Some insights into Jacobi's form of least action principle (Q5944917)

The author discusses in detail the Jacobi form of least action principle of stationary action. It is generally accepted that this principle, which treats only the trajectories in configuration space without refering to the position of the system at a given time instant, is a stationary variational principle. Here the author investigates the question: under which condition the Jacobi form of least action will be a principle of minimum action? Using modern functional analysis, it is demonstrated that in many important cases in which the configuration space has given boundaries and the trajectories are not smooth, the conditions for minimality can be fulfilled. The author also discusses an analogy with geometric optics.