How does a charge or a Noether charge create a continuous symmetry?

Question

I recently searched up the definition for a more generalized definition of charge whether it be color charge or electric charge . And I found this but i fail to understand what this statement actually means :

A charge is any generator of a continuous symmetry of the physical system under study. When a physical system has a symmetry of some sort, Noether's theorem implies the existence of a conserved current. The thing that "flows" in the current is the "charge", the charge is the generator of the (local) symmetry group. This charge is sometimes called the Noether charge.--- Source Wikipedia

Taking this statement " A charge is any generator of a continuous symmetry of the physical system under study."

What do continuous symmetry means and generator mean? How does a charge create a continuous symmetry and how does it impact the physical system?

Answer 1

1. OP's question is related to the converse/inverse Noether's theorem, cf. e.g. this Phys.SE post.

2. Let us for simplicity only consider the Hamiltonian formulation. In the Hamiltonian formulation a constant of motion/conserved quantity/charge $Q(q,p,t)$ is the generator of a Hamiltonian vector field $X_Q$, which in turn generates a finite continuous Hamiltonian flow. The corresponding infinitesimal transformation is $$\delta z^I ~=~\epsilon\{z^I, Q\}, \qquad I~\in~\{1,\ldots, 2n\}, \qquad \delta t ~=~0,\tag{A}$$ where $\epsilon$ is an infinitesimal parameter. One may show that this is a symmetry of the Hamiltonian action, cf. statement 3 in my Phys.SE answer here.