Symplectic geometry and conservation laws

Question

following these notes (concretely on page 5), regarding the conservation laws derived from the action, it states that they are of the form

$$ \Theta_a\eta^a (0) = \Theta_a\eta^a (t) .\tag{1}$$

now, in the same page it says that $ \eta $ is a variation of the coordinates $ t \mapsto \xi(t) $ to $$ t \mapsto \xi(t)+\eta(t).\tag{2} $$

My intuition tells me that the physics shouldn't depend on this variation, since it's arbitrary, however the conservation laws depend explicitly on it, what is the problem here with my view? Supposing i'm wrong and the conservation laws do depend on the variation, and won't this break some symmetry, since it would render the physics dependant on some component of the coordinates?

Answer 1

OP's eq. (1) appears to be the standard formula for the bare Noether charge $$Q_{(0)}~=~\frac{\partial L_H}{\partial \dot{\xi}^a}\eta^a~=~\Theta_a\eta^a$$ for a vertical symmetry vector field $\eta$ for the Hamiltonian action $$A~=~\int L_H\mathrm{d}t~=~\int (\Theta -H\mathrm{d}t).$$

See also this related Phys.SE post.

References:

1. Josh Powell, Aspects of Symplectic Geometry in Physics; chapter 1 p.5.