Editing Superposition principle (section)

{{short description|Fundamental physics principle stating that physical solutions of linear systems are linear}}
{{About|the superposition principle in linear systems|the geologic principle|law of superposition|other uses|Superposition (disambiguation)}}
[[File:Anas platyrhynchos with ducklings reflecting water.jpg|thumb|right|Superposition of almost [[plane wave]]s (diagonal lines) from a distant source and waves from the [[Wake (physics)|wake]] of the [[duck]]s. [[Linearity]] holds only approximately in water and only for waves with small amplitudes relative to their wavelengths.]]
[[File:Rolling animation.gif|right|thumb| [[Rolling]] motion as superposition of two motions. The rolling motion of the wheel can be described as a combination of two separate motions: [[translation (geometry)|translation]] without [[rotation]], and rotation without translation.]]

The '''superposition principle''',<ref>The Penguin Dictionary of Physics,  ed. Valerie Illingworth, 1991, Penguin Books, London.</ref> also known as '''superposition property''', states that, for all [[linear system]]s, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input ''A'' produces response ''X'', and input ''B'' produces response ''Y'', then input (''A'' + ''B'') produces response (''X'' + ''Y'').

A [[function (mathematics)|function]] <math>F(x)</math> that satisfies the superposition principle is called a [[linear function]]. Superposition can be defined by two simpler properties: [[additive map|additivity]]  
<math display="block">F(x_1 + x_2) = F(x_1) + F(x_2)</math>
and [[homogeneous function|homogeneity]]
<math display="block">F(ax) = a F(x)</math>
for [[scalar (mathematics)|scalar]] {{mvar|a}}.

This principle has many applications in [[physics]] and [[engineering]] because many physical systems can be modeled as linear systems. For example, a [[beam (structure)|beam]] can be modeled as a linear system where the input stimulus is the [[structural load|load]] on the beam and the output response is the [[Deflection (engineering)|deflection]] of the beam. The importance of linear systems is that they are easier to analyze mathematically; there is a large body of mathematical techniques, [[frequency-domain]] [[linear transform]] methods such as [[Fourier transform|Fourier]] and [[Laplace transform|Laplace]] transforms, and [[linear operator]] theory, that are applicable. Because physical systems are generally only approximately linear, the superposition principle is only an approximation of the true physical behavior.

The superposition principle applies to ''any'' linear system, including [[algebraic equation]]s, [[linear differential equations]], and [[system of equations|systems of equations]] of those forms. The stimuli and responses could be numbers, functions, vectors, [[vector field]]s, time-varying signals, or any other object that satisfies [[vector space|certain axioms]]. Note that when vectors or vector fields are involved, a superposition is interpreted as a [[vector sum]]. If the superposition holds, then it automatically also holds for all linear operations applied on these functions (due to definition), such as gradients, differentials or integrals (if they exist).