How to model a symmetry using Lie Groups?

I have been reading lately about Lie groups, and although all books keep listing the groups, and talk about Lie algebras and all that, one thing I still don't know how is it made, and I guess it's the most important question as much as my understanding goes, but most textbooks keep ignoring.

My question is, Now if I have a symmetry, and I could put it into equation (symmetry of sea waves - simplified for example, or symmetry of planetary motion), how to use Lie groups to express this symmetry? How to identify which Lie group responsible for that?

I am still at the very beginning but for some reason whatever I research I cannot find an answer to this question. I hope I would find help here!

Answer

A group is a set of transformations that don't change the internal relationship in "something", whether it's another mathematical structure or a set of equations describing a physical object or the physical object itself.

So putting a symmetry into equations means to find a set of functions or maps $f$ such that $$ x \text{ is OK} \Leftrightarrow f(x) \text{ is OK}$$ where "is OK" means that some conditions are satisfied, for example it may mean that the equations of motion are solved. To identify the group of symmetries of a physical system means to find all the maps $f$ for which the equivalence above is obeyed – and to compare the multiplicative table with possible tables for different groups, and find which one is the right one.

Lie group is just a special subset of groups in which the functions or maps $f$ make up a differentiable manifold, i.e. in which one needs continuous parameters such as real numbers to uniquely pinpoint an element of the group. A special mathematical toolkit, one that involves Lie algebras and their classification and representations etc., is useful to analyze Lie groups and systems with Lie group symmetries.

The very defining property of the groups is that their inner structure – especially the composition rule $(g_1,g_2)\mapsto g_1\cdot g_2$ – is independent from the system on which the group acts. This allows us to treat many aspects of these systems with the same symmetries by universal tools.

I am afraid my answer won't satisfy what you want to hear and I am also afraid that the reason is that you want to hear something that isn't true. ;-) A group itself isn't an equation.