Theoretical Mechanics IPSP

Jürgen Vollmer, Universität Leipzig

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book:chap2:2.8_cartesian_coordinates [2021/11/06 17:01] – created abrilbook:chap2:2.8_cartesian_coordinates [2022/04/01 21:13] (current) – jv
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 +[[forcestorques|2. Balancing Forces and Torques]]
 +  * [[  2.1 Motivation and Outline| 2.1 Motivation and outline: forces are vectors ]]
 +  * [[  2.2 Sets| 2.2 Sets ]]
 +  * [[  2.3 Groups| 2.3 Groups ]]
 +  * [[  2.4 Fields| 2.4 Fields ]]
 +  * [[  2.5 Vector spaces| 2.5 Vector spaces ]]
 +  * [[  2.6 Physics application balancing forces| 2.6.  Physics application: balancing forces]]
 +  * [[  2.7 The inner product | 2.7 The inner product]]
 +  * ** 2.8 Cartesian coordinates **
 +  * [[  2.9 Cross products --- torques| 2.9 Cross products — torques ]]
 +  * [[ 2.10 Worked example Calder's mobiles| 2.10 Worked example: Calder's mobiles ]]
 +  * [[ 2.11 Problems| 2.11 Problems ]]
 +  * [[ 2.12 Further reading| 2.12 Further reading ]]
 +
 +----
 +
 ==== 2.8 Cartesian coordinates ===== ==== 2.8 Cartesian coordinates =====
  
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 <WRAP 120pt left #fig_vector-2DcoordinatesE> <WRAP 120pt left #fig_vector-2DcoordinatesE>
-{{vector-2Dcoordinates_E.png}} Figure 2.14: Representation of the vector $\mathbf c$ in terms of the orthogonal unit vectors $(\mathbf e_1, \mathbf e_2)$.+{{vector-2Dcoordinates_E.png}}  
 + 
 +Figure 2.14: Representation of the vector $\mathbf c$ in terms of the orthogonal unit vectors $(\mathbf e_1, \mathbf e_2)$.
 </WRAP> </WRAP>
  
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 </WRAP> </WRAP>
  
-<WRAP 120pt left #fig_vector-2DcoordinatesAlt> +<WRAP 200pt left #fig_vector-2DcoordinatesAlt> 
-{{vector-2Dcoordinates2.png}} Representation of the vector $\mathbf c$ of [[#fig_vector-2DcoordinatesE |Figure 2.14]] in terms of the bases $(\mathbf e_1, \mathbf e_2)$ and $(\mathbf n_1, \mathbf n_2)$.+{{vector-2Dcoordinates2.png}}  
 + 
 +Representation of the vector $\mathbf c$ of [[#fig_vector-2DcoordinatesE |Figure 2.14]] in terms of the bases $(\mathbf e_1, \mathbf e_2)$ and $(\mathbf n_1, \mathbf n_2)$.
 </WRAP> </WRAP>
  
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 <wrap lo>**Remark 2.22** <wrap lo>**Remark 2.22**
 For a given basis the representation in terms of coordinates is unique. For a given basis the representation in terms of coordinates is unique.
-</wrap>\\+</wrap>
  
 //Proof.// //Proof.//
 1. The coordinates $a_i$ of a vector $\mathbf a$ are explicitly given by $a_i = \langle a \mid \mathbf e_i\rangle$. 1. The coordinates $a_i$ of a vector $\mathbf a$ are explicitly given by $a_i = \langle a \mid \mathbf e_i\rangle$.
-This provides unique numbers for a given basis set.\\+This provides unique numbers for a given basis set.
  
 2. Assume now that two vectors $\mathbf a$ and $\mathbf b$ have the same coordinate representation. 2. Assume now that two vectors $\mathbf a$ and $\mathbf b$ have the same coordinate representation.
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 ==== 2.8.1 Self Test ==== ==== 2.8.1 Self Test ====
  
-<wrap #quest_functions-01>Problem 2.19: ** Cartesian coordinates in the plane ** </wrap>\\+<wrap #quest_functions-01>Problem 2.19: ** Cartesian coordinates in the plane ** </wrap>
  
 a) Mark the following points in a Cartesian coordinate system: a) Mark the following points in a Cartesian coordinate system:
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       (0, \, 0) \quad (0, \, 3) \quad (2, \, 5) \quad (4, \, 3) \quad (4, \,0)       (0, \, 0) \quad (0, \, 3) \quad (2, \, 5) \quad (4, \, 3) \quad (4, \,0)
 \end{align*} \end{align*}
-Add the points $(0, \, 0) \; (4, \, 3) \; (0, \, 3) \; (4, \, 0)$, and connect the points in the given order. What do you see?\\+Add the points $(0, \, 0) \; (4, \, 3) \; (0, \, 3) \; (4, \, 0)$, and connect the points in the given order. What do you see?
  
 b) What do you find when drawing a line segment connecting the following points? b) What do you find when drawing a line segment connecting the following points?
 \begin{align*} \begin{align*}
       (0, \, 0) \quad (1, \, 4) \quad (2, \, 0) \quad (-1, \, 3) \quad (3, \, 3) \quad (0, \, 0)       (0, \, 0) \quad (1, \, 4) \quad (2, \, 0) \quad (-1, \, 3) \quad (3, \, 3) \quad (0, \, 0)
-\end{align*}\\+\end{align*}
  
  
-<wrap #quest_forces-2> Problem 2.20: ** Geometric and algebraic form of the scalar product ** </wrap>\\+<wrap #quest_forces-2> Problem 2.20: ** Geometric and algebraic form of the scalar product ** </wrap>
  
-<WRAP 120pt left>+<WRAP 200pt left>
 {{02_scalarProduct.png}} {{02_scalarProduct.png}}
 </WRAP> </WRAP>
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       \mathbf a \cdot \mathbf b = a \, b \: \cos(\theta_a -\theta_b)       \mathbf a \cdot \mathbf b = a \, b \: \cos(\theta_a -\theta_b)
 \end{align*} \end{align*}
-**Hint: ** Work backwards, expressing $\cos(\theta_a -\theta_b)$ in terms of $\cos\theta_a$, $\cos\theta_b$, $\sin\theta_a$, and $\sin\theta_b$.\\+++ Hint: | $\quad$ Work backwards, expressing $\cos(\theta_a -\theta_b)$ in terms of $\cos\theta_a$, $\cos\theta_b$, $\sin\theta_a$, and $\sin\theta_b$. ++
  
 b) As a shortcut to the explicit calculation of a) one can introduce the coordinates $a_1 = a \, \cos\theta_a$ and $a_2 =  a \, \sin\theta_a$, and write $\mathbf a$ as a tuple of two numbers. Proceeding analogously for $\mathbf b$ one obtains b) As a shortcut to the explicit calculation of a) one can introduce the coordinates $a_1 = a \, \cos\theta_a$ and $a_2 =  a \, \sin\theta_a$, and write $\mathbf a$ as a tuple of two numbers. Proceeding analogously for $\mathbf b$ one obtains
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 \mathbf b = \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} \mathbf b = \begin{pmatrix} b_1 \\ b_2 \end{pmatrix}
 \]    \]   
-How does the product $\mathbf a \cdot \mathbf b$ look like in terms of these coordinates?\\+How does the product $\mathbf a \cdot \mathbf b$ look like in terms of these coordinates?
  
-c) How do the arguments in a) and b) change for $D$ dimensional vectors that are represented as linear combinations of a set of orthonormal basis vectors $\hat{\boldsymbol e}_1, \dots, \hat{\boldsymbol e}_D$?\\+c) How do the arguments in a) and b) change for $D$ dimensional vectors that are represented as linear combinations of a set of orthonormal basis vectors $\hat{\boldsymbol e}_1, \dots, \hat{\boldsymbol e}_D$?
  
-:!: What changes when the basis is not orthonormal? What if it is not even orthogonal?\\+:!: What changes when the basis is not orthonormal? What if it is not even orthogonal?
  
 <wrap #quest_forces-scalarProductR>Problem 2.21: </wrap> <wrap #quest_forces-scalarProductR>Problem 2.21: </wrap>
 ** Scalar product on $\mathbb{R}^D$** \\ ** Scalar product on $\mathbb{R}^D$** \\
-Show that the scalar product on $\mathbb{R}^D$ takes exactly the same form as for the complex case, [[#Thm_ScalarProduct |Theorem 2.3]]. However, complex conjugation is not necessary in that case.\\+Show that the scalar product on $\mathbb{R}^D$ takes exactly the same form as for the complex case, [[#Thm_ScalarProduct |Theorem 2.3]]. However, complex conjugation is not necessary in that case.
  
 <wrap #quest:forces-PaliMatrices>Problem 2.22: </wrap> <wrap #quest:forces-PaliMatrices>Problem 2.22: </wrap>
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                                                     a_{ij} \in \mathbb{C}  \; \land \; a_{ij}=a_{ji}^*                                                     a_{ij} \in \mathbb{C}  \; \land \; a_{ij}=a_{ji}^*
 \end{align*} \end{align*}
-Show to that end:\\+Show to that end:
  
 a) The matrices $\sigma_0$, $\dots$, $\sigma_4$ are linearly independent.\\ a) The matrices $\sigma_0$, $\dots$, $\sigma_4$ are linearly independent.\\
 b) $\displaystyle x_0, \dots, x_4 \in \mathbb{R}  \quad\Rightarrow\quad \sum_{i=0}^4 x_i \, \sigma_i \in \mathbb{H}$\\ b) $\displaystyle x_0, \dots, x_4 \in \mathbb{R}  \quad\Rightarrow\quad \sum_{i=0}^4 x_i \, \sigma_i \in \mathbb{H}$\\
 c) $\displaystyle M  \in \mathbb{H}  \quad\Rightarrow \quad \exists x_0, \dots, x_4 \in \mathbb{R} : M = \sum_{i=0}^4 x_i \, \sigma_i$\\ c) $\displaystyle M  \in \mathbb{H}  \quad\Rightarrow \quad \exists x_0, \dots, x_4 \in \mathbb{R} : M = \sum_{i=0}^4 x_i \, \sigma_i$\\
-:!: What about linear combinations with coefficients $z_1, \dots, z_4 \in \mathbb{C}$? Is $\sum_{i=0}^4 z_i \, \sigma_i$ Hermitian? Do these matrices form a vector space?\\+:!: What about linear combinations with coefficients $z_0, \dots, z_3 \in \mathbb{C}$? Is $\sum_{i=0}^3 z_i \, \sigma_i$ Hermitian? Do these matrices form a vector space?
  
 ~~DISCUSSION|Questions, Remarks, and Suggestions~~ ~~DISCUSSION|Questions, Remarks, and Suggestions~~
 +
book/chap2/2.8_cartesian_coordinates.1636214488.txt.gz · Last modified: 2021/11/06 17:01 by abril