In
physics, equations of motion are
equations that describe the behavior of a
physical system in terms of its
motion 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 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 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 classical m ...
, but are replaced by
curved space
Curved space often refers to a spatial geometry which is not "flat", where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. Cu ...
s 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 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 su ...
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 motion ...
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 values, 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 (the first time
derivative of , ), and its acceleration (the second
derivative of , ), and time .
Euclidean vectors 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 ,
:
where is time, and each overdot denotes one
time derivative. The
initial conditions are given by the ''constant'' values at ,
:
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 syste ...
, 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 and is more likely to be exactly solvable. In general, the equation will be
non-linear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
, and cannot be solved exactly so a variety of approximations must be used. The solutions to nonlinear equations may show
chaotic
Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program was able to be seen on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kid ...
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,
priests,
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. Dif ...
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 obse ...
s predicted solar and lunar
eclipses, 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. 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
Merton College (in full: The House or College of Scholars of Merton in the University of Oxford) is one of the constituent colleges of the University of Oxford in England. Its foundation can be traced back to the 1260s when Walter de Merton, c ...
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 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'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 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 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. 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 parallel ...
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
Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician, scientist and music theorist. He made various contributions in many areas of science and engineering, both theoretical and practical. He also translated vario ...
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,
Isaac Newton,
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 mat ...
, 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
In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
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
spacetime meant the classical equations of motion were also modified to account for the finite
speed of light, and
curvature of spacetime
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. G ...
. 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 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 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, ]
:
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 curvature
In geometry, the center of curvature of a curve is found at a point that is at a distance from the curve equal to the radius of curvature lying on the normal vector. It is the point at infinity if the curvature is zero. The osculating ci ...
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 :
:
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 vec ...
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 :
:
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, 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 the ...
.
[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.
:
where:
* is the particle's initial
position
* is the particle's final position
* is the particle's initial
velocity
* 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 the ...
* 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 ;
:
in magnitudes,
:
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
:
and substituting into
:
then simplifying to get
:
or in magnitudes
:
From
:
substituting for in
:
From
:
substituting into
:
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, 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:
:
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 algeb ...
. The derivations are essentially the same as in the collinear case,
:
although the
Torricelli 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 ...
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 arithmeti ...
of the dot product as follows:
:
:
:
Applications
Elementary and frequent examples in kinematics involve
projectile
A projectile is an object that is propelled by the application of an external force and then moves freely under the influence of gravity and air resistance. Although any objects in motion through space are projectiles, they are commonly found in ...
s, 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
Scalar may refer to:
* Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
, 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 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:
:
Substituting and cancelling minus signs gives:
:
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,
:
where is the constant
angular acceleration
In physics, angular acceleration refers to the time rate of change of angular velocity. As there are two types of angular velocity, namely spin angular velocity and orbital angular velocity, there are naturally also two types of angular accelera ...
, is the
angular velocity
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
, 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 a ...
), 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
:
where and are the
polar unit vectors. Differentiating with respect to time gives the velocity
:
with radial component and an additional component due to the rotation. Differentiating with respect to time again obtains the acceleration
:
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 the ...
,
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
:
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 motion ...
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, ]
:
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 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 continuum, 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
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 and ...
; 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 (see
four-momentum).
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 influenced by other particles is
:
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 forc ...
. 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,
:
by equating the
torque 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 syste ...
. Analogous to mass times acceleration, the
moment of inertia tensor 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 is
:
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 examples
[The 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 the ...
,
:
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 thrown in the air, in air currents (such as wind) described by a vector field of resistive forces ,
:
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,
:
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 , 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 derivatives of the generalized coordinates are the ''generalized velocities''
:
The
Euler–Lagrange equations are
[Classical Mechanics (second edition), T.W.B. Kibble, European Physics Series, 1973, ]
:
where the ''Lagrangian'' is a function of the configuration and its time rate of change (and possibly time )
:
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 are
:
where the Hamiltonian
:
is a function of the configuration and conjugate ''
"generalized" momenta''
:
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 mult ...
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 is
:
where
:
is ''Hamilton's principal function'', also called the ''
classical action'' is a
functional
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional sy ...
of . 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
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
''
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 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
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, c ...
, a theorem due to
Emmy Noether.
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
:
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
In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
:
:
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 and charge :
:
where and are the electromagnetic
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
*Scalar (physics), a physical quantity that can be described by a single element of a number field such a ...
and
vector 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 cl ...
in Lagrangian mechanics is given by:
:
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):
:
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 ...
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
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
s with a
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
, the metric provides the notion of arc length (see
line element for details). The
differential arc length is given by:
:
and the geodesic equation is a second-order differential equation in the coordinates. The general solution is a family of geodesics:
:
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 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 ...
'':
:
where is the separation vector between two geodesics, (''not'' just ) is the
covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differen ...
, and 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.
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 q ...
, which enter the equations of motion under
covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differen ...
s 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 b ...
. The
Mathisson–Papapetrou–Dixon equations 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
waves 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 equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
s, since the waves or fields are functions of space and time. For a particular solution,
boundary conditions
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to t ...
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
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called ''stateful systems''). In this formulation, ''time'' is not required to be a continuous parameter, but may be disc ...
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 th ...
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 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).
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 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 equations''. 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 ampl ...
. 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 equations can be derived. The general linear wave equation in 3D is:
:
where is any mechanical or electromagnetic field amplitude, say:
* the
transverse or
longitudinal
Longitudinal is a geometric term of location which may refer to:
* Longitude
** Line of longitude, also called a meridian
* Longitudinal engine, an internal combustion engine in which the crankshaft is oriented along the long axis of the vehicle, ...
displacement of a vibrating rod, wire, cable, membrane etc.,
* the fluctuating
pressure of a medium,
sound pressure
Sound pressure or acoustic pressure is the local pressure deviation from the ambient (average or equilibrium) atmospheric pressure, caused by a sound wave. In air, sound pressure can be measured using a microphone, and in water with a hydrophone ...
,
* the
electric fields or , or the
magnetic fields or ,
* the
voltage 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 which ...
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, 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 ...
, 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 is the
wavefunction of the system, is the quantum
Hamiltonian operator, rather than a function as in classical mechanics, and is the
Planck constant 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 physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers. In other words, it says ...
, 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
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. T ...
by the
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
,
*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 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 that ...
to quantum mechanics and field theory, placing emphasis on the use of a Lagrangians rather than Hamiltonians.
See also
*
Scalar (physics)
*
Vector
*
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
*
Velocity
*
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 the ...
*
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 a ...
*
Angular speed
Angular may refer to:
Anatomy
* Angular artery, the terminal part of the facial artery
* Angular bone, a large bone in the lower jaw of amphibians and reptiles
* Angular incisure, a small anatomical notch on the stomach
* Angular gyrus, a region ...
*
Angular velocity
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
*
Angular acceleration
In physics, angular acceleration refers to the time rate of change of angular velocity. As there are two types of angular velocity, namely spin angular velocity and orbital angular velocity, there are naturally also two types of angular accelera ...
*
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 i ...
*
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
*
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
*
Euler–Lagrange equation
*
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)
*
Newton–Euler laws of motion for a rigid body
References
{{reflist
Classical mechanics
Equations of physics