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If momentum is to be conserved over the volume V over a region Q, changes in the momentum of matter through the Lorentz force must be balanced by changes in the momentum of the electromagnetic field and outflow of momentum. If P mech is the momentum of all the particles in Q, and the particles are treated as a continuum, then Newton's second ...
The energy and momentum of an object measured in two inertial frames in energy–momentum space – the yellow frame measures E and p while the blue frame measures E ′ and p ′. The green arrow is the four-momentum P of an object with length proportional to its rest mass m 0 .
Momentum space is the set of all momentum vectors p a physical system can have; the momentum vector of a particle corresponds to its motion, with units of [mass][length][time] −1. Mathematically, the duality between position and momentum is an example of Pontryagin duality .
The precision of the position is improved, i.e. reduced σ x, by using many plane waves, thereby weakening the precision of the momentum, i.e. increased σ p. Another way of stating this is that σ x and σ p have an inverse relationship or are at least bounded from below. This is the uncertainty principle, the exact limit of which is the ...
Although the quantity p kin is the "physical momentum", in that it is the quantity to be identified with momentum in laboratory experiments, it does not satisfy the canonical commutation relations; only the canonical momentum does that. This can be seen as follows.
Log-log plot of γ (blue), v/c (cyan), and η (yellow) versus proper velocity w/c (i.e. momentum p/mc).Note that w/c is tracked by v/c at low speeds and by γ at high speeds. The dashed red curve is γ − 1 (kinetic energy K/mc 2), while the dashed magenta curve is the relativistic Doppler factor.
This is different from the parabolic energy-momentum relation for classical particles. Thus, in practice, the linearity or the non-parabolicity of the energy-momentum relation is considered as a key feature for relativistic particles. These two types of relativistic particles are remarked as massless and massive, respectively.
In the simple case of a single particle moving with a constant velocity (thereby undergoing uniform linear motion), the action is the momentum of the particle times the distance it moves, added up along its path; equivalently, action is the difference between the particle's kinetic energy and its potential energy, times the duration for which ...