Degree of Freedom: Exploring Its Meaning & Implications

In summary, the conversation discusses the concept of degrees of freedom and how it applies to describing a system uniquely. It is defined as the minimum number of parameters needed to describe the system, with examples given for a point in 3-dimensional space and an object in 3-dimensional space. There is also a discussion about whether 3 or 2 angles are needed to uniquely describe an object's rotation, with some disagreement on the matter.
  • #1
cks
165
0
What do you understand about this?

From high school, I just memorize the definition of it, the number of ways of obtaining energy independently. Well, it's pretty unclear by this definition.
 
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  • #2
Perhaps you could tell us more precisely the problem you have encountered with using degrees of freedom. To put it simply, the number of degrees of freedom of a system are the minimum number of parameters you need in order to describe the system uniquely. So, for example, a point in 3 dimensional space has 3 degrees of freedom (one for each of the three dimensions). If we state which value in each dimension the point has (i.e. define the coordinate (x,y,z)) then we have uniquely specified where in space that point is. An object in 3 dimensional space has 6 degrees of freedom: three defining its position in space, and 3 defining the angles of rotation. Again, if we specify these 6 parameters, then we have uniquely described the system.
 
  • #3
For your second explanation, why there are 3 angles of rotation to specify the system. ?
 
  • #4
cristo said:
An object in 3 dimensional space has 6 degrees of freedom: three defining its position in space, and 3 defining the angles of rotation. Again, if we specify these 6 parameters, then we have uniquely described the system.
Surely there are only five. I think you can uniquely describe the angle of rotation of an object using only two angles of rotation.
 
  • #5
DaveC426913 said:
Surely there are only five...

For a rigid body in three-space there are indeed six possible degrees of freedom.

DaveC426913 said:
I think you can uniquely describe the angle of rotation of an object using only two angles of rotation.

In most cases I would agree that you are correct indeed.
 
  • #6
I think it's helpful to distinguish between 'quadratic' DOF and 'free' DOF.

Yes- rotation of an object is in general determined by 3 DOF, which could be expressed as 2 polar coordinates and 1 rotation about the axis. A diatomic molecule only has 2 rotational axes because it is symmetric about the last rotation.
 
  • #7
I meant objects that have no rotational symmetry. They can be oriented in any of three directions by rotation aroind only two axes.

Wait... I just demonstrated that it DOES take 3 DoFs to uniquely describe an object. (though it may only require two to GET it there.)
 
  • #8
Yaw, pitch, roll. There are three. I think you were probably just forgetting roll.
 
  • #10
cks said:
What do you understand about this?

From high school, I just memorize the definition of it, the number of ways of obtaining energy independently. Well, it's pretty unclear by this definition.
By definition, the degrees of freedom is the minimum number of variables required to uniquely define the mechanical configuration of thhe system. E.g. a double pendulum has two degress of freedom. Two variables that might be chosen are the angle each pendulum makes with the vertical.

Best wishes

Pete
 

1. What is the definition of degree of freedom?

The degree of freedom refers to the number of independent variables or parameters that can vary in a statistical analysis. It represents the number of values that are free to vary in the data without violating any constraints or assumptions.

2. Why is degree of freedom important in statistical analysis?

Degree of freedom is important because it affects the accuracy and reliability of statistical tests. It allows us to determine the appropriate sample size for a study and helps in estimating the variability of the data. It also ensures that our results are not affected by any confounding variables.

3. How does the degree of freedom affect the outcome of a statistical test?

The degree of freedom affects the outcome of a statistical test by influencing the critical values used to determine statistical significance. As the degree of freedom increases, the critical values decrease, making it more difficult to reject the null hypothesis. This means that having more degrees of freedom can lead to a more conservative interpretation of the data.

4. Can the degree of freedom be negative?

No, the degree of freedom cannot be negative. It is a non-negative integer that represents the number of independent observations in a sample. If the degree of freedom is calculated to be negative, it is most likely due to an error in the data or calculation.

5. How is degree of freedom calculated in different statistical tests?

The calculation of degree of freedom varies depending on the statistical test being performed. In general, it is calculated as the number of observations minus the number of parameters or constraints. For example, in a t-test, the degree of freedom is equal to the sample size minus one.

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