What is a Simple Example of a Functional and Lagrangian in Gravity?

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Discussion Overview

The discussion revolves around the concept of functionals and Lagrangians in the context of gravity, specifically seeking simple examples and clarifications related to Hartle's book "Gravity." Participants explore definitions, examples, and the mathematical framework surrounding functionals and their applications in physics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant seeks a simple example of a functional or Lagrangian, expressing difficulty in understanding the material in Hartle's book.
  • Another participant defines a functional as a mapping from a function to a number, providing an example related to the action integral for a particle's trajectory.
  • A different participant emphasizes that a functional is an integral where the integrand is treated as a variable, not a fixed function.
  • Some participants debate the definitions of functionals, with one asserting that the standard definition is not being accurately represented.
  • One participant describes an example from Hartle involving a free particle and its action, discussing the calculation of extremal paths.
  • Another participant introduces the Euler-Lagrange equations as a method to find functions that minimize the action functional.
  • There is a discussion about the nature of functionals, with some participants arguing about the distinction between a functional and a number derived from it.
  • One participant notes the importance of the unknown function in the action integral, suggesting that it is crucial to the definition of a functional.
  • Another participant comments on the terminology used in the discussion, drawing parallels between functions and functionals.

Areas of Agreement / Disagreement

Participants express differing views on the definitions and nature of functionals, with no clear consensus reached. Some agree on certain aspects of functionals, while others contest these definitions and interpretations.

Contextual Notes

Participants reference specific examples and definitions from functional analysis and classical mechanics, but there are unresolved nuances regarding the definitions and applications of functionals in the context of gravity.

  • #31


I got it now.

Thanks a lot for all of your help.

Matt
 
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  • #32


I recommend that you also try to derive the Euler-Lagrange equations using the method I described earlier. It would give you a deeper understanding of these things, and I also find it much easier to remember the derivation than the actual equation for some reason.
 

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