 13 March 2016 (UTC)

Definition in lead seems to exclude rotation
As I understand it, a position vector only includes information about location w.r.t. a reference point, without specifying the attitude or orientation of the object. A quick reading of the lead paragraph offers no hint that rotation may also be described as motion. (There are other issues with clarity and quality of sourcing, but this may be enough to open discussion with...) Just plain Bill (talk) 14:29, 15 November 2016 (UTC)

WHAT IS A MIRROR WORLD ?
IT IS A BELIEF OF SOME SCIENTISTS LIKE EINSTIEN THAT THERE EXISTS A OTHER WORLD AND THIS WORLD IS ITS COPY. IT IS GIVEN AS PER THE THEORY OF "MANY WORLDS" BY EINSTEIN Aabina shah (talk) 07:21, 15 December 2017 (UTC)

External links modified (January 2018)
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Motion
I think you should give a little more about the motion because it will help us to understand it briefly Sarajit roy 678 (talk) 04:10, 6 October 2018 (UTC)

Requested move 2 February 2019

 * The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this section. 

The result of the move request was: moved (closed by non-admin page mover)  SITH   (talk)   10:10, 9 February 2019 (UTC)

– Obviously the primary topic. Previously moved a decade ago to make room for a disambiguation page. That's not needed, many articles with motion in the title are referring to this, some are much less important, and we have other physical primary topics like Velocity. w umbolo  ^^^  10:21, 2 February 2019 (UTC)
 * Motion (physics) → Motion
 * Motion → Motion (disambiguation)


 * Support - fundamental concept and primary topic. -- Netoholic @ 16:07, 2 February 2019 (UTC)
 * Support - per nom and Netoholic. No other topics by this name are as fundamental. Gap9551 (talk) 17:05, 2 February 2019 (UTC)
 * Support. This is clearly the primary topic per |Motion_(legal)|Motion_(physics)|Motion_(parliamentary_procedure)|Motion_(democracy) page views.  Calidum   19:48, 2 February 2019 (UTC)
 * Support Per nom, the others in the "Procedures" section surely derive from the physics topic.  Crouch, Swale  ( talk ) 19:51, 2 February 2019 (UTC)
 * Comment. There's also Movement, a disambiguation page, but I believe that it should stay as such. Off-topic, but there are wikidata duplicates of that disambiguation page, see Q1223600 and Q3326911. w umbolo   ^^^  20:53, 2 February 2019 (UTC)
 * Support as primary topic. Rreagan007 (talk) 19:56, 4 February 2019 (UTC)


 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

Classical Mechanics
The entire section titled "Classical Mechanics" is just a different (and slightly expanded) version of the preceding text. Be it the significance of classical mechanics to the laws of motion to what classical mechanics describes; it's the same thing. Emdosis (talk) 01:13, 10 October 2021 (UTC)

Wiki Education assignment: PHY 381 History of Modern Physics
— Assignment last updated by Janyahmercedes (talk) 02:52, 26 January 2023 (UTC)

ListSorter
I'm undoing Special:Diff/1131381527 "(Sorted bullet lists using ListSorter)" since it broke the formatting. It moved most, but not all, of a long reference from the text into the "See also" section, leaving the behind, and moved bulleted items from the "Types of motion" section up to where the reference used to be. This looks odd and was certainly not a minor edit. I believe the following two edits were only trying to fix up the problems caused by the first edit, so I'm going to restore the version before the first edit. -- John of Reading (talk) 11:09, 12 January 2023 (UTC)