If I gave a problem to one of my friends who aren’t in physics or mathematics, they’d probably say that it’s way too complicated for them to solve. What’s amazing to me though, is how they are so often wrong about that assumption. Truthfully, many of the problems that I tackle at school (not actual scientific problems) are relatively easy and just require transforming the problem. What I mean by this is that our first line of attack for a new situation is to try and transform it into an old situation that we know how to do.
For example, when I first learned about Lagrange multipliers this year, we started solving a problem using the method that came into our minds first. This involved solving for a variable in one function and then substituting it into the other.
Only at that point did my professor show us the motivation behind using Lagrange multipliers. In the particular problem we were dealing with, we were trying to find the maximize the sum of three numbers while ensuring their product was a certain value. I learnt how Lagrange multipliers preserve the symmetry in a situation and made it relatively more straightforward and systematic to solve.
At the same time, we didn’t just jump into this new method. We started by transforming it into a manner that we could solved and worked from there. After that, we were shown how the alternative method was generally easier to work with. The crucial point is that I didn’t start from absolutely nothing. The Lagrange method was shown after the “regular” method we had used, making it easier to follow.
Remember: a lot of the “complicated” things you will learn are just variations on a theme, adjusted slightly for your present situation.