# Please provide the simplest example you can think of, of generators of time evolution and generalized coordinates

I was reading the Wikipedia article about Noether's theorem and this thing popped out:

Then the resultant perturbation can be written as a linear sum of the individual types of perturbations

$$\delta t = \sum_r \epsilon_r T_r \!$$ $$\delta \mathbf{q} = \sum_r \epsilon_r \mathbf{Q}_r ~$$

where $\epsilon_r$ are infinitesimal parameter coefficients corresponding to each:

• generator $T_r$ of time evolution, and
• generator $Q_r$ of the generalized coordinates.

But, when I clic on "generator" it leads me to the article about Lie Groups which is by itself a topic in which I could spend weeks, to say the least.

I had once in my past discrete groups in a lecture and I am more or less conversant with elementary lagrangian mechanics, matrix calculus and elementary non-relativistic quantum mechanics (however not with that SU(?) stuff about rotation invariances yet). Is it possible that someone here provides a very short explanation of what those generators are, and the simplest mathematical example you can think of?

Let us for simplicity address OP's question in the context of point mechanics where $q^i$ are generalized position coordinates on some manifold $M$ [instead of considering field theory with fields $\phi^{\alpha}(x)$]. Let us also for simplicity consider the special case where there are no horizontal/time transformations

$$\delta t ~=~ \sum_a \epsilon_{(a)} X_{(a)} ~=~ 0,$$

but only vertical transformations

$$\delta q^i ~=~ \sum_a \epsilon_{(a)} Y^i_{(a)},$$

where $\epsilon_{(a)}$ are (infinitesimal) parameters.$^1$

In a nutshell, the generators $Y_{(a)}$ are vector fields

$$Y_{(a)}~=~ \sum_i Y_{(a)}^i\frac{\partial}{\partial q^i}.$$

The Lie bracket of two vector fields $[Y_{(a)},Y_{(b)}]$ is again a vector field. In certain cases, the generators/vector fields form a Lie algebra.

Example: The generator for a translation in the $a$'th direction of position space $\mathbb{R}^3$ is

$$Y_{(a)}~=~\frac{\partial}{\partial q^a},$$

with constant vector field components

$$Y^i_{(a)}~=~\delta^i_a.$$

In this case, the generators $Y_{(a)}$ commute, so they form an Abelian Lie algebra.

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$^1$ See also this Phys.SE answer.