book:chap6:6.1_motivation_and_outline
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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 | ||
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<WRAP box round # | <WRAP box round # | ||
We describe the position of the mass in a mathematical pendulum by the angle $\theta(t)$, | We describe the position of the mass in a mathematical pendulum by the angle $\theta(t)$, | ||
- | as introduced in [[book: | + | as introduced in [[book: |
The position of the mass in the 2D pendulum plane is thus described by the vector | The position of the mass in the 2D pendulum plane is thus described by the vector | ||
\begin{align*} | \begin{align*} | ||
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\hat{\boldsymbol\theta}(\theta(t)) = \begin{pmatrix} \cos\theta(t) \\ \sin\theta(t) \end{pmatrix} | \hat{\boldsymbol\theta}(\theta(t)) = \begin{pmatrix} \cos\theta(t) \\ \sin\theta(t) \end{pmatrix} | ||
\end{align*} | \end{align*} | ||
+ | </ | ||
+ | <wrap lo>** Remark 6.1. ** | ||
Note that $\hat{\mathbf R}(\theta)$ and $\hat{\boldsymbol\theta}(\theta)$ are orthonormal vectors | 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 | that describe the position of the mass in terms of polar coordinates | ||
rather than fixed-in-space Cartesian coordinates. | rather than fixed-in-space Cartesian coordinates. | ||
+ | </ | ||
+ | |||
+ | <WRAP box round # | ||
+ | Let $\{ \hat{\boldsymbol x}, \hat{\boldsymbol z}\}$ be a basis of $\mathbb R^2$, and | ||
+ | $(R, | ||
+ | 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/ | ||
+ | We denote the vector from the origin to $(R, | ||
+ | Then the following statements hold | ||
+ | - $\hat{\boldsymbol R}(\theta)$ is a normal vector at the position $(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 | ||
</ | </ | ||
- | <WRAP box round> | + | <wrap lo>** Remark 6.2. ** |
+ | The coordinate representation of $\hat{\boldsymbol R}(\theta)$ and $\hat{\boldsymbol | ||
+ | </ | ||
+ | |||
+ | <wrap lo>** Remark 6.3. ** <wrap # | ||
+ | The assertions of [[# | ||
+ | The different reference axis //only// changes the coordinate representation of the vectors. | ||
+ | </ | ||
+ | |||
+ | <WRAP 120pt right # | ||
+ | {{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, | ||
+ | and another angle $\psi$ that describes the position of point that is not on the axis (blue point). | ||
+ | </ | ||
+ | <WRAP box round> | ||
A ping-pong ball consists of $N$ atoms located in the three-dimensional space. | 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. | 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. | 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: | + | Rather than specifying the position of each atom one can therefore specify the position of the ball in terms of six generalized coordinates ([[# |
Three coordinates provide its center of mass. | 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, | 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 | and a third angle specifies the orientation of a point on its equator | ||
when rotating the ball around the axis. | when rotating the ball around the axis. | ||
- | </ | ||
- | |||
- | <WRAP 120pt right # | ||
- | {{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, | ||
- | and another angle $\psi$ that describes the position of point that is not on the axis (blue point). | ||
</ | </ | ||
Line 98: | Line 123: | ||
In all cases we will eventually reduce the dynamics to one-dimensional problems. | In all cases we will eventually reduce the dynamics to one-dimensional problems. | ||
The final [[book: | The final [[book: | ||
+ | |||
+ | |||
+ | ==== 6.1.1 Self Test ==== | ||
+ | |||
+ | ---- | ||
+ | |||
+ | <wrap # | ||
+ | \\ | ||
+ | Verify the assertions of [[# | ||
+ | |||
+ | - Specify the coordinate representation of $\{ \hat{\boldsymbol R}(\theta), \hat{\boldsymbol | ||
+ | - Verify the assertion of [[# | ||
+ | |||
+ | ---- | ||
+ | |||
+ | <wrap # | ||
+ | \\ | ||
+ | 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 [[# | ||
+ | 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~~ | ~~DISCUSSION~~ |
book/chap6/6.1_motivation_and_outline.1643713090.txt.gz · Last modified: 2022/02/01 11:58 by abril