This paper presents a unified framework for constructing symplectic manifolds from systems with symmetries. It provides a general method for reducing phase spaces when a group acts symplectically on a symplectic manifold. The framework is applied to various physical and mathematical examples, including Kostant's result on the symplectic structure of coadjoint orbits of a Lie group, and Smale's criterion for relative equilibria. The paper also discusses the symplectic structure on the reduced phase space, showing that it is well-defined and closed. The reduced phase space is defined as the quotient of the preimage of a moment map by the isotropy group. The paper includes several examples, such as the symplectic action on the cotangent bundle of a manifold, the reduction of the rigid body, fluid flow, and general relativity. It also discusses the stability of relative equilibria and their relation to Hamiltonian systems on the reduced phase space. The paper concludes with a theorem on the stability of relative equilibria and references to related works by other authors.This paper presents a unified framework for constructing symplectic manifolds from systems with symmetries. It provides a general method for reducing phase spaces when a group acts symplectically on a symplectic manifold. The framework is applied to various physical and mathematical examples, including Kostant's result on the symplectic structure of coadjoint orbits of a Lie group, and Smale's criterion for relative equilibria. The paper also discusses the symplectic structure on the reduced phase space, showing that it is well-defined and closed. The reduced phase space is defined as the quotient of the preimage of a moment map by the isotropy group. The paper includes several examples, such as the symplectic action on the cotangent bundle of a manifold, the reduction of the rigid body, fluid flow, and general relativity. It also discusses the stability of relative equilibria and their relation to Hamiltonian systems on the reduced phase space. The paper concludes with a theorem on the stability of relative equilibria and references to related works by other authors.