Equations of motion

Kinematic quantities of a classical particle of mass m: 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;[7]

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

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. Still for curved paths, the acceleration is directed towards the center of curvature of the path. Again, loosely speaking, second order derivatives are related to curvature.

The rotational analogues are the "angular vector" (angle the particle rotates about some axis) 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:[8]

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.

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.

These equations apply to a particle moving linearly, in three dimensions in a straight line with constant acceleration.[9] 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.

${\displaystyle {\begin{aligned}v&=at+v_{0}\quad [1]\\\end{aligned}}}$

${\displaystyle {\begin{aligned}r&=r_{0}+v_{0}t+{\tfrac {1}{2}}{a}t^{2}\quad [2]\\\end{aligned}}}$

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

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 elementary physics the same formulae are frequently written in different notation as:

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

where u has replaced v₀, 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.[10][11]

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,

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

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

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

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

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

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 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:

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

Substituting and cancelling minus signs gives:

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

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,

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

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.

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).[12] 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

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

where ê_r and ê_θ are the polar unit vectors. Differentiating with respect to time gives the velocity

${\displaystyle \mathbf {v} =\mathbf {\hat {e}} _{r}{\frac {dr}{dt}}+r\omega \mathbf {\hat {e}} _{\theta }}$

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

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

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

Special cases of motion described 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.

In 3D space, the equations in spherical coordinates (r, θ, φ) with corresponding unit vectors ê_r, ê_θ and ê_φ, the position, velocity, and acceleration generalize respectively to

${\displaystyle {\begin{aligned}\mathbf {r} &=\mathbf {r} \left(t\right)=r\mathbf {\hat {e}} _{r}\\\mathbf {v} &=v\mathbf {\hat {e}} _{r}+r\,{\frac {d\theta }{dt}}\mathbf {\hat {e}} _{\theta }+r\,{\frac {d\varphi }{dt}}\,\sin \theta \mathbf {\hat {e}} _{\varphi }\\\mathbf {a} &=\left(a-r\left({\frac {d\theta }{dt}}\right)^{2}-r\left({\frac {d\varphi }{dt}}\right)^{2}\sin ^{2}\theta \right)\mathbf {\hat {e}} _{r}\\&+\left(r{\frac {d^{2}\theta }{dt^{2}}}+2v{\frac {d\theta }{dt}}-r\left({\frac {d\varphi }{dt}}\right)^{2}\sin \theta \cos \theta \right)\mathbf {\hat {e}} _{\theta }\\&+\left(r{\frac {d^{2}\varphi }{dt^{2}}}\,\sin \theta +2v\,{\frac {d\varphi }{dt}}\,\sin \theta +2r\,{\frac {d\theta }{dt}}\,{\frac {d\varphi }{dt}}\,\cos \theta \right)\mathbf {\hat {e}} _{\varphi }\end{aligned}}\,\!}$

In the case of a constant φ this reduces to the planar equations above.

The first general equation of motion developed was Newton's second law of motion. In its most general form it 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,[13]

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

${\displaystyle \mathbf {F} =m\mathbf {a} }$

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 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 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, such as special and general relativity (see four-momentum).[13] 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[7][14]

${\displaystyle {\frac {d\mathbf {p} _{i}}{dt}}=\mathbf {F} _{E}+\sum _{i\neq j}\mathbf {F} _{ij}\,\!}$

where p_i is the momentum of particle i, F_ij is the force on particle i by particle j, and F_E 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,[15]

${\displaystyle {\boldsymbol {\tau }}={\frac {d\mathbf {L} }{dt}}\,,}$

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,

${\displaystyle {\boldsymbol {\tau }}=\mathbf {I} \cdot {\boldsymbol {\alpha }}.}$

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[16]

${\displaystyle {\frac {d\mathbf {L} _{i}}{dt}}={\boldsymbol {\tau }}_{E}+\sum _{i\neq j}{\boldsymbol {\tau }}_{ij}\,,}$

where L_i is the angular momentum of particle i, τ_ij the torque on particle i by particle j, and τ_E is resultant external torque (due to any agent not part of system). Particle i does not exert a torque on itself.

Some examples[17] of Newton's law include describing the motion of a simple pendulum,

${\displaystyle -mg\sin \theta =m{\frac {d^{2}(l\theta )}{dt^{2}}}\quad \Rightarrow \quad {\frac {d^{2}\theta }{dt^{2}}}=-{\frac {g}{l}}\sin \theta \,,}$

and a damped, sinusoidally driven harmonic oscillator,

${\displaystyle F_{0}\sin(\omega t)=m\left({\frac {d^{2}x}{dt^{2}}}+2\zeta \omega _{0}{\frac {dx}{dt}}+\omega _{0}^{2}x\right)\,.}$

For describing the motion of masses due to gravity, Newton's law of gravity can be combined with Newton's second law. For two examples, a ball of mass m thrown in the air, in air currents (such as wind) described by a vector field of resistive forces R = R(r, t),

${\displaystyle -{\frac {GmM}{|\mathbf {r} |^{2}}}\mathbf {\hat {e}} _{r}+\mathbf {R} =m{\frac {d^{2}\mathbf {r} }{dt^{2}}}+0\quad \Rightarrow \quad {\frac {d^{2}\mathbf {r} }{dt^{2}}}=-{\frac {GM}{|\mathbf {r} |^{2}}}\mathbf {\hat {e}} _{r}+\mathbf {A} \,\!}$

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,

${\displaystyle {\frac {d^{2}\mathbf {r} _{i}}{dt^{2}}}=G\sum _{i\neq j}{\frac {m_{i}m_{j}}{|\mathbf {r} _{j}-\mathbf {r} _{i}|^{3}}}(\mathbf {r} _{j}-\mathbf {r} _{i})}$

where i = 1, 2, …, N labels the quantities (mass, position, etc.) associated with each particle.

Geodesics on a sphere are arcs of great circles (yellow curve). On a 2D–manifold (such as the sphere shown), the direction of the accelerating geodesic is uniquely fixed if the separation vector ξ is orthogonal 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).[22]

The above equations are valid in flat spacetime. In curved 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:[23]

and the geodesic equation is a second-order differential equation in the coordinates. The general solution is a family of geodesics:[24]

${\displaystyle {\frac {d^{2}x^{\mu }}{ds^{2}}}=-\Gamma ^{\mu }{}_{\alpha \beta }{\frac {dx^{\alpha }}{ds}}{\frac {dx^{\beta }}{ds}}}$

where Γ ^μ_αβ 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 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:

${\displaystyle {\frac {D^{2}\xi ^{\alpha }}{ds^{2}}}=-R^{\alpha }{}_{\beta \gamma \delta }{\frac {dx^{\alpha }}{ds}}\xi ^{\gamma }{\frac {dx^{\delta }}{ds}}}$

where ξ^α = x₂^α − x₁^α is the separation vector between two geodesics, D/ds (not just d/ds) is the covariant derivative, and R^α_βγδ 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.[25]

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.

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.

Equations that describe the spatial dependence and time evolution of fields are called field equations. These include

• 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).

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.

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:

${\displaystyle {\frac {1}{v^{2}}}{\frac {\partial ^{2}X}{\partial t^{2}}}=\nabla ^{2}X}$

where X = X(r, t) is any mechanical or electromagnetic field amplitude, say:[26]

• 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,

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.

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:

${\displaystyle i\hbar {\frac {\partial \Psi }{\partial t}}={\hat {H}}\Psi \,,}$

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 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 their quantum 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 to quantum mechanics and field theory, placing emphasis on the use of a Lagrangians rather than Hamiltonians.