Mathematical definition of a rigid body

[wrobel]

I am teaching classical mechanics for students from mathematical dept. Students demand mathematically rigor definition of a rigid body. It is strange but I have not found the one in textbooks.

My attempt :

Df. (kinematical version): Take a non-void open set $D\subset\mathbb{R}^3$ and a pair of functions:
$$ \boldsymbol r_A\in C^2\big([0,T],\mathbb{R}^3\big),
\quad U\in C^2\big([0,T],\mathrm{SO}(3)\big),\quad U(0)=E,\quad \boldsymbol r_A(0)\in \overline D.$$
We shall say that a point $B$ belongs to the rigid body if its motion is described by the following position vector
$$\boldsymbol r_B(t)=\boldsymbol r_A(t)+U(t)\big(\boldsymbol r_B(0)-\boldsymbol r_A(0)\big),\quad t\in[0,T], \qquad(*)$$
where the initial value $\boldsymbol r_B(0)\in\overline D.$
By definition the rigid body consists of such points $B$.

 

[Hill]

I am teaching classical mechanics for students from mathematical dept. Students demand mathematically rigor definition of a rigid body. It is strange but I have not found the one in textbooks.

My attempt :

Df. (kinematical version): Take a non-void open set $D\subset\mathbb{R}^3$ and a pair of functions:
$$ \boldsymbol r_A\in C^2\big([0,T],\mathbb{R}^3\big),
\quad U\in C^2\big([0,T],\mathrm{SO}(3)\big),\quad U(0)=E,\quad \boldsymbol r_A(0)\in \overline D.$$
We shall say that a point $B$ belongs to the rigid body if its motion is described by the following position vector
$$\boldsymbol r_B(t)=\boldsymbol r_A(t)+U(t)\big(\boldsymbol r_B(0)-\boldsymbol r_A(0)\big),\quad t\in[0,T], \qquad(*)$$
where the initial value $\boldsymbol r_B(0)\in\overline D.$
By definition the rigid body consists of such points $B$.

Doesn't the definition above assume a mathematically rigorous definition of "motion"?

 

[wrobel]

I am afraid I do not understand the question

 

[PeroK]

I would have said that a rigid body is a set of particles whose pairwise distances from each other are constant.

 

[Hill]

I would have said that a rigid body is a set of particles whose pairwise distances from each other are constant.

This definition allows for reflection, while rigid body does not.

 

[PeroK]

This definition allows for reflection, while rigid body does not.

Reflection is impossible if each particle obeys Newton's laws of motion.

 

[wrobel]

I would have said that a rigid body is a set of particles whose pairwise distances from each other are constant.

Sure we must just add that this set contains at least three points that do not belong to the same line. Else the angular velocity vector is not defined uniquely.
I usually use the definition you quoted but I do not know a completely formal way to pass from this definition to derivation of the Euler formula:
$$\boldsymbol v_B=\boldsymbol v_A+\boldsymbol\omega\times \boldsymbol{AB}$$

 

[Frabjous]

You could define strain and set it to zero. This will take care of translation and rotation.

 

[wrobel]

You could define strain and set it to zero. This will take care of translation and rotation.

Yes, but the main problem is a way from a definition to the Euler formula.

 

[Frabjous]

Just some random thoughts
1) Choose a point on the body (not space) as the origin.
2) After motion, this point on the body is still the origin. In space, this point has undergone a translation only.
3) Since the body is rigid, every other point on the body can only undergo a rotation about the origin point.

 

[wrobel]

1) Choose a point on the body (not space) as the origin.
2) After motion, this point on the body is still the origin. In space, this point has undergone a translation only.
3) Since the body is rigid, every other point on the body can only undergo a rotation about the origin point.

isn't it a verbal description of the formula (*) from the initial post?

 

[Frabjous]

isn't it a verbal description of the formula (*) from the initial post?

In my mind, I was laying out a proof in reponse to post #7.

I know that for an arbitrary displacement, a rigid body can undergo translation and rotation. I believe my post showed how to handle the translation. I believe the rotation can be handled formally as described here https://en.wikipedia.org/wiki/Euler's_rotation_theorem

I think you know all this, so I am unclear on what you asking for.

 

[wrobel]

I just noted that if we want to have a mathematically consistent theory without gaps filled with physical intuition, we have to formalize properly the definition of a rigid body and there is a formalization that gives apparently the shortest way to the main formula of the rigid body's kinematic.