Why does a fixed distance reduce degrees of freedom?

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Discussion Overview

The discussion centers around the concept of degrees of freedom in a system of particles, specifically addressing how a fixed distance between two particles affects their degrees of freedom. The scope includes theoretical reasoning and conceptual clarification regarding coordinate systems and constraints in physics.

Discussion Character

  • Exploratory, Conceptual clarification, Technical explanation

Main Points Raised

  • One participant notes that a particle has 3 degrees of freedom, leading to the conclusion that two particles would typically have 6 degrees of freedom, but a fixed distance reduces this to 5 degrees of freedom.
  • Another participant explains that knowing the coordinates of one particle allows for determining the position of the second particle using two angles, thus requiring only 5 pieces of information in total.
  • A participant questions whether using different types of coordinates (Cartesian and polar) is appropriate when discussing degrees of freedom, suggesting that the coordinates should be of the same type.
  • A later reply argues that it does not matter which coordinate system is used, as the physical equations remain valid across different representations, and some constraints may be easier to handle in certain coordinate systems.
  • The same participant mentions that the equations of motion for maintaining a fixed distance are simpler in spherical coordinates compared to Cartesian coordinates.

Areas of Agreement / Disagreement

Participants express differing views on the appropriateness of using mixed coordinate systems and the implications of fixed distances on degrees of freedom, indicating that multiple competing perspectives remain in the discussion.

Contextual Notes

There are unresolved considerations regarding the choice of coordinate systems and the implications of constraints on the equations of motion, which may vary depending on the representation used.

ajayguhan
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A particle has 3 degrees of freedom, therefore N particles have 3N degrees of freedom. But a two particle whose distance between them is constant has 5 degrees of freedom instead of 6. I know that the fixed distance is constarin so it reduced the degrees of freedom, but why does the constarin reduces the degrees of freedom?


Thanks in advance.
 
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Because if I tell you the coordinates of one particle, I only need to tell you two bits of information (usually two angles) to be able to know where the other particle is, as the distance is fixed.

So the total information needed is 3 coordinates and 2 angles, hence 5 degrees of freedom.
 
3 co ordinates are cartesian while other two is polar doesn't it bother? I mean the co ordinates should of the same type or can they vary?
 
ajayguhan said:
3 co ordinates are cartesian while other two is polar doesn't it bother? I mean the co ordinates should of the same type or can they vary?
I doens't matter. You don't even have to have cartesian coordinates to start with. The same situation can be described using spherical coordinates for the first particle. You just usually use the simplest representation for the problem: the physical equations are equally valid whichever system you use. It just happens that most of the time, some representations lead to simpler equations.

It is common for n-body problems to separate the centre of mass motion from the relative motion of the bodies. If the centre of mass is rotating, it makes more sense to describe it using spherical coordinates, while the relative motion can be described with cartesian coordinates or spherical coordinates or cylindrical coordinates or anything else.

Also, some constraints are difficult to work with in certain coordinate systems. In your example, keeping the distance fixed between the two particles, I wouldn't even know how to write the equations of motion preserving that distance in cartesian coordinates, whereas they are very simple in spherical coordinates.
 
Got it, thanks.
 

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