The Principle of Least Action in Special Relativity posits that a free particle traverses the path between two spacetime events which maximizes proper time, known geometrically as the length of its world line under the Minkowski metric. Formulated by setting potential energy to zero and utilizing the invariant interval $ds^2 = -c^2dt^2 + dx^2$, this relativistic action leads directly to Newton's second law in the low-velocity limit while reducing to a constant term for high-energy trajectories that preserve relativistic momentum conservation. This framework extends classical Lagrangian mechanics into the domain of special relativity, establishing that spacetime geometry dictates particle dynamics without external forces by treating straight world lines as geodesics in flat space.
The Principle of Least Action in Special Relativity posits that a free particle traverses the path between two spacetime events which maximizes proper time, known geometrically as the length of its w…