Isaac Newton's “quantity of motion,” defined as the product of a particle's mass and velocity, whose time rate of change equals the applied force; in modern physics, the rate of change of energy with respect to velocity. The law of conservation of linear momentum states that the total linear momentum of an isolated system remains constant. See also: Angular momentum; Conservation of momentum; Energy; Momentum
Role in classical mechanics
The linear momentum of a particle is the vector quantity p given by Eq. (1),
Newton's second law of motion states that the rate of change of linear momentum is proportional to the applied force F, Eq. (2).
See also: Force
This is actually the definition of force in terms of the fundamental quantities of physics: mass, length (here, position), and time. If more than one force acts on the particle, then F is the (vector) sum of the forces. Newton's first law of motion is implied because if F = 0, then p is constant. See also: Dimensions (mechanics); Dynamics; Equilibrium of forces; Inertia; Resultant of forces
Newton based his laws on a number of assumptions he took for granted. These are (1) total mass is constant (this is sometimes called Newton's zeroth law of motion); (2) masses (which are scalar quantities) add linearly; (3) velocities (which are vector quantities) and forces (also vectors) add linearly, implying that momenta must add linearly; (4) forces do not depend on velocity; (5) forces between particles are directed along the line that connects them; (6) all physical interactions are completely deterministic; and (7) mass, energy, and momentum are infinitely subdivisible. See also: Conservation of mass; Determinism; Relative motion
In addition to the above assumptions are others regarding the reference frames: the coordinate systems for specifying positions and velocities. By definition, inertial reference frames are those in which Newton's laws are true; this implies that the frames are neither rotating nor accelerating, but are only moving at a constant velocity with respect to the observer. Newton's laws also assume that all the reference frames are Newtonian, meaning that measurements of position and time are completely independent; there is one absolute universal time measurement; and influences act across all distances instantaneously (the speed of light is infinite). See also: Frame of reference
The various branches of modern physics are characterized by the extent to which one or more of the above assumptions are dropped. For example, assumptions (1), (2), and (3) are not true for relativistic velocities; assumptions (4) and (5) are not true for magnetic forces between particles; (6) is not true for quantum-mechanical systems; and (7) is not true for atomic, nuclear, and subnuclear physics. General relativity describes physics in accelerating frames of reference.
The total linear momentum of a multiparticle system equals the sum of the linear momenta of its constituent particles. If the system's configuration is not important, then it can be modeled as a single particle, having the same mass as the system, located at the system's center of mass, and subjected to the sum of the external forces. If there are no external forces (or if they always sum to zero), then by Newton's first law, the center of mass will move at a constant velocity. See also: Center of mass; Collision (physics)
Expanding Eq. (3) by the usual rules
of differentiation shows that, in general, change in linear momentum can be caused by (nonrelativistic) change in mass. An example would be in the analysis of rocket flight wherein the total mass of the rocket decreases as its propellants are ejected.
When mass is constant,
The units of measure for linear momentum are the Newton second (in SI units) and pound-force second (in U.S Customary units). See also: Units of measurement
Linear momentum in continuous media
Continuous media include deformable solids, fluids (liquids and gases), plasmas, and “empty” space. For such media, it is more appropriate to use momentum density, defined as momentum per unit volume of the medium. Two examples are given here.
The Navier-Stokes equations describe the dynamics of fluids in terms of conservation of momentum. Linear momentum appears as the density (mass per unit volume) times velocity at a given point in the fluid, that is, the momentum density at that point. See also: Fluid-flow principles; Navier-stokes equation
Classical electromagnetic theory
Electromagnetic waves transport energy according to the Poynting vector S = E × H, where E and H are the electric and magnetic field vectors, respectively. The energy propagates in the direction of S at the speed of light, suffusing the (lossless) medium with electromagnetic momentum g = (1/c2) S, measured in N-s/m3, that will exert radiation pressure on matter. The pressure is very small: for sunlight, it is about 0.44 kg-force/(km)2, or 2.5 lb-force/(mi)2, on the Earth's surface. See also: Poynting's vector; Radiation pressure
Role in relativistic mechanics
Relativistic effects become important when velocities approach the speed of light c. In particular, a particle moving with speed v appears more massive by the factor γ(v), Eq. (5).
However, an isolated system will retain the same form of law of conservation of linear momentum provided we use the relativistic momentum, Eq. (6), of each constituent particle.
Relativistic momentum is sometimes called the 3-momentum because its three components plus an energy term constitute the momentum-energy 4-vector, or 4-momentum, much used in high-energy particle physics. Although different observers will obtain different measurements of a particle's energy and momentum, they will all agree on the value of its rest mass. Mathematically stated, the magnitude of the 4-momentum c2 m0 (the momentum-energy, or momenergy), given by Eq. (7),
is invariant and so relates any observer's own measurements of the particle's energy E and momentum p. For a particle at rest in the observer's frame, Eq. (7) yields Einstein's famous equation for rest energy: Erest = m0 c2. See also: Relativistic mechanics; Relativity
Linear momentum in quantum theory
Quantum mechanics differs radically from classical mechanics in several ways. With specific regard to linear momentum in nonrelativistic quantum theory:
1. A particle's linear momentum and position cannot both be precisely measured at the same time. Heisenberg's uncertainty principle specifies the lower limit on the product of the uncertainties (probable range of errors) of the two measurements by Eq. (8)
2. Electromagnetic radiation is quantized into
photons whose linear momentum is given by Eq. (9), where λ is the electromagnetic wavelength. When a photon loses some of its momentum to a subatomic particle, its wavelength increases correspondingly. See also: Compton effect; Quantum (physics)
4. The Lagrangians and Hamiltonians of classical mechanics are invalid since they are premised on the observer being able to accurately measure momentum and position simultaneously. However, the same form of Hamiltonian is used, but with the dynamic variables replaced by their analogous Hamiltonian operators. For example, the momentum operator in the one-dimensional form of Schrödinger's equation is given by Eq. (10), where
By applying the momentum operator in the appropriate way to the wave function, one obtains the probability of obtaining any given value of measured momentum. Although repetition of the experiment will yield different results in detail, nevertheless the total linear momentum will always be found to be conserved. See also: Nonrelativistic quantum theory; Quantum mechanics; Schrödinger's wave equation
Role in relativistic quantum theory
When relativistic effects become important, the total momentum-energy of a system is still conserved. However, tools such as Schrödinger's equation become invalid because they use time independently of space and one must instead use the methods of quantum field theories. Among these methods are the diagrammatic momentum-space Feynman rules of quantum electrodynamics. See also: Feynman diagram; Quantum electrodynamics; Quantum field theory; Quantum gravitation; Relativistic quantum theory