Theoretical Mechanics IPSP

Jürgen Vollmer, Universität Leipzig

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book:chap5:5.1_motivation_and_outline [2022/01/04 02:59] – [5.1 Motivation and outline: How do particles collide?] abrilbook:chap5:5.1_motivation_and_outline [2022/01/04 03:15] (current) – abril
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-FIXME draft with missing figures and references :!: 
- 
 ===== 5.1  Motivation and outline: How do particles collide?  ===== ===== 5.1  Motivation and outline: How do particles collide?  =====
  
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 </WRAP> </WRAP>
  
-<WRAP 120pt right #fig_CoulombScattering> +<WRAP 120pt left #fig_CoulombScattering> 
-{{EOM_CoulombScattering_phaseSpace.png}} +{{EOM_CoulombScattering.png}}
-{{EOM_CoulombScattering_TrajectoryShape.png}}+
  
 Figure 5.4: Phase-space flow and the shape of trajectories for scattering with a repulsive Coulomb potential. Figure 5.4: Phase-space flow and the shape of trajectories for scattering with a repulsive Coulomb potential.
 </WRAP> </WRAP>
  
-The phase-space portrait and the shape of the orbits for repulsive interactions are plotted in [[#fig_CoulombScattering|Fig 5.4]].+The phase-space portrait and the shape of the orbits for repulsive interactions are plotted in [[#fig_CoulombScattering|Figure 5.4]].
 We observe that the trajectory shape describes the approach of the other particle from a perspective of an observer We observe that the trajectory shape describes the approach of the other particle from a perspective of an observer
 that sits on a particle located in the origin. that sits on a particle located in the origin.
 When the observer sits on a particle that has a much larger mass than the approaching particle,  When the observer sits on a particle that has a much larger mass than the approaching particle, 
-then an outside observer will see virtually no motion of the mass-rich particle and the lines in [[#fig_CoulombScattering|Fig 5.4]] describe the lines of the trajectories of the light particle in  a plane selected by the initial angular momentum of the scattering problem.+then an outside observer will see virtually no motion of the mass-rich particle and the lines in [[#fig_CoulombScattering|Figure 5.4]] describe the lines of the trajectories of the light particle in  a plane selected by the initial angular momentum of the scattering problem.
 In general, two particles of masses $m_1$ and $m_2$ will be at opposite sides of the center of mass. In general, two particles of masses $m_1$ and $m_2$ will be at opposite sides of the center of mass.
-In  a coordinate system with its origin at the center of mass the lines in [[#fig_CoulombScattering|Fig 5.4]]+In  a coordinate system with its origin at the center of mass the lines in [[#fig_CoulombScattering|Figure 5.4]]
 describe the particle trajectories up to describe the particle trajectories up to
 factors $m_1/(m_1+m_2)$ factors $m_1/(m_1+m_2)$
 and $-m_2/(m_1+m_2)$ for the first and second particle, respectively. and $-m_2/(m_1+m_2)$ for the first and second particle, respectively.
-A pair of trajectories for $m_1 = 0.3  \, (m_1+m_2)$ and $\epsilon=1.2$ is shown in [[#fig_CoulombTrajectory|Fig 5.5]].+A pair of trajectories for $m_1 = 0.3  \, (m_1+m_2)$ and $\epsilon=1.2$ is shown in [[#fig_CoulombTrajectory|Figure 5.5]].
 The approximation as point particles is well justified when the sum of the particle radii is much smaller than their closest approach $R_0 / ( \epsilon - 1)$. The approximation as point particles is well justified when the sum of the particle radii is much smaller than their closest approach $R_0 / ( \epsilon - 1)$.
  
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 ==== 5.1.1 Self Test ==== ==== 5.1.1 Self Test ====
  
-<WRAP 120pt right #fig_CoulombScatteringAngle>+<WRAP 120pt left #fig_CoulombScatteringAngle>
 {{EOM_CoulombScattering_ScatteringAngle.png}} {{EOM_CoulombScattering_ScatteringAngle.png}}
  
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 For the choice of coordinates adopted in  For the choice of coordinates adopted in 
-[[#fig_CoulombScattering|Fig 5.4]] and [[#fig_CoulombTrajectory|Fig 5.5]]+[[#fig_CoulombScattering|Figure 5.4]] and [[#fig_CoulombTrajectory|Figure 5.5]]
 the trajectories have an asymptotic angle $\theta$ with the $\hat x$-axis when they approach each other the trajectories have an asymptotic angle $\theta$ with the $\hat x$-axis when they approach each other
 and they separate with an asymptotic angle$-\theta$. and they separate with an asymptotic angle$-\theta$.
  
 **a)** Show that **a)** Show that
-\begin{align} \label{eq:CoulombScatteringAngle} +<wrap #eq_CoulombScatteringAngle></wrap> 
-      \tan^2\theta = \frac{ 2\, E\, L^2 }{\mu \, C^2}+\begin{align} 
 +      \tan^2\theta = \frac{ 2\, E\, L^2 }{\mu \, C^2} \tag{5.1.1}
 \end{align} \end{align}
  
book/chap5/5.1_motivation_and_outline.1641261543.txt.gz · Last modified: 2022/01/04 02:59 by abril