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[1] In classical mechanics, the kinetic energy of a non-rotating object of mass m traveling at a speed v is . [2] The kinetic energy of an object is equal to the work, or force in the direction of motion times its displacement , needed to accelerate the object from rest to its given speed.
Its initial value is 1 (when v = 0); and as velocity approaches the speed of light (v → c) γ increases without bound (γ → ∞). α (Lorentz factor inverse) as a function of velocity—a circular arc. In the table below, the left-hand column shows speeds as different fractions of the speed of light (i.e. in units of c). The middle column ...
The following notations are used very often in special relativity: Lorentz factor = where = and v is the relative velocity between two inertial frames.. For two frames at rest, γ = 1, and increases with relative velocity between the two inertial frames.
In the fundamental branches of modern physics, namely general relativity and its widely applicable subset special relativity, as well as relativistic quantum mechanics and relativistic quantum field theory, the Lorentz transformation is the transformation rule under which all four-vectors and tensors containing physical quantities transform from one frame of reference to another.
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.
Continuous charge distribution. The volume charge density ρ is the amount of charge per unit volume (cube), surface charge density σ is amount per unit surface area (circle) with outward unit normal n̂, d is the dipole moment between two point charges, the volume density of these is the polarization density P.
This equation holds for a body or system, such as one or more particles, with total energy E, invariant mass m 0, and momentum of magnitude p; the constant c is the speed of light. It assumes the special relativity case of flat spacetime [1] [2] [3] and that the particles are free.
The relativistic Lagrangian can be derived in relativistic mechanics to be of the form: = (˙) (, ˙,). Although, unlike non-relativistic mechanics, the relativistic Lagrangian is not expressed as difference of kinetic energy with potential energy, the relativistic Hamiltonian corresponds to total energy in a similar manner but without including rest energy.