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

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SUMMARY

The discussion defines "degrees of freedom" in a physical system as the minimum number of independent coordinates required to describe its state. Using the example of two identical simple pendulums, it is established that while one angle can describe their position when oscillating in phase, each pendulum retains its own degree of freedom, resulting in a total of two degrees of freedom. The conversation emphasizes the importance of recognizing independent systems and their respective coordinates, even when they exhibit synchronized behavior.

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  • Knowledge of independent and dependent variables in physics.
  • Basic grasp of ordinary differential equations (ODEs) related to motion.
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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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