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I'm confused here. I have a three particle (rigid) system. What would be the degree of freedom? I found out five. 3 coordinates for center of mass and 2 for describing orientation.
But we have only three constraints, i.e. three equations that reduce 9 coordinates by 3, 9 - 3 = 6, which gives 6 degrees of freedom?? Did I miss something above?

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You need at least three parameters to describe the orientation of a rigid body. There are many parameterizations: – mmc Sep 8 '12 at 21:26
@mmc that would be good information to include in an answer – David Z Sep 9 '12 at 5:28
Possible duplicates: and links therein. – Qmechanic Sep 9 '12 at 7:36
up vote 2 down vote accepted

You missed that to specify orientation, you not only need an axis, but how far the body rotated around the axis. The two axis angles and the angle of rotation is the Euler angle parametrization, and I find it unweildy because the relation between this and position involves transcendental functions.

The nicest way to give the rotation part is to specify a rotation matrix R which has the property that $R^TR=I$. This has 3 parameters, since you have three orthogonal unit vectors inside, which is 2 components for the first (it's unit length), one component for the second (it's perpendicular to the first and unit length) and none for the third. This is most convenient for pencil and paper and computer calculations both, this is why it is hardly ever presented in textbooks.

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Hi @Ron ... could you add picture please. I find it easy to picture it geometrically. And thank you very much for response. – Monkey D. Luffy Sep 9 '12 at 12:17
@MonkeyD.Luffy: It takes time, and I will not spend time on this. What I said is sufficient. – Ron Maimon Sep 9 '12 at 15:40
Can you give me direct answer? What is the degree of freedom for a rigid body in 3 dimensional space? Hints are misleading me. Maybe I'll work out myself to fit in your answer. – Monkey D. Luffy Sep 9 '12 at 16:04
@MonkeyD.Luffy: The number is six, not five, it is six. Three translational, three rotational, altogether six, as I said in the answer. – Ron Maimon Sep 9 '12 at 16:29
all right ... thanks!! – Monkey D. Luffy Sep 9 '12 at 19:24

Every rigid body has 3 translational dof. In addition, there are 0, 2, or 3 rotational dof, depending on the geometry, giving a total of 3, 5, or 6 dof.

A spherically symmetric rigid body has no rotational dof.

A rigid body with rotational symmetry around an axis has 2 rotational dof, namely two angles for orienting the symmetry axis along a direction.

All other rigid bodies have 3 rotational dof, namely two angles with respect to an arbitrary axis attached to the body, and an angle for rotationg around this axis. This gives the Euler angle parameterization of the manifold of orientations (algebraically an $SO(3)$.) An important alternative parameterization is the quaternion parameterization, especially useful in computational geometry. It has a parameter vector $u$ with 4 components whose length is 1, leaving 3 dof. ($u$ and $-u$ describe the same rotation.)

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Good answer. Note that in a quaternion (4 numbers) not all the components are independent - 4 values but 3 degrees of freedom. – Floris Oct 29 '14 at 18:31

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