# Principle of superposition

It is said that a system is linear or raises of the principle of superposition so to the sum of two excitations corresponds the sum of the two corresponding answers.

More precisely, the excitations being noted F (by reference to the forces of mechanics) and the answers X (by reference to the movements), the system is linear if:

• the answer $x_1 \left(T\right) \,$ being associated with the excitation $f_1 \left(T\right) \,$ and the answer $x_2 \left(T\right) \,$ associated with the excitation $f_2 \left(T\right) \,$,

• $\ lambda_1 \,$ and $\ lambda_2 \,$ being two unspecified numbers,
• the answer $\ lambda_1 x_1 + \ lambda_2 x_2 \,$ is associated with the excitation $\ lambda_1 f_1 + \ lambda_2 f_2 \,$.

This result spreads then with an unspecified number of excitations. In other words, if one can break up an excitation into a sum of simple functions, it will be possibly possible to explicitly calculate the corresponding answer while adding with the calculable individual answers.

In fact, the concrete systems having this property are extremely rare, not to say non-existent. Extremely fortunately, good number of systems can be reasonably linearized, either by being unaware of small non-linearities on the assumption of the small movements (see oscillating Systèmes with a degree of freedom), or while proceeding to a linearization optimized in the contrary case.

## Examples

In the case of the electrical circuits made up exclusively of linear elements (resistances, capacities, inductances, generators of independent tension or current or linearly depending on a current, a tension, etc), the answer in a branch is equal to the sum of the answers by each independent generator taken separately, by inactivant all the other independent generators (generating of tension replaced by wire and generators of current by open switches).

In Mechanical quantum, the atomic particles can exist in several superimposed and simultaneous states, for example a electron which can be in two places at the same time, or a Photon which passes by both Fentes of Young at the same time.

## See too

### Geology

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