  Let $G$ be a Lie group acting properly on a smooth manifold $M$. If $M/G$ is
connected, then we exhibit some simple and basic constructions for proper
actions. In particular, we prove that the reduction principle in compact
transformation groups holds for proper actions. As an application, we prove
that a reduction principle holds for polar actions and for the integral
invariant for isometric actions of Lie groups, called copolarity, which
measures how far from being polar the action is. We also investigate symplectic
actions. Hence we assume that $(M,\omega)$ is a symplectic manifold and the $G$
action on $M$ preserves $\omega$. %If $G$ is Abelian, we generalize results
proved in \cite{Ben,DP1,DP2} The main result is the Equivalence Theorem for
coisotropic actions, generalizing \cite[Theorem 3 p.267]{HW}. Finally, we
completely characterize asystatic actions generalizing results proved in
\cite{pg}.
