A rigid body is defined from the idealized concept that, distance between any two mass points remains constant throughout the motion.
Since, it is satisfied by the most of the objects we use in real life (Not absolutely, but perfectly applicable), the kinematics of these rigid bodies plays significant role in many areas.
The special property of this rigid body is that, we need only 6 independent coordinates to completely define the state of a rigid body in 3-Dimensional Space. No matter how many particle it contains, it can be always applied, entirely due to the constraints.
Other than the distance constraints, it is also possible to add additional constraints in any rigid body motion, and so the number of independent coordinates will be more reduced.
It is customary to use the first set of 3 independent coordinates as the "Space fixed coordinates" and the second set of 3 coordinates as the "Body fixed coordinates" but with the same origin.
So that, you can explain the state of the body (general position) with the Space fixed coordinates and its orientation relative to this Space fixed coordinates using the Body fixed coordinates.
Specifying one set of coordinates relative to another needs the basic rules of "Coordinate Transformation".
Let us say the $(x_1,x_2,x_3)$ and $(x'_1,x'_2,x'_3)$ are the components of same vector in two sets of orthogonal coordinate system with same origin. If $ \{ \hat{e_1}, \hat{e_2}, \hat{e_3} \}$ and $\{ \hat{e'_1} , \hat{e'_2} , \hat{e'_3}\}$ are the corresponding unit vectors,
Direction cosines are defined by,
$$ cos(\theta)_{ij} = cos (\hat{e'_i}\cdot\hat{e_j}) = \hat{e'_i}\cdot\hat{e_j} \,\,\,\ldots...eq.(1)$$
Using the direction cosines, new primed unit vectors in terms of old non primed unit vectors written by,
$$ \hat{e'_i} = \sum_j (\hat{e'_i}\cdot\hat{e_j}) \hat{e_j} = \sum_j cos(\theta)_{ij} \hat{e_j} \,\,\ldots...eq.(2)$$
Let me write the general vector in cartesian coordinates as,
$$ \vec{r} = x_1\hat{e_1} + x_2 \hat{e_2} + x_3\hat{e_3} = x'_1\hat{e'_1} + x'_2\hat{e'_2} + x'_3\hat{e'_3} \,\,\ldots...eq.(3) $$
Then each of the new coordinates in general can be written using eq.(2),
$$ x'_i = \vec{r}\cdot\hat{e'_i} = \sum_j x_j\hat{e_j} \cdot \hat{e'_i} = \sum_j x_j cos(\theta)_{ij} \,\,\ldots...eq.(4)$$
The domain of this definition can be expanded for any general vector.
Note: In three dimension, all indices run from 1 to 3
Accordingly in 3 dimension, we need 9 direction cosines to make the transformation from one to another namely,
$$ cos(\theta)_{ij} = \hat{e_j}\cdot\hat{e'_i} $$ where "i and j" each runs from 1 to 3 and so it gives 9 components.
But only three of them are needed to specify the orientation. What about the others?
It so happens, there are quite few extra relations exists due to the orthogonality property of the coordinates.
The orthogonality relations are given by,
$$ \hat{e_i}\cdot\hat{e_j} = \delta_{ij} \\~\\ \hat{e'_i}\cdot\hat{e'_j} = \delta_{ij} \,\,\, eq.(5) $$
Expanding the unit vector in one system in terms of the unit vectors of other system using eq.(2) and making use of direction cosines, it can be rewritten as,
$$ \sum_{j=1}^3 cos(\theta)_{i'j} cos(\theta)_{ij} = \delta_{ii'} \,\,\, eq.(6)$$
where $\delta $ is the Kronecker-delta symbol.
To make life simpler, we use the Einstein summation convention, and we denote the direction cosines $$ cos(\theta)_{ij} = a_{ij} $$
Now, eq.(4) becomes,
$$ x'_i = a_{ij} x_j \,\,\,\,\,\,\ldots...eq.(7) $$
where summation is assumed.
The magnitude of the vector $\vec{r} $ in both cases are the same. Using that property with the Pythagoras relation for orthogonal coordinates, we can write,
$$ x'_{i} x'_i = a_{ij} x_j a_{ik} x_j x_k $$
therefore, $$ a_{ij} a_{ik} = \delta_{jk} \,\,\,\,\,\ldots...eq.(8)$$
where both j,k runs from 1 to 3.
This is the exact same condition obtained from orthogonality relation but in a new representation.
Since both i and j runs from 1 to 3 in discrete sense, we can write the set of direction cosines in a general matrix form where i used for row and j for column. Let me call the matrix as A,
$$ A = \left[\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} \right] $$
The general transformation from one system to other can be thought of as an Matrix operation on one system which results the coordinates of other system.
Using matrices,
$$ \left[\begin{matrix} x'_1 \\ x'_2 \\ x'_3 \end{matrix}\right] = \left[\begin{matrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} \right] \left[\begin{matrix} x_1 \\ x_2 \\ x_3 \end{matrix} \right] \,\,\,\ldots...eq.(9)$$
Euler Angles:
Lagrangian formalism is created based on the concept of degrees of freedom and constraints. Lagrangian itself is defined in terms of Independent coordinates.
Finding the independent coordinates for a system in motion is essential for solving the Lagrangian.
From orthogonality relations, we can be sure that, we don't need all the 9 direction cosines as independent coordinates. All we need is some three independent functions of these direction cosines.
It will therefore necessary to define a set of three new independent functions to describe the orientation of rigid body in space using the Lagrangian.
[But there is also another condition for our Matrices, which says the determinant of the matrix should be +1. Otherwise it would be an inversion of the coordinates.]
There are really a number of different independent functions obtained in a number of different ways to describe the Lagrangian. But Euler angles is the customary one, where we will make out transform from one to another by making three successive rotations in a specific way.
The sequence in each step can be thought of a matrix operating on the coordinate system at that particular instant.
Let me start the initial transformation from the $\{ \hat{e_1},\hat{e_2},\hat{e_3}\}$ coordinate system, by making angle $\phi$ counter clock wise about $\hat{e_3}$ axis. And let me denote the resultant coordinate system by, $\{\hat{e_1}^1, \hat{e_2}^1,\hat{e_3}^1\}$
In the second stage, we make the transformation of $\{\hat{e_1}^1, \hat{e_2}^1,\hat{e_3}^1\}$ by rotating it about $\hat{e_1}^1 $ axis in the counter clock wise direction by an angle $\theta$
Consequently we arrive at a newer system denoted by $ \{\hat{e_1}^2, \hat{e_2}^2, \hat{e_3}^2\}$.
Now, we make the final transformation by rotating $\{\hat{e_1}^2,\hat{e_2}^2,\hat{e_3}^2\}$ by an angle $\psi$ with respect to $\hat{e_3}^2$ axis.
Thus, we finally arrived our desired transformation that is $$\{\hat{e_1}^3,\hat{e_2}^3,\hat{e_3}^3\} = \{\hat{e'_1},\hat{e'_2},\hat{e'_3}\} $$
In Matrix Notation, each transformation can be written as follows,
First transformation, $$ E_1 = A_1 E \,\,\,\,\ldots...eq.(10)$$
where "E" is the set of coordinate elements and "A" is the transformation Matrix.
Second transformation, $$ E_2 = A_2 E_1 \,\,\,\,\ldots...eq.(11)$$
Final transformation, $$ E_3 = A_3 E_2 = E' \,\,\,ldots...eq.(12)$$
Hence therefore, the combined transformation is denoted by the product of respective matrices,
$$ E' = A_3 A_2 A_1 E = R E \,\,\,\,\ldots...eq.(13) $$
where $ A_3 A_2 A_1 = R $
Writing each transformation in terms of its matrix element values i.e. direction cosine values,
Since $A_1$ represents the counter clock wise rotation of "E" by an angle $\phi$ about $\hat{e_3}$ axis, it can be written in the matrix form as,
$$ A_1 = \left[ \begin{matrix} cos\phi &sin\phi &0\\ -sin\phi & cos\phi &0 \\ 0 & 0 & 1 \end{matrix}\right] \,\,\,\ldots...eq.(14) $$
Similarly, $A_2$ is the rotation of $E_1$ by an angle $\theta$ with respect to $\hat{e_1}^1$ axis,
$$ A_2 = \left[ \begin{matrix} 1&0&0\\ 0& cos\theta &sin\theta\\ 0 & -sin\theta & cos\theta\end{matrix} \right] \,\,\,\ldots...eq.(15) $$
Finally, $A_3 $ is the rotation of $ E_2$ by angle $\psi$ with respect to $\hat{e_3}^2 $ axis,
$$ A_3 = \left[ \begin{matrix} cos\psi &\sin\psi & 0\\ -sin\psi & cos\psi & 0 \\ 0 &0&1 \end{matrix}\right] \,\,\,\ldots...eq.(16)$$
Combining together, three transformations can be written using a single matrix R as follows,
$$ R = A_1A_2A_3 $$
subsituting for $ A_1, A_2, A_3 $ gives,
$$ R= \left[ \begin{matrix} cos{\psi} cos{\phi} - cos{\theta} sin{\phi} sin{\psi} & cos{\psi} sin{\phi}+ cos{\theta} cos{\phi} sin{\psi} & sin{\psi}sin{\theta}\\ -sin{\psi}cos{\phi} - cos{\theta}sin{\phi}cos{\psi} & -sin{\psi}sin{\phi}+cos{\theta}cos{\phi}cos{\psi} & cos{\psi}sin{\theta}\\ sin{\theta}sin{\phi} & -sin{\theta}cos{\phi} & cos{\theta} \end{matrix}\right] \,\, \ldots...eq.(17)$$
The inverse transformation from body fixed to space fixed coordinates is just given by the inverse of R i.e. $R^{-1}$ which is the transpose of R.
You can ask, why I choose this specific order of rotations. It needn't be. You can choose many other ways. But the only condition is, two consecutive rotations shouldn't be about the same axis.
Since, it is satisfied by the most of the objects we use in real life (Not absolutely, but perfectly applicable), the kinematics of these rigid bodies plays significant role in many areas.
The special property of this rigid body is that, we need only 6 independent coordinates to completely define the state of a rigid body in 3-Dimensional Space. No matter how many particle it contains, it can be always applied, entirely due to the constraints.
Other than the distance constraints, it is also possible to add additional constraints in any rigid body motion, and so the number of independent coordinates will be more reduced.
It is customary to use the first set of 3 independent coordinates as the "Space fixed coordinates" and the second set of 3 coordinates as the "Body fixed coordinates" but with the same origin.
So that, you can explain the state of the body (general position) with the Space fixed coordinates and its orientation relative to this Space fixed coordinates using the Body fixed coordinates.
Specifying one set of coordinates relative to another needs the basic rules of "Coordinate Transformation".
Let us say the $(x_1,x_2,x_3)$ and $(x'_1,x'_2,x'_3)$ are the components of same vector in two sets of orthogonal coordinate system with same origin. If $ \{ \hat{e_1}, \hat{e_2}, \hat{e_3} \}$ and $\{ \hat{e'_1} , \hat{e'_2} , \hat{e'_3}\}$ are the corresponding unit vectors,
Direction cosines are defined by,
$$ cos(\theta)_{ij} = cos (\hat{e'_i}\cdot\hat{e_j}) = \hat{e'_i}\cdot\hat{e_j} \,\,\,\ldots...eq.(1)$$
Using the direction cosines, new primed unit vectors in terms of old non primed unit vectors written by,
$$ \hat{e'_i} = \sum_j (\hat{e'_i}\cdot\hat{e_j}) \hat{e_j} = \sum_j cos(\theta)_{ij} \hat{e_j} \,\,\ldots...eq.(2)$$
Let me write the general vector in cartesian coordinates as,
$$ \vec{r} = x_1\hat{e_1} + x_2 \hat{e_2} + x_3\hat{e_3} = x'_1\hat{e'_1} + x'_2\hat{e'_2} + x'_3\hat{e'_3} \,\,\ldots...eq.(3) $$
Then each of the new coordinates in general can be written using eq.(2),
$$ x'_i = \vec{r}\cdot\hat{e'_i} = \sum_j x_j\hat{e_j} \cdot \hat{e'_i} = \sum_j x_j cos(\theta)_{ij} \,\,\ldots...eq.(4)$$
The domain of this definition can be expanded for any general vector.
Note: In three dimension, all indices run from 1 to 3
$$ cos(\theta)_{ij} = \hat{e_j}\cdot\hat{e'_i} $$ where "i and j" each runs from 1 to 3 and so it gives 9 components.
But only three of them are needed to specify the orientation. What about the others?
It so happens, there are quite few extra relations exists due to the orthogonality property of the coordinates.
The orthogonality relations are given by,
$$ \hat{e_i}\cdot\hat{e_j} = \delta_{ij} \\~\\ \hat{e'_i}\cdot\hat{e'_j} = \delta_{ij} \,\,\, eq.(5) $$
Expanding the unit vector in one system in terms of the unit vectors of other system using eq.(2) and making use of direction cosines, it can be rewritten as,
$$ \sum_{j=1}^3 cos(\theta)_{i'j} cos(\theta)_{ij} = \delta_{ii'} \,\,\, eq.(6)$$
where $\delta $ is the Kronecker-delta symbol.
To make life simpler, we use the Einstein summation convention, and we denote the direction cosines $$ cos(\theta)_{ij} = a_{ij} $$
Now, eq.(4) becomes,
$$ x'_i = a_{ij} x_j \,\,\,\,\,\,\ldots...eq.(7) $$
where summation is assumed.
The magnitude of the vector $\vec{r} $ in both cases are the same. Using that property with the Pythagoras relation for orthogonal coordinates, we can write,
$$ x'_{i} x'_i = a_{ij} x_j a_{ik} x_j x_k $$
therefore, $$ a_{ij} a_{ik} = \delta_{jk} \,\,\,\,\,\ldots...eq.(8)$$
where both j,k runs from 1 to 3.
This is the exact same condition obtained from orthogonality relation but in a new representation.
Since both i and j runs from 1 to 3 in discrete sense, we can write the set of direction cosines in a general matrix form where i used for row and j for column. Let me call the matrix as A,
$$ A = \left[\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} \right] $$
The general transformation from one system to other can be thought of as an Matrix operation on one system which results the coordinates of other system.
Using matrices,
$$ \left[\begin{matrix} x'_1 \\ x'_2 \\ x'_3 \end{matrix}\right] = \left[\begin{matrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} \right] \left[\begin{matrix} x_1 \\ x_2 \\ x_3 \end{matrix} \right] \,\,\,\ldots...eq.(9)$$
Euler Angles:
Lagrangian formalism is created based on the concept of degrees of freedom and constraints. Lagrangian itself is defined in terms of Independent coordinates.
Finding the independent coordinates for a system in motion is essential for solving the Lagrangian.
From orthogonality relations, we can be sure that, we don't need all the 9 direction cosines as independent coordinates. All we need is some three independent functions of these direction cosines.
[But there is also another condition for our Matrices, which says the determinant of the matrix should be +1. Otherwise it would be an inversion of the coordinates.]
There are really a number of different independent functions obtained in a number of different ways to describe the Lagrangian. But Euler angles is the customary one, where we will make out transform from one to another by making three successive rotations in a specific way.
The sequence in each step can be thought of a matrix operating on the coordinate system at that particular instant.
Let me start the initial transformation from the $\{ \hat{e_1},\hat{e_2},\hat{e_3}\}$ coordinate system, by making angle $\phi$ counter clock wise about $\hat{e_3}$ axis. And let me denote the resultant coordinate system by, $\{\hat{e_1}^1, \hat{e_2}^1,\hat{e_3}^1\}$
In the second stage, we make the transformation of $\{\hat{e_1}^1, \hat{e_2}^1,\hat{e_3}^1\}$ by rotating it about $\hat{e_1}^1 $ axis in the counter clock wise direction by an angle $\theta$
Consequently we arrive at a newer system denoted by $ \{\hat{e_1}^2, \hat{e_2}^2, \hat{e_3}^2\}$.
Now, we make the final transformation by rotating $\{\hat{e_1}^2,\hat{e_2}^2,\hat{e_3}^2\}$ by an angle $\psi$ with respect to $\hat{e_3}^2$ axis.
Thus, we finally arrived our desired transformation that is $$\{\hat{e_1}^3,\hat{e_2}^3,\hat{e_3}^3\} = \{\hat{e'_1},\hat{e'_2},\hat{e'_3}\} $$
In Matrix Notation, each transformation can be written as follows,
First transformation, $$ E_1 = A_1 E \,\,\,\,\ldots...eq.(10)$$
where "E" is the set of coordinate elements and "A" is the transformation Matrix.
Second transformation, $$ E_2 = A_2 E_1 \,\,\,\,\ldots...eq.(11)$$
Final transformation, $$ E_3 = A_3 E_2 = E' \,\,\,ldots...eq.(12)$$
Hence therefore, the combined transformation is denoted by the product of respective matrices,
$$ E' = A_3 A_2 A_1 E = R E \,\,\,\,\ldots...eq.(13) $$
where $ A_3 A_2 A_1 = R $
Writing each transformation in terms of its matrix element values i.e. direction cosine values,
Since $A_1$ represents the counter clock wise rotation of "E" by an angle $\phi$ about $\hat{e_3}$ axis, it can be written in the matrix form as,
$$ A_1 = \left[ \begin{matrix} cos\phi &sin\phi &0\\ -sin\phi & cos\phi &0 \\ 0 & 0 & 1 \end{matrix}\right] \,\,\,\ldots...eq.(14) $$
Similarly, $A_2$ is the rotation of $E_1$ by an angle $\theta$ with respect to $\hat{e_1}^1$ axis,
$$ A_2 = \left[ \begin{matrix} 1&0&0\\ 0& cos\theta &sin\theta\\ 0 & -sin\theta & cos\theta\end{matrix} \right] \,\,\,\ldots...eq.(15) $$
Finally, $A_3 $ is the rotation of $ E_2$ by angle $\psi$ with respect to $\hat{e_3}^2 $ axis,
$$ A_3 = \left[ \begin{matrix} cos\psi &\sin\psi & 0\\ -sin\psi & cos\psi & 0 \\ 0 &0&1 \end{matrix}\right] \,\,\,\ldots...eq.(16)$$
Combining together, three transformations can be written using a single matrix R as follows,
$$ R = A_1A_2A_3 $$
subsituting for $ A_1, A_2, A_3 $ gives,
$$ R= \left[ \begin{matrix} cos{\psi} cos{\phi} - cos{\theta} sin{\phi} sin{\psi} & cos{\psi} sin{\phi}+ cos{\theta} cos{\phi} sin{\psi} & sin{\psi}sin{\theta}\\ -sin{\psi}cos{\phi} - cos{\theta}sin{\phi}cos{\psi} & -sin{\psi}sin{\phi}+cos{\theta}cos{\phi}cos{\psi} & cos{\psi}sin{\theta}\\ sin{\theta}sin{\phi} & -sin{\theta}cos{\phi} & cos{\theta} \end{matrix}\right] \,\, \ldots...eq.(17)$$
The inverse transformation from body fixed to space fixed coordinates is just given by the inverse of R i.e. $R^{-1}$ which is the transpose of R.
You can ask, why I choose this specific order of rotations. It needn't be. You can choose many other ways. But the only condition is, two consecutive rotations shouldn't be about the same axis.
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