The Principle of Stationary Action posits that the trajectory between two states is determined by making a functional called the action stationary under arbitrary variations in position, formalized via the Euler-Lagrange differential equations $\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0$. In classical mechanics, this framework utilizes a Lagrangian $L$ defined as the difference between kinetic and potential energy to derive equations of motion that are invariant under coordinate transformations, providing an abstract formulation equivalent to Newton's laws but superior for handling non-inertial frames and complex systems.
The Principle of Stationary Action posits that the trajectory between two states is determined by making a functional called the action stationary under arbitrary variations in position, formalized v…