Undergrad Is the Action Always a Minimum in the Principle of Least Action?

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SUMMARY

The principle of least action states that the action S must be an extremum, specifically a minimum, for a physical path. This convention stems from historical parallels drawn with Fermat's principle of least time, which inspired the formulation of the principle of least action in mechanics. While both minimum and maximum actions can yield valid results, the term "stationary" is more accurate to describe the nature of action in this context. The discussion highlights the importance of terminology and its implications in physics.

PREREQUISITES
  • Understanding of the principle of least action in classical mechanics
  • Familiarity with Fermat's principle of least time
  • Basic knowledge of variational principles in physics
  • Introduction to path integrals in quantum mechanics
NEXT STEPS
  • Research the mathematical formulation of the principle of least action
  • Explore the implications of stationary action in classical and quantum mechanics
  • Study the relationship between Fermat's principle and the principle of least action
  • Learn about path integrals and their applications in quantum field theory
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Physicists, students of mechanics, and anyone interested in the foundational principles of physics and their historical context.

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Hello, When we applying the principle of least action, we require ##\delta S=0##, which corresponding to the action S being an extremum. I am just wondering why do we say that the action is a minimum instead of a maximum for a physical path? Can I use the path integral to explain this problem?
Thanks for all responses.
 
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I think that long ago, when Fermat's principle of least time came about, that people so liked the idea that they asked themselves why couldn't such a thing work for mechanics. The principle of least action was then made intentionally analogous to Fermat's principle. These optical considerations would then inspire others to develop wave mechanics.
 
If S is minimum , -S is maximum and anyway the both work. Stationary is more appropriate word to say.
 

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