Principle of Least Action OR Hamilton's Principle

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Discussion Overview

The discussion revolves around the relationship between the principle of least action and Hamilton's principle, exploring whether they are equivalent and how one can transition from one to the other. The scope includes theoretical considerations and mathematical reasoning related to mechanics.

Discussion Character

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

Main Points Raised

  • Some participants propose that Hamilton's principle generalizes the principle of least action.
  • One participant presents a mathematical derivation showing that a specific case of Hamilton's principle arises when potential energy is constant, leading to a particular form of the Lagrangian.
  • Another participant questions the assumption of constant potential energy in celestial mechanics, suggesting that potential energy typically varies in such contexts.
  • There is mention of the freedom to include a total time derivative of a function in the Lagrangian, with uncertainty expressed about its implications for orbits and potential energy representation.
  • References to external sources are provided to clarify concepts and support claims made in the discussion.

Areas of Agreement / Disagreement

Participants express differing views on the equivalence of the principles, with some asserting a generalization while others highlight specific conditions under which the principles may differ. The discussion remains unresolved regarding the exact relationship between the two principles.

Contextual Notes

Participants note limitations in their assumptions, particularly regarding the constancy of potential energy in specific scenarios, and the implications of including total time derivatives in the Lagrangian formulation.

shehry1
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Are the principle of least action(http://astro.berkeley.edu/~converse/Lagrange/Kepler%27sFirstLaw.htm) and the hamilton principle 'exactly' the same? As far as I know, yes. How do I go from one to the other
 
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i believe 'Hamilton Principle' generalizes the 'Least Action principle'
 
It looks like the first equation that you refer to is a special case of Hamilton's principle, where the potential energy is taken to be constant. In the case of [tex]U = c[/tex], the Lagrangian is simply [tex](1/2)mv^2 + c[/tex], and hamilton's principle becomes

[tex]\delta \int \frac{1}{2} mv^2 {\mathrm d}t = 0 \Longrightarrow \delta \int mv^2 {\mathrm d}t = 0[/tex].

Since [tex]dt = {dx}/{v}[/tex], this is equivalent to

[tex]\delta \int mv {\mathrm d}x = 0[/tex].

This is usually written in the form

[tex]\delta \int p {\mathrm d}q = 0[/tex]

to emphasize that q is a generalized coordinate.
 
Last edited:
dx said:
It looks like the first equation that you refer to is a special case of Hamilton's principle, where the potential energy is taken to be constant. In the case of [tex]U = c[/tex], the Lagrangian is simply [tex](1/2)mv^2 + c[/tex], and hamilton's principle becomes

[tex]\delta \int \frac{1}{2} mv^2 {\mathrm d}t = 0 \Longrightarrow \delta \int mv^2 {\mathrm d}t = 0[/tex].

Since [tex]dt = {dx}/{v}[/tex], this is equivalent to

[tex]\delta \int mv {\mathrm d}x = 0[/tex].

This is usually written in the form

[tex]\delta \int p {\mathrm d}q = 0[/tex]

to emphasize that q is a generalized coordinate.

I wondered whether that was the case. However, this being celestial mechanics, it is obvious that the potential is not constant. In fact, only one or two lines down, the energy is written as a sum of KE and Potential (as you would expect)
 
There's also a freedom of a total time derivative of a function of (q,t) in the Lagrangian, so maybe in the case orbits the potential can be written in this way? I'm not sure.
 
dx said:
There's also a freedom of a total time derivative of a function of (q,t) in the Lagrangian, so maybe in the case orbits the potential can be written in this way? I'm not sure.

http://en.wikipedia.org/wiki/Reduced_action
Check out the disambiguation
 

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