Another way to teach the least-action principle

Abstract We present another way to teach the least-action principle in classical mechanics. Beginning with Newton’s second law of motion, we keep finding equivalent propositions based mainly on the method of undetermined coefficients in a multivariable-dependent identity to arrive at the Euler–Lagrange equations and the conventional principle of stationary action $$\delta S=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mi>S</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> with the boundary condition for the variation of the physical path. Pedagogically, by following the logical steps that we show, students can easily learn that Lagrangian mechanics naturally arises from Newtonian mechanics. This approach makes it further clear that Newton’s laws do not require the minimum action—true for a free particle, just as the shortest path between two points is a straight line. However, the action is not always the minimum once the potential energy is turned on. The derivation of the least-action principle from Newton’s second law may resemble Euclid’s approach to geometric axioms.