# How do non-linear equations lead to self-interaction?

If I go to a shop and buy $$5$$ apples and $$10$$ bananas then I can usually take the price of one apple $$a$$ and the price of one banana $$b$$ and add these together to get a total cost of $$5a+10b$$. And I pay the same total amount if I buy apples and bananas at the same time or I buy apples, then go back to the shop later and buy bananas - my purchases do not interact with one another. This is a linear system.

But if there is an offer of "$$5$$ apples for the price of $$3$$" or "one free banana with every $$5$$ apples" or "$$10\%$$ off if you spend more than $$\5$$" then the cost of $$5$$ apples and $$10$$ bananas will no longer be $$5a+10b$$. This is a non-linear system, and there is an interaction between my different purchases.

In the context of a conventional Lagrangian formulation, the main points go as follows:

• Linear EOMs have a quadratic Lagrangian.

• $$n$$ coupled linear EOMs can be diagonalized into $$n$$ uncoupled linear EOMs in 1 variable each.

• Non-linear EOMs have a Lagrangian, which has cubic or higher-order terms, aka. interaction terms, which correspond to vertices in Feynman diagrams.

• If an interaction term only depends on 1 variable, it is called a self-interaction.

• The perturbative solution to the EOMs can be represented as a directed rooted tree, where branch points/vertices are interactions, cf. e.g. eq. (6) in my Phys.SE answer here.

• For linear EOMs, the tree has no branch points/vertices/interactions.