Jacobi least time vs. Fermat Hamilton

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    Hamilton Jacobi Time
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Discussion Overview

The discussion revolves around the comparison between the Fermat/Hamilton principle and the Jacobi least time principle, focusing on their generality and applicability in variational methods. Participants seek to understand the differences between these principles and the implications of using one over the other.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant asks for a simple explanation of why the Fermat/Hamilton principle is considered more general than the Jacobi least time principle, expressing a desire to understand the implications of choosing one principle over the other.
  • Another participant notes difficulty in finding information on the Jacobi least time principle and requests a description, indicating potential confusion over terminology.
  • A participant shares a link to lecture notes that may clarify the concepts involved, suggesting that "Jacobi least time" might refer to "Fermat's least time" or Jacobi's analogy of Fermat's principle for particles.
  • There is uncertainty expressed about whether "least time" could ever be inferior to "least action," indicating a need for further exploration of the relationship between these principles.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the generality of the principles discussed, and multiple competing views remain regarding their definitions and implications.

Contextual Notes

There are limitations in the discussion regarding the clarity of terminology and the definitions of the principles, as well as unresolved questions about their comparative generality.

andrewr
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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.
 
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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 said:
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.
 

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