# [Physics] How do non-linear equations lead to self-interaction

classical-mechanicsinteractionsnon-linear-systems

In my life I hear/read this statement a lot:

A non-linear equation or theory leads to self-interactions.

For example in GR, we say that gravity can interact with itself because it is non-linear.
For some reason I always assumed it was correct. However now I think about it, I can not see a clear reason in the maths for why this statement holds. Can someone help me out? ðŸ˜€

Edit$$_1$$: As Vadim pointed out. The statement should be the other way around.

A self interacting physical system leads to non-linear equations.

Edit$$_2$$: The questions is beautifully answered by @gandalf61 for 2 variable system. However, still do not really understand what is going on for 1 variable system, e.g. in general relativity. Could someone maybe also give an example there? Thank you in advance. ðŸ˜€

In the comments on the answer of @gandalf61, you will also find the answer of edit$$_2$$.

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.