How Many Degrees of Freedom Does an Object in 3D Space Have?

[e2m2a]

I see a lot of ambiguous explanations of degrees of freedom on the web and I need clarification. Suppose there is an object in space that can move freely along either the x,y, or z axis. Do we say it has six degrees of freedom because it can move along the x-axis one way or the opposite way, and one way or the other way on the y-axis and one way or the other way on the z axis or do we just say it has 3 degrees of freedom relating to the number of axes of motion?

 

[BvU]

A zero of an axis is a pretty arbitrary choice -- so in that respect: three. However, depending on the context, six might also be a good choice (three position components, three velocity components -- and then higher order derivatives are taken care of by equations of motion).

All this for a pointlike object. For an extended object there are some more degrees of freedom (rotational).

[edit] "depending on the context" should be explained. For the physics I bow to @wrobel below. My DOF habitat is in equation solving -- xqq for possible confusion...

 

[SchroedingersLion]

Intuitively, the degrees of freedom describe the fundamental independent ways a particle can move. All movements are superpositions of such fundamental movements.
E.g. in two dimensions, there might be two degrees of freedom, one for the x-, one for the y- direction. All other directions are superpositions of these two basic movements.
However, if you consider a pendulum in two dimensions, the same is not true. The pendulum can move in x- and in y- direction, but not independently (the x-position already defines the respective y-position). Thus, there is just one degree of freedom for the pendulum: Either the x- or the y- direction (or more conveniently the angle). That means the amount of degrees of freedom depends on the situation at hand.

 

[wrobel]

Degrees of freedom never characterize velocities in a system. They characterize set of possible positions of the system. Formal definition is as follows. Let a system consists of $N$ mass points with radius-vectors $\boldsymbol r_1,\ldots,\boldsymbol r_N$. Assume that this system is subordinated to the following constraints
$$\sum_{i=1}^N\Big(\boldsymbol a_{ij}(t,\boldsymbol r_1,\ldots,\boldsymbol r_N),\boldsymbol{\dot r}_i\Big)+b_j(t,\boldsymbol r_1,\ldots,\boldsymbol r_N)=0,\quad j=1,\ldots, n<3N.$$
The vectors $\xi_j=(\boldsymbol a_{1j},\ldots, \boldsymbol a_{Nj})\in \mathbb{R}^{3N}$ are linearly independent.
If the constraints are holonomic: $f_j(t,\boldsymbol r_1,\ldots,\boldsymbol r_N)=0$ then this case is reduced to the previous one by differentiation in $t$:
$$\sum_{i=1}^N\Big(\frac{\partial f_j}{\partial \boldsymbol r_i},\boldsymbol{\dot r}_i\Big)+\frac{\partial f_j}{\partial t}=0.$$

By definition, the vector space of virtual displacements consists of vectors $(\delta \boldsymbol r_1,\ldots,\delta\boldsymbol r_N)\in\mathbb{R}^{3N}$ such that
$$\sum_{i=1}^N\Big(\boldsymbol a_{ij},\delta\boldsymbol r_i\Big)=0.$$

By definition the number of degrees of freedom equals dimension of the space of virtual displacements. It is easy to see that the number of degrees of freedom is equal to $3N-n$

 

[e2m2a]

Thanks for the responses. There is much to ponder.

 

[Mech_Engineer]

I'm a little surprised no one stated it specifically, but the reason objects in 3-dimensional space have 6 degrees of freedom is because there are 3 translational (e.g. X, Y, Z) and 3 rotational (e.g. Rot_X, Rot_Y, and Rot_Z) degrees of freedom. A rigid unconstrained part can translate along any axis, and also rotate about any axis.

More reading here: https://en.wikipedia.org/wiki/Six_degrees_of_freedom

Six degrees of freedom (6DoF) refers to the freedom of movement of a rigid body in three-dimensional space. Specifically, the body is free to change position as forward/backward (surge), up/down (heave), left/right (sway) translation in three perpendicular axes, combined with changes in orientation through rotation about three perpendicular axes, often termed yaw (normal axis), pitch (transverse axis), and roll (longitudinal axis).