This is not a homework problem and I am not taking a class in classical mechanics right now.

I am using L Hand and J Finch, a textbook on analytical mechanics (based on a series of lectures from Cornell).

Question: Show that a rigid body with three or more mass points has six degrees of freedom.

Attempt: of course we are talking about a 3 dimensional eucldiean space.

The formula that the book provides: N = 3M - j

Where M is the number of the particles. j is the number of constraints. N is the number of degrees of freedom.

For now the coordinates only represent the position. They do not mention momentum yet. So for instance, a rigid body composed of two point masses has 5 degrees of freedom. since there is one constraint connecting the two point masses and there are 6 coordinates in total.

I attempted to use combinatorics to generate a formula for M point masses (M is greater than or equal to 3). N = 3M -MC2 (M choose 2).

It clearly fails.

I am guessing that some constraints are dependent on others.

I tried an example of 5 point masses. there are 5C2 = 10 constraints. And I calculated 3*5 - 10 = 5 degrees of freedom. There is something wrong.

I need some help to PHYSICALLY understand how constraints are dependent on each other and how to mathematically derive a formula.

Thank you