the effective potential

A trick that turns a two- or three-dimensional problem into a one-dimensional one by absorbing the conserved quantities into a modified potential. In the classic case of a planet orbiting the Sun, you fold the conserved angular momentum into an extra repulsive centrifugal term, and the messy orbital problem becomes a single particle sliding in a one-dimensional valley.

For central-force motion the angular momentum L is conserved because the angle is cyclic, so the radial energy equation reads E = (1/2) m r_dot^2 + V_eff(r) with V_eff(r) = V(r) + L^2 / (2 m r^2). That second term is the centrifugal barrier. The radial motion is then exactly that of a one-dimensional particle in V_eff: circular orbits sit at the minima of V_eff, and the shape of the well tells you at a glance whether orbits are bound, unbound, or stable.

You meet the effective potential in Kepler and central-force orbits, in scattering theory, in the innermost stable circular orbit of general relativity (where V_eff gains an extra relativistic term), and in the radial Schrodinger equation, whose l(l+1) hbar^2 / (2 m r^2) centrifugal term is the quantum echo of exactly this idea. It relies on having a cyclic coordinate whose conserved momentum you can eliminate.