What is the motivation for principle of stationary action

Click For Summary

Discussion Overview

The discussion revolves around the motivations for the principle of stationary action in physics, exploring whether its basis is empirical, theoretical, or a combination of both. Participants examine historical contexts, mathematical frameworks, and implications in various physical theories, including classical mechanics and quantum mechanics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants note that empirical evidence, such as Fermat's principle in optics and Maupertius's studies, contributes to the motivation for the action principle.
  • Others argue that the beauty of the least-action principle lies in its ability to generalize Newton's third law of motion, particularly in complex interactions between subsystems.
  • A participant highlights the connection between the principle of stationary action and the calculus of variations, questioning the universal applicability of assigning an action to physical systems.
  • Another participant discusses the deep physical reasons for the Hamilton principle of least action as seen in Feynman's path-integral formulation of quantum mechanics, suggesting that stationary action corresponds to classical trajectories.
  • Some express confusion regarding the motivation for insisting that physical systems should follow an extremal path of action, beyond the fact that it works.
  • A participant mentions that the principle serves as a formalism that provides insights and alternative methods for deriving equations across various fields of physics.

Areas of Agreement / Disagreement

Participants express a mix of agreement and disagreement regarding the motivations for the principle of stationary action. While some acknowledge empirical and theoretical foundations, others question its universal applicability and the reasons behind its effectiveness.

Contextual Notes

Participants note that the principle of stationary action is a postulate used to derive equations in various physical theories, but there is uncertainty about its foundational motivations and the necessity of assigning an action to physical systems.

  • #31
marcusl said:
I think one reason for heavy reliance on variational approaches early on was computational. In an era before computers (or even slide rules), solving equations of motion at 1000's of sequential time or space points required heroism. A variational calculation found the path all at once. Variational computations of, e.g., electromagnetic systems (and even circuits) remained popular until computers came onto the scene, as did conformal mapping and the like.
True enough. + I think they (i.e. variational methods and principles) are also related and connected to numeric computational methods even today. For that interesting connection between Variational Calculus and Numerical Methods, as well as Optimization Methods and Mathematical Programming see e.g. :

Encyclopedia of Mathematics [Variational calculus, numerical methods of]

https://www.google.gr/url?sa=t&sour...ggdMAA&usg=AFQjCNGyCnW-qeLyrtWU6zxnHOcXhfcNQg

I quote here a relevant to your comment part, (but perhaps the other way around connection) e.g.:

"The first numerical methods of the calculus of variations appeared in the work of L. Euler. However, their most rapid development took place in the mid-20th century as a result of the spread of computers and the possibility, afforded by these techniques, to solve complicated problems in technology. The development of numerical methods in variational calculus mostly concerned the theory of optimal control, which, from the aspect of practical applications, is the most important (cf. Optimal control, mathematical theory of)."
 
Last edited:
  • Like
Likes   Reactions: vanhees71
Physics news on Phys.org
  • #32
ljagerman said:
If you seek a discussion of the principle ... than it actually does.
Acknowledged. Interesting.
 

Similar threads

Graduate Why does the principle of stationary action work?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Action in Lagrangian Mechanics
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Variational operator in the least action principle
  • · Replies 4 ·
Replies
4
Views
1K
Undergrad Is Hamilton's principle an empirical statement?
  • · Replies 24 ·
Replies
24
Views
2K
Graduate An ab initio Hilbert space formulation of Lagrangian mechanics
  • · Replies 10 ·
Replies
10
Views
2K
Undergrad End point information in lagrangain variation principle
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Principle of Stationary Action - Intuition
  • · Replies 13 ·
Replies
13
Views
3K
Graduate Variation of Lagrangian w/r to canonical momenta
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Principle of Least Action for optics?
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Is there a principle of stationary action for QFT?
  • · Replies 8 ·
Replies
8
Views
2K