In a system obeying Newton's laws, which principle best explains the trajectory of a particle under conservative forces?

The principle that best explains the path a particle follows when only conservative forces act is the principle of least (or stationary) action, which states that the actual trajectory makes the action integral stationary. Using this principle, the Euler–Lagrange equations give Newton’s second law as a consequence, so the motion follows from the geometry of the action. Because the forces are conservative, the potential energy is a function of position only, and the action integral contains kinetic minus potential energy. For example, a particle in a gravitational field follows a parabolic arc because that path makes the action integral minimal among all possible curves. Thus the principle of least action provides the most general and powerful explanation of the trajectory under conservative forces.