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


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? 😀

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. 😀

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

Best Answer

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.

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