Why does a fixed distance reduce degrees of freedom?

In summary: 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?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
  • #1
ajayguhan
153
1
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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  • #2
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
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?
 
  • #4
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.
 
  • #5
Got it, thanks.
 

1. What is the concept of degrees of freedom in science?

Degrees of freedom refer to the number of independent variables in a system that can vary without affecting the overall behavior or state of the system. In other words, it represents the number of ways a system can move or change.

2. Why is the number of degrees of freedom important?

The number of degrees of freedom is important because it helps determine the complexity and behavior of a system. Systems with a higher number of degrees of freedom are generally more complex and require more variables to be considered in their analysis.

3. What is the difference between 3N and 5 degrees of freedom?

In scientific terms, 3N and 5 degrees of freedom refer to the number of degrees of freedom in a system with 3N and 5 variables, respectively. The number of degrees of freedom can vary depending on the number of variables and their interdependencies in a system.

4. How are degrees of freedom calculated?

Degrees of freedom are calculated by subtracting the number of constraints or limitations in a system from the total number of variables. This gives the number of independent variables that can vary without affecting the overall state of the system.

5. Can the number of degrees of freedom change in a system?

Yes, the number of degrees of freedom in a system can change if there are changes in the number of variables or constraints. For example, if a new variable is introduced or a constraint is removed, the number of degrees of freedom will change accordingly.

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