Where is the equivalence between laws and symmetries present in Noether's theorem?

Noether's theorem is a fundamental principle in theoretical physics that links symmetries associated with physical systems to conservation laws. According to the theorem, for every continuous symmetry of the action of a physical system, there is a corresponding conservation law.

The essence of Noether's theorem lies in the action, which is a functional that summarizes the dynamics of the system. The action is defined over a mathematical structure known as symplectic space, which deals with the phase space of dynamical systems. When a physical system exhibits a symmetry in its action—such as translational symmetry (invariance under shift in position) or rotational symmetry (invariance under changes in orientation)—there are corresponding conserved quantities, like momentum and angular momentum.

This deep connection is crucial because it establishes a framework for understanding why certain quantities remain constant over time despite the evolution of the system. So, when focusing on the equivalence between physical laws and symmetries represented in Noether's theorem, the critical aspect is the formulation of the laws via the action principle, particularly in the space defined by the system's phase space, which is inherently linked to symplectic geometry. Therefore, the option mentioning physical laws based on action defined over symplectic space accurately captures the essence