A foundational quantity in physics that is always conserved in isolated systems, defined in classical mechanics as the product of mass and velocity. An equivalent definition that is better suited to modern physics is that momentum is the rate of change of energy with respect to velocity: this allows extending the conservation laws to include entities of zero mass. Whenever energy is transported from one system to another, it is always associated with some form of momentum. See also: Classical mechanics
The word momentum (Latin for motion) by itself usually means linear momentum, the vector quantity obtained by multiplying (or scaling) the particle's velocity vector by its mass. The linear momentum of a system is the sum of the linear momenta of the particles composing it. By Newton's second law of motion, a particle's linear momentum increases with time in direct proportion to the applied force. See also: Dynamics; Force; Linear momentum; Motion; Newton's laws of motion
A particle's angular momentum is a vector quantity defined as its moment of (linear) momentum about a specified axis. The angular momentum of a system is the sum of the angular momenta (about the same axis) of the particles composing it. A system's angular momentum increases with time in direct proportion to the total applied moment of force, or torque, about the axis. See also: Angular momentum; Rotational motion; Torque
The concept of momentum is of fundamental importance because in isolated physical systems (that is, ones not acted on by some outside influence) both linear and angular momentum do not change with time: they are said to be conserved. (This statement also applies to total system energy.) The law of conservation of linear momentum comes from the fact that the laws of physics do not depend on where the system is located (the universe is homogeneous). The law of conservation of angular momentum comes from the fact that the laws of physics do not depend on the orientation of the system (the universe is isotropic). The two conservation laws of momentum along with the law of conservation of energy are the three foundational axioms of classical physics. See also: Conservation laws (physics); Conservation of energy; Conservation of momentum; Kinematics; Mechanics; Symmetry laws (physics)
For continuous systems such as semirigid bodies, fluids, and electromagnetic radiation, momentum is often expressed in the form of momentum density, that is, momentum per unit volume.
When velocities approach the speed of light, observed mass increases and Newton's laws of motion must give way to Einstein's more accurate laws of relativity. Nevertheless, the same form of law of conservation of linear momentum remains valid if the relativistic increase of mass is taken into account. Relativity views space and time united into a single entity, called spacetime, and correspondingly unites momentum and energy into momentum-energy, or momenergy. Just as the spacetime interval between events is invariant (the same for all observers), the rest-mass component of a particle's momentum-energy is also. This invariance implies that particles with finite momentum but no mass, such as photons, can travel only at the speed of light. See also: Photon; Relativity
In systems where quantum-mechanical effects are significant, the exact outcome of an experiment cannot be forecast, only the probability of obtaining any one of all the possible outcomes. Nevertheless, each experiment will show that linear and angular momentum are conserved. See also: Quantum mechanics
The conservation laws of classical mechanics are embodied in the powerful analysis methods of J. L. Lagrange and W. R. Hamilton, applicable to constrained dynamical systems whose degrees of freedom are limited (or at least countable). See also: Constraint; Degree of freedom (mechanics)
The state of a constrained system can be uniquely specified by an appropriate choice of generalized coordinates represented by qj, one for each degree of freedom. In Lagrangian analysis, the system's total kinetic energy T is written as a function of all the qj and the generalized velocities j ≙ dqj/dt. Inserting T into Lagrange's equations (1)
yields one differential equation (of second degree in time) for each degree of freedom, j = 1, 2, 3,…, and so forth.
The resulting set of simultaneous differential equations are then solved for the desired dynamic variables. The term ∂T/∂j is called the generalized momentum; it is conserved in isolated systems. For systems that do not dissipate energy (through friction, say), T is replaced by the Lagrangian function L ≡ T − V, where V is the system's total potential energy. See also: Lagrange's equations
As a simple example, consider a bead of mass m constrained to move on a fixed circle of radius R; this system has only one degree of freedom. A natural (but not unique) choice of generalized coordinate is the angle θ, as seen from the circle's center, between the bead and a fixed reference direction. It can be shown that the bead's kinetic energy is T = ½m(Rθ˙)2. Differentiating T with respect to ˙θ gives the generalized momentum: ∂T/∂θ˙ = mR2θ˙. For this particular case, the familiar formula for angular momentum is obtained.
It is sometimes convenient to convert the Lagrangian into the Hamiltonian function H, a function of the generalized coordinates and their generalized momenta, represented by pj. In conservative systems, H equals the total energy T + V. Analysis proceeds by plugging H into Hamilton's equations (2),
which generate two differential equations (of first degree in time) for each degree of freedom j.
This set of simultaneous differential equations is then solved for the desired results. Because of the symmetry of these equations, pj and qj are said to be canonically conjugate, and the pj are called canonical momenta or conjugate momenta. See also: Canonical coordinates and transformations; Hamilton's equations of motion
In the equations of nonrelativistic quantum mechanics, the generalized coordinates are replaced formally by their corresponding operators. See also: Nonrelativistic quantum theory; Schrödinger's wave equation
Momentum in unified theories
A major area of research is to reconcile the current contradictions between quantum theory and general relativity. Just as Newton's laws of motion are approximations to those of relativity, it may turn out that the laws of relativity are themselves approximations. For example, the hypothesis of loop quantum gravity, that spacetime is quantized, carries the implication that the speed of light is not constant. All candidate theories, however, must be compatible with current conservation laws in the limit of everyday mass, time, and length scales. Conversely, any future experiments revealing situations in which momentum (or energy) is not conserved will illuminate the way to a self-consistent unified theory. See also: Quantum gravitation