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--- book:chap6:6.1_motivation_and_outline [2022/02/01 11:58] – created abril
+++ book:chap6:6.1_motivation_and_outline [2022/04/15 15:41] (current) – [6.0.1 Self Test] jv
@@ Line 37: / Line 37: @@
 <WRAP box round #bsp_LagrangeParameterizationPendulum>**Example 6.1** <wrap em>Generalized coordinates for a pendulum</wrap> \\
 We describe the position of the mass in a mathematical pendulum by the angle $\theta(t)$,
-as introduced in [[book:chap1:1.3_order-of-magnitude_guesses #bsp_pendulum-omg |Example 1.10]].
+as introduced in [[book:chap1:1.3_order-of-magnitude_guesses#bsp_pendulum-omg|Example 1.10]].
 The position of the mass in the 2D pendulum plane is thus described by the vector
 \begin{align*}
@@ Line 53: / Line 53: @@
       \hat{\boldsymbol\theta}(\theta(t)) =  \begin{pmatrix} \cos\theta(t) \\ \sin\theta(t) \end{pmatrix}
 \end{align*}
+</WRAP>
+<wrap lo>** Remark 6.1. **
 Note that $\hat{\mathbf R}(\theta)$ and $\hat{\boldsymbol\theta}(\theta)$ are orthonormal vectors
 that describe the position of the mass in terms of polar coordinates
 rather than fixed-in-space Cartesian coordinates.
+</wrap>
+<WRAP box round #Thm_Uvectors4polarCoordinates>**Theorem 6.1** <wrap em>Basis vectors for polar coordinates</wrap> \\
+Let $\{ \hat{\boldsymbol x}, \hat{\boldsymbol z}\}$ be a basis of $\mathbb R^2$, and
+$(R,\theta)$ be the polar coordinates((The choice of the axes and the angle reflects the notations adopted in [[:book:chap1:1.2_dimensional_analysis #fig_pendulum-nodim|Figure 1.2]] and [[:book:chap1:1.3_order-of-magnitude_guesses #fig_pendulum-omg|Figure 1.3]].))
+associated to a point with cartesian coordinates $(x,z)$.
+Then
+  *  $R = \sqrt{x^2+z^2}$ is the distance of the point from the origin
+  *  $\theta = -\arctan(x/z)$ the angle with respect to $-\hat{\boldsymbol z}$,
+We denote the vector from the origin to $(R,\theta)$ as $R \: \hat{\boldsymbol R}(\theta)$.
+Then the following statements hold
+  -  $\hat{\boldsymbol R}(\theta)$ is a normal vector at the position $(R,\theta)$ of a circle $\mathsf{C}_R$ with center at the origin at radius $R$.
+  -  $\hat{\boldsymbol \theta} = \partial_\theta \hat{\boldsymbol R}$ is a vector tangential to $\mathsf{C}_R$ at the position $(R, \theta)$.
+  -  $\partial_\theta \hat{\boldsymbol \theta} = - \hat{\boldsymbol R}$.
+  -  For every $\theta \in [0, 2\pi)$ the vectors $\{ \hat{\boldsymbol R}(\theta), \hat{\boldsymbol  \theta}(\theta)\}$ form an orthonormal basis of  $\mathbb R^2$.
 </WRAP>
-<WRAP box round>**Example 5.1** <wrap em>Generalized coordinates for a ping-pong ball</wrap> \\
+<wrap lo>** Remark 6.2. **
+The coordinate representation of $\hat{\boldsymbol R}(\theta)$ and $\hat{\boldsymbol  \theta}(\theta)$ in cartesian coordinates is provided in [[#bsp__LagrangeParameterizationPendulum|Example 6.2]].
+</wrap>
+<wrap lo>** Remark 6.3. **  <wrap #remark_Uvectors4polarCoordinates />
+The assertions of [[#thm_Uvectors4polarCoordinates|Theorem 6.1]] also apply when the unit vectors of $\mathbb R^2$ are denoted as $\{ \hat{\boldsymbol x}, \hat{\boldsymbol y}\}$, and when the angle $\theta$ denotes the angle with respect of the $\hat{\boldsymbol x}$ axis.
+The different reference axis //only// changes the coordinate representation of the vectors.
+</wrap>
+<WRAP 120pt right #fig_pingPongCoordinates>
+{{pingPongCoordinates.png}}
+Figure 6.3: The position of a ball in space can be described in terms of a 3D vector $\mathbf Q$ that describes the center of the ball (red dot),
+angles $\theta,\phi$ that describe the orientation in space of a fixed axes in the ball (green line),
+and another angle $\psi$ that describes the position of point that is not on the axis (blue point).
+</WRAP>
+<WRAP box round>**Example 6.2** <wrap em>Generalized coordinates for a ping-pong ball</wrap> \\
 A ping-pong ball consists of $N$ atoms located in the three-dimensional space.
 During a match they follow an intricate trail in the vicinity of the ping-pong players.
 At any time during their motion the atoms are located on a thin spherical shell with fixed positions with respect to each other.
-Rather than specifying the position of each atom one can therefore specify the position of the ball in terms of six generalized coordinates (\cref{fig:pingPongCoordinates}):
+Rather than specifying the position of each atom one can therefore specify the position of the ball in terms of six generalized coordinates ([[#fig_pingPongCoordinates|Figure 6.3]]):
 Three coordinates provide its center of mass.
 The orientation of the ball can be provided by specifying the orientation of a body fixed axis in terms of its polar and azimuthal angle,
 and a third angle specifies the orientation of a point on its equator
 when rotating the ball around the axis.
-</WRAP>
-<WRAP 120pt right #fig_pingPongCoordinates>
-{{pingPongCoordinates.png}}
-Figure 6.3: The position of a ball in space can be described in terms of a 3D vector $\mathbf Q$ that describes the center of the ball (red dot),
-angles $\theta,\phi$ that describe the orientation in space of a fixed axes in the ball (green line),
-and another angle $\psi$ that describes the position of point that is not on the axis (blue point).
 </WRAP>
@@ Line 98: / Line 123: @@
 In all cases we will eventually reduce the dynamics to one-dimensional problems.
 The final [[book:chap6:lagrange|Section 6.6]] deals with the relation between continuous symmetries of the dynamics and conservation laws.
+==== 6.1.1 Self Test ====
+----
+<wrap #quest_setSelftest-polarCoords> Problem 6.1: </wrap>** Different reference axes for polar coordinates **
+\\
+Verify the assertions of [[#Thm_Uvectors4polarCoordinates|Theorem 6.1]] and [[#remark_Uvectors4polarCoordinates|Remark 6.3]]:
+  -  Specify the coordinate representation of $\{ \hat{\boldsymbol R}(\theta), \hat{\boldsymbol  \theta}(\theta)\}$ for the case where $\theta$ denoes the angle with respect to the positive $\hat{\boldsymbol x}$ axis.
+  -  Verify the assertion of [[#Thm_Uvectors4polarCoordinates|Theorem 6.1]].
+----
+<wrap #quest_setSelftest-LagrangeDieCoordinates > Problem 6.2: </wrap>**Describing the orientation of dice**
+\\
+We place a die on the table such that its center lies at the origin of a 3D cartesian coordinate frame,
+and its axes are aligned with the coordinate axes.
+We characterize the configuration of the die by the number of dots on the faces pointing in the three positive coordinate directions.
+  -  Show that there are 24 different possibilities to place the die.
+  -  Determine the angles $(\theta, \phi, \psi)$ (cf [[#fig_pingPongCoordinates|Figure 6.3]])
+that will turn a die from the configuration be $(1,2,3)$ to
+\begin{align*}
+      \text{b1) } (2,3,4) &&
+      \text{b2) } (4,6,2) &&
+      \text{b3) } (1,3,5) &&
+\end{align*}
 ~~DISCUSSION~~