![]() ![]() These are the cases where the Lagrangian is not hyperregular. Conversely, the cases where the Lagrangian does not admit a Hamiltonian are exactly the cases where the principle of stationary action fails to yield a unique solution, thus does not hold. That is, the principle holds on all Hamiltonian systems, whether or not they admit a Lagrangian. Note that the true physical content is in the displacement field \(\vec ,p,t)\) depends on conjugate momentum as well. This allows us to fully understand both the physical and geometric significance of the principle in a much more precise way. We have found that the principle of stationary action is equivalent to assumptions ( DR), ( IND) and ( KE) and that the variation of the action can be understood as the flow of evolutions through the surface delimited by the path and its variation. The main article will present all the key points needed to follow the argument and its physical and geometric meaning, leaving the mathematical details and calculations to the Supplementary Information. We then proceed to the case of multiple independent degrees of freedom using tools from differential geometry. To make the result accessible to the widest audience, we first cover the case of a single degree of freedom using standard vector calculus. The two aspects that have been sorely lacking, and that we provide, are (1) a clear geometric interpretation of the action principle and (2) a tight connection between the math and the physics it represents. ![]() ![]() The mathematics needed to run the argument is well established 8, 9, 10, 11. The argument can proceed in the reverse direction: assuming the principle of stationary action recovers a dynamical system that exhibits those three physical assumptions. we can reconstruct the dynamical state simply by looking at the trajectory) is what allows us to express the principle in the usual form. ![]() The assumption of equivalence between kinematics and dynamics (i.e. What we find is that this arises as a general mathematical feature of divergence-free fields (and closed two-forms), which are the appropriate tools to describe a flow that conserves the number of states. This physically motivated understanding of the classical theory can be used to characterize both the physics and the geometry underlying the principle of stationary action. We have found that Lagrangian mechanics is equivalent to three assumptions: determinism/reversibility, independence of degrees of freedom and kinematics/dynamics equivalence 7. We are left to wonder: what exactly is the action and why is it stationary for actual trajectories?Īs part of our larger project Assumptions of Physics, we developed an approach, called Reverse Physics 6, which examines current theories to find a set of starting physical assumptions that are sufficient to rederive them. Moreover, the Lagrangian for a system is not uniquely defined, making the actual value of the action for a path not directly physically significant. First of all, the typical characterization of the Lagrangian as the difference between kinetic and potential energy fails even for simple systems, like a charged particle under a magnetic field. While the principle of stationary action is regarded by many as one of the most important tools in physics, its physical meaning is not completely clear 1, 2, 3, 4, 5. ![]()
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