Equations of motion

In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. 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. 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 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 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 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 (s), initial velocity (u), final velocity (v), acceleration (a), and time (t). A differential equation of motion, usually identified as some physical law (for example, F = ma), and applying definitions of physical quantities, is used to set up an equation to solve a kinematics problem. Solving the differential equation will lead to a general solution with arbitrary constants, the arbitrariness corresponding to a set of solutions. A particular solution can be obtained by setting the initial values, which fixes the values of the constants. Stated 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 second derivative 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,

M [ r ( t ) , r ˙ ( t ) , r ¨ ( t ) , t ] = 0 , {\displaystyle M\left[\mathbf {r} (t),\mathbf {\dot {r}} (t),\mathbf {\ddot {r}} (t),t\right]=0\,,}

where t is time, and each overdot denotes one time derivative. The initial conditions are given by the constant values at t = 0,

r ( 0 ) , r ˙ ( 0 ) . {\displaystyle \mathbf {r} (0)\,,\quad \mathbf {\dot {r}} (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.

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 and astronomers predicted solar and lunar eclipses, the solstices and the equinoxes of the Sun 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 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 was not 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. Galileo deduced the equation s = ⁠1/2⁠gt2 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 and gave a correct definition of momentum. This emphasis of momentum as a fundamental quantity in dynamics is of prime importance. He measured momentum by the product of velocity and weight; mass is a later concept, developed by Huygens and Newton. In the swinging of a simple pendulum, Galileo says in Discourses that "every momentum acquired in the descent along an arc is equal to that which causes the same moving body to ascend through the same arc." His analysis on projectiles indicates that Galileo had grasped the first law and the second law of motion. He did not generalize and make them applicable to bodies not subject to the earth's gravitation. That step was Newton's contribution. The term "inertia" was used by Kepler who applied it to bodies at rest. (The first law of motion is now often called the law of inertia.) Galileo did not fully grasp the third law of motion, the law of the equality of action and reaction, though he corrected some errors of Aristotle. With Stevin and others Galileo also wrote on statics. He formulated the principle of the parallelogram of forces, but he did not fully recognize its scope. Galileo also was interested by the laws of the pendulum, his first observations of which were as a young man. In 1583, while he was praying in the cathedral at Pisa, his attention was arrested by the motion of the great lamp lighted and left swinging, referencing his own pulse for time keeping. To him the period appeared the same, even after the motion had greatly diminished, discovering the isochronism of the pendulum. More careful experiments carried out by him later, and described in his Discourses, revealed the period of oscillation varies with the square root of length but is independent of the mass the pendulum. Thus we arrive at René Descartes, 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 spacetime meant 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. 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 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;

v = d r d t , a = d v d t = d 2 r d t 2 {\displaystyle \mathbf {v} ={\frac {d\mathbf {r} }{dt}}\,,\quad \mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}\mathbf {r} }{dt^{2}}}}

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. For motion with constant speed, the acceleration is directed towards the center of curvature of the path. Again, loosely speaking, second order derivatives are related to curvature. The rotational analogues are the "angular vector" (angle the particle rotates about some axis) θ = θ(t), angular velocity ω = ω(t), and angular acceleration α = α(t):

θ = θ n ^ , ω = d θ d t , α = d ω d t , {\displaystyle {\boldsymbol {\theta }}=\theta {\hat {\mathbf {n} }}\,,\quad {\boldsymbol {\omega }}={\frac {d{\boldsymbol {\theta }}}{dt}}\,,\quad {\boldsymbol {\alpha }}={\frac {d{\boldsymbol {\omega }}}{dt}}\,,}

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:

v = ω × r {\displaystyle \mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {r} }

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 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. 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.

v = v 0 + a t [ 1 ] r = r 0 + v 0 t + 1 2 a t 2 [ 2 ] r = r 0 + 1 2 ( v + v 0 ) t [ 3 ] v 2 = v 0 2 + 2 a ( r − r 0 ) [ 4 ] r = r 0 + v t − 1 2 a t 2 [ 5 ] {\displaystyle {\begin{aligned}v&=v_{0}+at&[1]\\r&=r_{0}+v_{0}t+{\tfrac {1}{2}}{a}t^{2}&[2]\\r&=r_{0}+{\tfrac {1}{2}}\left(v+v_{0}\right)t&[3]\\v^{2}&=v_{0}^{2}+2a\left(r-r_{0}\right)&[4]\\r&=r_{0}+vt-{\tfrac {1}{2}}{a}t^{2}&[5]\\\end{aligned}}}

where:

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

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 some programs, such as the IGCSE Physics and IB DP Physics programs (international programs but especially popular in the UK and Europe), the same formulae would be written with a different set of preferred variables. There u replaces v0 and s replaces r - r0. They are often referred to as the SUVAT equations, where "SUVAT" is an acronym from the variables: s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time. In these variables, the equations of motion would be written

v = u + a t [ 1 ] s = u t + 1 2 a t 2 [ 2 ] s = 1 2 ( u + v ) t [ 3 ] v 2 = u 2 + 2 a s [ 4 ] s = v t − 1 2 a t 2 [ 5 ] {\displaystyle {\begin{aligned}v&=u+at&[1]\\s&=ut+{\tfrac {1}{2}}at^{2}&[2]\\s&={\tfrac {1}{2}}(u+v)t&[3]\\v^{2}&=u^{2}+2as&[4]\\s&=vt-{\tfrac {1}{2}}at^{2}&[5]\\\end{aligned}}}

Constant linear acceleration in any direction

The initial position, initial velocity, and acceleration vectors need not be collinear, and the equations of motion 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,

v = a t + v 0 [ 1 ] r = r 0 + v 0 t + 1 2 a t 2 [ 2 ] r = r 0 + 1 2 ( v + v 0 ) t [ 3 ] v 2 = v 0 2 + 2 a ⋅ ( r − r 0 ) [ 4 ] r = r 0 + v t − 1 2 a t 2 [ 5 ] {\displaystyle {\begin{aligned}\mathbf {v} &=\mathbf {a} t+\mathbf {v} _{0}&[1]\\\mathbf {r} &=\mathbf {r} _{0}+\mathbf {v} _{0}t+{\tfrac {1}{2}}\mathbf {a} t^{2}&[2]\\\mathbf {r} &=\mathbf {r} _{0}+{\tfrac {1}{2}}\left(\mathbf {v} +\mathbf {v} _{0}\right)t&[3]\\\mathbf {v} ^{2}&=\mathbf {v} _{0}^{2}+2\mathbf {a} \cdot \left(\mathbf {r} -\mathbf {r} _{0}\right)&[4]\\\mathbf {r} &=\mathbf {r} _{0}+\mathbf {v} t-{\tfrac {1}{2}}\mathbf {a} t^{2}&[5]\\\end{aligned}}}

although the Torricelli equation [4] can be derived using the distributive property of the dot product as follows:

v 2 = v ⋅ v = ( v 0 + a t ) ⋅ ( v 0 + a t ) = v 0 2 + 2 t ( a ⋅ v 0 ) + a 2 t 2 {\displaystyle v^{2}=\mathbf {v} \cdot \mathbf {v} =(\mathbf {v} _{0}+\mathbf {a} t)\cdot (\mathbf {v} _{0}+\mathbf {a} t)=v_{0}^{2}+2t(\mathbf {a} \cdot \mathbf {v} _{0})+a^{2}t^{2}}

( 2 a ) ⋅ ( r − r 0 ) = ( 2 a ) ⋅ ( v 0 t + 1 2 a t 2 ) = 2 t ( a ⋅ v 0 ) + a 2 t 2 = v 2 − v 0 2 {\displaystyle (2\mathbf {a} )\cdot (\mathbf {r} -\mathbf {r} _{0})=(2\mathbf {a} )\cdot \left(\mathbf {v} _{0}t+{\tfrac {1}{2}}\mathbf {a} t^{2}\right)=2t(\mathbf {a} \cdot \mathbf {v} _{0})+a^{2}t^{2}=v^{2}-v_{0}^{2}}

∴ v 2 = v 0 2 + 2 ( a ⋅ ( r − r 0 ) ) {\displaystyle \therefore v^{2}=v_{0}^{2}+2(\mathbf {a} \cdot (\mathbf {r} -\mathbf {r} _{0}))}

Applications Elementary and frequent examples in kinematics involve projectiles, for example a ball thrown upwards into the air. Given initial velocity u, one can calculate how high the ball will travel before it begins to fall. The acceleration is local acceleration of gravity g. 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 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 v = 0. Using equation [4] in the set above, we have:

s = v 2 − u 2 − 2 g . {\displaystyle s={\frac {v^{2}-u^{2}}{-2g}}.}

Substituting and cancelling minus signs gives:

s = u 2 2 g . {\displaystyle s={\frac {u^{2}}{2g}}.}

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,

ω = ω 0 + α t θ = θ 0 + ω 0 t + 1 2 α t 2 θ = θ 0 + 1 2 ( ω 0 + ω ) t ω 2 = ω 0 2 + 2 α ( θ − θ 0 ) θ = θ 0 + ω t − 1 2 α t 2 {\displaystyle {\begin{aligned}\omega &=\omega _{0}+\alpha t\\\theta &=\theta _{0}+\omega _{0}t+{\tfrac {1}{2}}\alpha t^{2}\\\theta &=\theta _{0}+{\tfrac {1}{2}}(\omega _{0}+\omega )t\\\omega ^{2}&=\omega _{0}^{2}+2\alpha (\theta -\theta _{0})\\\theta &=\theta _{0}+\omega t-{\tfrac {1}{2}}\alpha t^{2}\\\end{aligned}}}

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

These are the kinematic equations for a particle traversing a path in a plane, described by position r = r(t). 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. These are instantaneous quantities which change with time. The position of the particle is

r = r ( r ( t ) , θ ( t ) ) = r e ^ r {\displaystyle \mathbf {r} =\mathbf {r} \left(r(t),\theta (t)\right)=r\mathbf {\hat {e}} _{r}}

where êr and êθ are the polar unit vectors. Differentiating with respect to time gives the velocity

v = e ^ r d r d t + r ω e ^ θ {\displaystyle \mathbf {v} =\mathbf {\hat {e}} _{r}{\frac {dr}{dt}}+r\omega \mathbf {\hat {e}} _{\theta }}

with radial component ⁠dr/dt⁠ and an additional component rω due to the rotation. Differentiating with respect to time again obtains the acceleration

a = ( d 2 r d t 2 − r ω 2 ) e ^ r + ( r α + 2 ω d r d t ) e ^ θ {\displaystyle \mathbf {a} =\left({\frac {d^{2}r}{dt^{2}}}-r\omega ^{2}\right)\mathbf {\hat {e}} _{r}+\left(r\alpha +2\omega {\frac {dr}{dt}}\right)\mathbf {\hat {e}} _{\theta }}

which breaks into the radial acceleration ⁠d2r/dt2⁠, centripetal acceleration –rω2, Coriolis acceleration 2ω⁠dr/dt⁠, and angular acceleration rα. 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