Theoretical Mechanics IPSP

Jürgen Vollmer, Universität Leipzig

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book:chap3:3.4_constants_of_motion_cm

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book:chap3:3.4_constants_of_motion_cm [2024/12/11 15:07] – jvbook:chap3:3.4_constants_of_motion_cm [2024/12/16 15:32] (current) – jv
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 \end{align*} \end{align*}
 The last equality should be understood here as a definition of the final expression that is interpreted here in the spirit of the substitution rule of integration. The last equality should be understood here as a definition of the final expression that is interpreted here in the spirit of the substitution rule of integration.
 +Moreover, $\mathbf F (t)$ denotes here the force acting on the particle at time $t$, where the particle is at the position $\mathbf q (t)$ and moving with velocity $\dot{\mathbf q} (t)$, 
 +i.e., one may also have $\mathbf F \bigl( \mathbf q(t) \bigr)$, or $\mathbf F \bigl( \mathbf q(t), \dot{\mathbf q} (t) \bigr)$, or an explicit time dependence on top of the dependence on the particle position.
 +
  
 <WRAP box round>**Definition 3.5** <wrap em>Work and Line Integrals</wrap> \\ <WRAP box round>**Definition 3.5** <wrap em>Work and Line Integrals</wrap> \\
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 = \int P(t) \; \mathrm{d} t = \int P(t) \; \mathrm{d} t
 \] \]
-Here, $\mathbf F (t)$ denotes the force acting on the particle at time $t$, where the particle is at the position $\mathbf q (t)$ and moving with velocity $\dot{\mathbf q} (t)$,  
-i.e., one may also have $\mathbf F \bigl( \mathbf q(t) \bigr)$,  $\mathbf F \bigl( \mathbf q(t), \dot{\mathbf q} (t) \bigr)$, and/or an explicit time dependence on top of the dependence on the particle position. 
 </WRAP> </WRAP>
- +
 <wrap lo>** Remark 3.5. ** <wrap lo>** Remark 3.5. **
 +Here, $\mathbf F (t)$ denotes the force that is acting on the particle at time $t$, irrespective of how it emerges. Specifically, the time dependence of $\mathbf F (t)$ accounts for changes of the position $\mathbf q(t)$, the velocity $\dot{\mathbf q}(t)$, and any additional time dependences due to external incluences (if applicable).
 +</wrap>
 +
 +<wrap lo>** Remark 3.6. **
 The scalar product $\mathbf F \cdot \mathrm{d}\mathbf q$ or $P(t) = \mathbf F (t) \cdot \dot{\mathbf q} (t)$ The scalar product $\mathbf F \cdot \mathrm{d}\mathbf q$ or $P(t) = \mathbf F (t) \cdot \dot{\mathbf q} (t)$
 singles out only the action of the force parallel to the trajectory. singles out only the action of the force parallel to the trajectory.
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 </wrap> </wrap>
  
-<wrap lo #remark_line-integral_length-parameterization>** Remark 3.6. **+<wrap lo #remark_line-integral_length-parameterization>** Remark 3.7. **
 The result of the integral does not rely on the parameterization of the path by time. The result of the integral does not rely on the parameterization of the path by time.
 For instance mathematicians prefer to use the length $\ell$ of the path. For instance mathematicians prefer to use the length $\ell$ of the path.
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 </wrap> </wrap>
  
-<wrap lo #remark_length-of-lines>** Remark 3.7. **+<wrap lo #remark_length-of-lines>** Remark 3.8. **
   Line integrals are also used to determine the length, $L$, of a path in space.   Line integrals are also used to determine the length, $L$, of a path in space.
   After all, the length amounts to the time integral of the speed, $\dot\ell(t)$,   After all, the length amounts to the time integral of the speed, $\dot\ell(t)$,
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 </WRAP> </WRAP>
  
-<wrap lo>** Remark 3.8. **+<wrap lo>** Remark 3.9. **
 Conservative forces only depend on position, $\mathbf F = \mathbf F (\mathbf q)$. Conservative forces only depend on position, $\mathbf F = \mathbf F (\mathbf q)$.
 They neither explicitly depend on time nor on the velocity $\mathbf q$. They neither explicitly depend on time nor on the velocity $\mathbf q$.
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 **qed** **qed**
    
-<wrap lo>** Remark 3.9. **+<wrap lo>** Remark 3.10. **
 The work performed along a closed path vanishes for conservative forces. The work performed along a closed path vanishes for conservative forces.
 After all, in that case $\mathbf q_1 = \mathbf q_0$ such that $W = \Phi (\mathbf q_0) - \Phi( \mathbf q_1) = 0$. After all, in that case $\mathbf q_1 = \mathbf q_0$ such that $W = \Phi (\mathbf q_0) - \Phi( \mathbf q_1) = 0$.
 </wrap> </wrap>
  
-<wrap lo>** Remark 3.10. **+<wrap lo>** Remark 3.11. **
 The potential in itself is not an observable. ((An //observable// is a quantity that can be measured by direct observation.)) The potential in itself is not an observable. ((An //observable// is a quantity that can be measured by direct observation.))
 One can only observe the work, which is the potential difference between two positions, One can only observe the work, which is the potential difference between two positions,
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 </WRAP> </WRAP>
  
-<wrap lo>** Remark 3.11. **+<wrap lo>** Remark 3.12. **
 According to [[#Thm_work |Theorem 3.3]] differences of the value of the potential between two positions According to [[#Thm_work |Theorem 3.3]] differences of the value of the potential between two positions
 amount to the work performed in the potential. amount to the work performed in the potential.
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 </wrap> </wrap>
  
-<WRAP lo>** Remark 3.12. **+<WRAP lo>** Remark 3.13. **
 One can make use of the properties of scalar products to reduce the computational work One can make use of the properties of scalar products to reduce the computational work
 to determine the force for a given potential to determine the force for a given potential
book/chap3/3.4_constants_of_motion_cm.1733926041.txt.gz · Last modified: 2024/12/11 15:07 by jv