I recently completed a summer project on applications of lie groups to differential equations under the supervision of Jason D. Lotay. You could find the report here.

The idea is that differential equations admit group actions that map solutions to other solutions. For example, if \(y=f(x)\) is a solution to \(\frac{d^2 y}{dx^2} = 0\). Then for any real constant \(\epsilon\), \(y=f(x) + \epsilon\) and \(y=e^{\epsilon} f(x)\) are going to be solutions as well. Intuitively it means that scaling and translations are symmetries of the differential equation. This idea extends to more complicated differential equations such as the heat and wave equation, the symmetries of which uncover a lot of the fundamental properties about them.

One could refer to chapter 8 and 9 of The Oxford Handbook of Philosophy of Physics for a philosophical introduction / discussion on the role of symmetries in physics.

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