Theoretical Mechanics IPSP

Jürgen Vollmer, Universität Leipzig

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book:chap5:5.1_motivation_and_outline [2022/01/03 14:28] – jvbook: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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   R_0 = \frac{L^2}{\mu \, C}   R_0 = \frac{L^2}{\mu \, C}
 \end{align*} \end{align*}
-agrees with\cref{eq:Kepler1} up to a change of the sign of the one in the denominator and the length unit $R_0$. +agrees with [[book:chap4:4.9_solving_odes_by_coordinate_transformations #eq_Kepler1|Equation 4.9.3]] up to a change of the sign of the one in the denominator and the length unit $R_0$. 
-<wrap lo>** Remark 5.1. **  + 
 +<WRAP lo>** Remark 5.1. **  
 It is illuminating to adopt a different perspective on the origin of the minus sign in front of the one. It is illuminating to adopt a different perspective on the origin of the minus sign in front of the one.
 Let us write the force on particle $1$ as $\mathbf F_1 = F_1 \: \hat{\boldsymbol e}(\theta)$ Let us write the force on particle $1$ as $\mathbf F_1 = F_1 \: \hat{\boldsymbol e}(\theta)$
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 negative for a repulsive force.  negative for a repulsive force. 
 In the dimensionless force $F t_0^2 / \mu\, R_0$ the change of sign is taken into account by the sign of $C$ in In the dimensionless force $F t_0^2 / \mu\, R_0$ the change of sign is taken into account by the sign of $C$ in
-$R_0 = L^2 / \mu \, C$ and the solution takes the form of \cref{eq:Kepler1}.+$R_0 = L^2 / \mu \, C$ and the solution takes the form of [[book:chap4:4.9_solving_odes_by_coordinate_transformations #eq_Kepler1|Equation 4.9.3]].
 In order to obtain a positive length scale $\left\lvert R_0 \right\rvert = \pm R_0$ In order to obtain a positive length scale $\left\lvert R_0 \right\rvert = \pm R_0$
 we multiply the numerator and denominator of the solution by the $\pm 1$ we multiply the numerator and denominator of the solution by the $\pm 1$
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 and mapping of parameters to a known problem, and mapping of parameters to a known problem,
 rather than going again through the involved analysis. rather than going again through the involved analysis.
- \manimpossiblecube +</WRAP>
-</wrap>+
  
-<WRAP right id=fig_CoulombScattering> +<WRAP 120pt left #fig_CoulombScattering> 
-{{./Sage/EOM_CoulombScattering__phaseSpace.png}} +{{EOM_CoulombScattering.png}}
-{{./Sage/EOM_CoulombScattering__TrajectoryShape.png}}+
  
-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 XX]].+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 XX]] 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 XX]]+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 [[#CoulombTrajectory|Fig XY]].+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 $R_0 / ( \epsilon - 1)$.
  
-<WRAP right id=fig_CoulombTrajectory> +<WRAP 120pt right #fig_CoulombTrajectory> 
-{{./Sage/EOM_CoulombScattering__ScatteringTrajectory.png}}+{{EOM_CoulombScattering_ScatteringTrajectory.png}}
  
-The two black lines show the scattering trajectories of two particles with $\epsilon=1.2$ and relative mass $m_1 = 0.3 \, (m_1+m_2)$. +Figure 5.5: The two black lines show the scattering trajectories of two particles with $\epsilon=1.2$ and relative mass $m_1 = 0.3 \, (m_1+m_2)$. 
 They approach each other along the solid gray line and separate along the dotted line. They approach each other along the solid gray line and separate along the dotted line.
 Particle $1$ is initially at the top right. Particle $1$ is initially at the top right.
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 ==== Outline ==== ==== Outline ====
  
-In\cref{section:particleScattering} we study the collision of spherical hard-ball particles+In [[book:chap5:5.2_collisions_of_particles|Section 5.2]] we study the collision of spherical hard-ball particles
 that only interact by a force kick vertical to the surfaces at their contact point when they touch. that only interact by a force kick vertical to the surfaces at their contact point when they touch.
 Then we compare the Coulomb case and the force-kick case in order to explore which features of the outgoing trajectories are provided by conservation laws, Then we compare the Coulomb case and the force-kick case in order to explore which features of the outgoing trajectories are provided by conservation laws,
 irrespective of the type of interaction.  irrespective of the type of interaction. 
-In\cref{section:VolumeMassDensity} we discuss the forces+In [[book:chap5:5.3_volume_integrals|Section 5.3]] we discuss the forces
 of an extended object (Earth) on a point particle moving without further interactions in its gravitational field.  of an extended object (Earth) on a point particle moving without further interactions in its gravitational field. 
-In\cref{section:particleExtension} we further explore the impact spatial extension+In [[book:chap5:5.4_center_of_mass_and_spin|Section 5.4]] we further explore the impact spatial extension
 of solid particles: of solid particles:
 How does their shape matter? How does their shape matter?
-How are particles set into spinning motion, and how does the spin evolve? +How are particles set into spinning motion, and how does the spin evolve? [[book:chap5:5.5_bodies_with_internal_degrees_of_freedom|Section 5.5]] addresses the motion of particles with internal degrees of freedom.  
-\cref{section:internalDOF} addresses the motion of particles with internal degrees of freedom.  +Finally, in [[book:chap5:5.6_worked_example|Section 5.6]] we wrap up the findings of this section
-Finally, in \cref{section:workedExample-ballReflections} we wrap up the findings of this section+
 by discussing the reflections of balls: by discussing the reflections of balls:
 How do balls pick up spin in collisions? How do balls pick up spin in collisions?
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 ==== 5.1.1 Self Test ==== ==== 5.1.1 Self Test ====
  
-<WRAP right id=fig_CoulombScatteringAngle> +<WRAP 120pt left #fig_CoulombScatteringAngle> 
-{{./Sage/EOM_CoulombScattering__ScatteringAngle.png}}+{{EOM_CoulombScattering_ScatteringAngle.png}}
  
-Scattering angle $\theta$ for a collision of two particles that interact by a repulsive Coulomb +Figure 5.6: Scattering angle $\theta$ for a collision of two particles that interact by a repulsive Coulomb 
 </WRAP> </WRAP>
- 
-----  
  
 <wrap #quest_ColoumbScatteringAngle > Problem 5.1: </wrap>** Scattering angle for the Coulomb potential ** <wrap #quest_ColoumbScatteringAngle > Problem 5.1: </wrap>** Scattering angle for the Coulomb potential **
  
 For the choice of coordinates adopted in  For the choice of coordinates adopted in 
-[[#fig_CoulombScattering|Fig AA]] and [[#fig_CoulombTrajectory|Fig AB]]+[[#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$-\theta$.
  
-  -  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}
-  -  The parameter dependence of the scattering angle $\theta$ is shown in [[#fig_CoulombScatteringAngle|Fig ??]]. + 
-What happens to the line for very large values of $ 2\, E\, L^2  / \mu \, C^2$? +**b)** The parameter dependence of the scattering angle $\theta$ is shown in [[#fig_CoulombScatteringAngle|Figure 5.6]]. 
-  -  How would the scattering trajectories in [[#fig_CoulombTrajectory|Fig ??]] look like for $\theta = \pi/2$? +What happens to the line for very large values of $ 2\, E\, L^2  / \mu \, C^2$?\\ 
-Does this comply with your finding in b)?+ 
 +**c)** How would the scattering trajectories in [[#fig_CoulombTrajectory|Figure 5.5]] look like for $\theta = \pi/2$? Does this comply with your finding in b)?
  
  
 ~~DISCUSSION~~  ~~DISCUSSION~~ 
  
book/chap5/5.1_motivation_and_outline.1641216526.txt.gz · Last modified: 2022/01/03 14:28 by jv