Jacobi least time vs. Fermat Hamilton

[andrewr]

Could anyone give me a simple explanation as to why the Fermat/Hamilton principle would be called more general than the Jacobi least time principle? I am trying to understand what differences would result from using the one principle vs. the other; eg: where/in what way would the Jacobi least time be inferior?

I know basic ODE, PDE, linear algebra, and variational methods. I have a BSEE background.

Thanks.

 

[RedX]

I googled "Jacobi least time" and didn't get any results. I know what the Fermat/Hamiltonian principle is though. Could you describe the Jacobi least time principle? Maybe people might know it by a more common name.

 

[RedX]

Oh I see. Here's some lecture notes that explains it from baez's website:

http://math.ucr.edu/home/baez/classical/cm05week05.pdf

 

[andrewr]

Oh I see. Here's some lecture notes that explains it from baez's website:

http://math.ucr.edu/home/baez/classical/cm05week05.pdf

Thank you, that helps. The literature I was reading was ambiguous, I think "Jacobi least time" is supposed to be "Fermat's least time" and the author switched the names, or meant Jacobi's analogy of Fermat's principle for particles.

I still don't understand why one principle (either) would be more general than the other. Is "least time" ever inferior to "least action"? or vice versa?

--Thanks.

 

[PJC01]

The Fermat/Hamilton principle and the Jacobi least time principle are both variational principles used in the study of mechanics and optics. However, the Fermat/Hamilton principle is considered to be more general because it can be applied to a wider range of physical systems and phenomena compared to the Jacobi least time principle.

One key difference between the two principles is that the Jacobi least time principle is limited to systems that can be described by a Lagrangian formalism, while the Fermat/Hamilton principle can be applied to both Lagrangian and non-Lagrangian systems. This means that the Fermat/Hamilton principle can be used to study a wider range of physical phenomena, such as electromagnetic fields, fluids, and quantum systems, which cannot be described by a Lagrangian formalism.

Another difference is that the Jacobi least time principle is based on the concept of minimizing the time taken to travel between two points, while the Fermat/Hamilton principle is based on the concept of minimizing the action, which is a more general quantity that takes into account both time and energy. This makes the Fermat/Hamilton principle more applicable to systems with varying energy levels, while the Jacobi least time principle may not be as effective in such cases.

In terms of practical applications, the Fermat/Hamilton principle has been widely used in the fields of optics and quantum mechanics, where the behavior of light and particles can be described using wave equations. On the other hand, the Jacobi least time principle has found more applications in classical mechanics and engineering, where systems can be described by a Lagrangian formalism.

Overall, while both principles have their own strengths and applications, the Fermat/Hamilton principle is considered to be more general and versatile in its application to a wider range of physical systems.