What Is the Definition of Degrees of Liberty in a System?

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Discussion Overview

The discussion revolves around the definition of "degrees of liberty" in a physical system, specifically in the context of pendulums. Participants explore the implications of constraints and independence of coordinates in determining the degrees of freedom for systems of oscillating pendulums.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant requests a formal definition of "number degrees of liberty of a system."
  • Another participant defines it as the minimum number of independent coordinates needed to describe the physical state of the system, using a mathematical pendulum as an example.
  • A participant suggests that for two identical simple pendulums oscillating in phase, one angle is sufficient to describe the system, implying a degree of liberty of 1.
  • Another participant counters that there are 2 degrees of freedom, one for each pendulum, regardless of their phase relationship.
  • One participant questions if the definition is lacking, suggesting that one angle suffices to describe the entire system.
  • A participant argues that since the pendulums are independent systems, each requires its own angle, thus maintaining 2 degrees of freedom.
  • Another participant acknowledges that while one angle may describe the system under specific conditions, the general case requires consideration of both angles as independent coordinates.
  • One participant concludes that although the system can be described by one angle over time, the number of degrees of freedom remains 2.

Areas of Agreement / Disagreement

Participants express disagreement regarding the number of degrees of freedom for the two pendulums, with some arguing for 1 degree based on the phase relationship and others asserting that there are 2 degrees due to the independence of the systems.

Contextual Notes

Participants discuss the implications of constraints and independence in defining degrees of freedom, but the discussion does not resolve the differing interpretations of how many degrees of freedom apply in the case of the two pendulums.

quasar987
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Could someone write a formal definition for "number degrees of liberty of a system". Thank you very much.
 
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The minimum number of INDEPENDENT coordinates to describe the physical state of the system.

For example,the mathematical pendulum:a priori 3 coordinates for the bob,but once u impose the 2 customary constraints,u'll end up with one degree of freedom.

Daniel.
 
So if we have 2 identical simple pendulums, separated from each other by a fixed distance L and oscillating in a plane at the same angular frquency and in phase, then the angle that one makes with the vertical is sufficient to give the exact position of each bob and so the degree of liberty of this system is 1. Correct?
 
Ther are 2 degrees of freedom (one for each pendulum).The fact that they oscillate in phase only says that the two functions (angle of time) are identical as functional dependence,but the # of degrees of freedom is still 2.

Daniel.
 
But isn't there something missing in the definition then? I mean, isn't it true that one angle sufice to describe the entire system ?
 
Nope,there are 2 independent systems.If the oscillators are not coupled (meaning that between the 2 bobs there's no rigid rod to ensure they will always oscillate in phase),then there are 2 systems,each of them described by an angle which is the degree of freedom.

Daniel.
 
Oh ok... right NOW (i.e. for these particular initial conditions) they are described accurately by one angle, but it's POSSIBLE that we might need 2.. and that's what's meant by "independent coordinates".
 
Yes,you could say that they can be described by one angle vs.time,which would mean solving only one ODE,but the # of degrees of freedom is still 2.

End of story.

Daniel.
 

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