Theoretical Mechanics IPSP

Jürgen Vollmer, Universität Leipzig

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book:chap4:4.5_linear_odes [2021/12/07 18:52] – abrilbook:chap4:4.5_linear_odes [2024/02/01 00:50] (current) – polish English jv
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-===== 4.5  Linear ODEs --- Particle suspended from a spring  =====+===== 4.5  Linear ODEs — Particle suspended from a spring  =====
  
 There are two forces acting on a particle is suspended from a spring: There are two forces acting on a particle is suspended from a spring:
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 $t^2 \, \exp(\kappa \,t)$ for triple roots, etc. $t^2 \, \exp(\kappa \,t)$ for triple roots, etc.
 In this course we only deal with second order ODEs, where at most double roots arise. In this course we only deal with second order ODEs, where at most double roots arise.
-The solution strategy for that case will be discussed in [[#453 |Section 4.5.3]]+The solution strategy for that case will be discussed in [[#section_453 |Section 4.5.3]]
 </wrap> </wrap>
  
-<wrap #452></wrap>+<wrap #section_452></wrap>
 ==== 4.5.2  Solving the ODE for the mass suspended from a spring  ==== ==== 4.5.2  Solving the ODE for the mass suspended from a spring  ====
  
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-<wrap #453></wrap>+<wrap #section_453></wrap>
 ==== 4.5.3  Solution for the damped harmonic oscillator  ==== ==== 4.5.3  Solution for the damped harmonic oscillator  ====
  
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 a pair of complex conjugated numbers, or a pair of complex conjugated numbers, or
 we have to deal with the case $\gamma^2=4\,k/m$ where there only is a single root. we have to deal with the case $\gamma^2=4\,k/m$ where there only is a single root.
-We treat the cases one after the other.\\+We treat the cases one after the other.
  
 **1.  Two real roots**\\ **1.  Two real roots**\\
-In this case $\gamma^2 < 4\,k/m$ such that $\lambda_\pm \in \mathbb{R}_-$.  +In this case $\gamma^2 > 4\,k/m$, and $\lambda_\pm \in \mathbb{R}_-$.  
-The motion of the oscillator is described by+Hence, the motion of the oscillator is described by
 \begin{align*} \begin{align*}
   x(t) =  A_+ \; \mathrm{e}^{ \lambda_+ \, (t-t_0) } + A_- \; \mathrm{e}^{ \lambda_- \, (t-t_0) }   x(t) =  A_+ \; \mathrm{e}^{ \lambda_+ \, (t-t_0) } + A_- \; \mathrm{e}^{ \lambda_- \, (t-t_0) }
 \end{align*} \end{align*}
 which is a real-valued function for amplitudes $A_\pm \in \mathbb{R}$. which is a real-valued function for amplitudes $A_\pm \in \mathbb{R}$.
-The solution for the initial conditions $x(t_0) = x_0$ and $\dot x(t_0)=v_0$ +Moreover, the solution for the initial conditions $x(t_0) = x_0$ and $\dot x(t_0)=v_0$ 
-is then found by solving the equations+is found by solving the equations
 \begin{align*} \begin{align*}
   \left .   \left .
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 **2.  Two complex roots**\\ **2.  Two complex roots**\\
-This discussion is analogous to the one provided in [[#452 |Section 4.5.2]].+This discussion is analogous to the one provided in [[#section_452 |Section 4.5.2]].
 One obtains One obtains
 <wrap #eq_EOMharmonicOscillator_dampedOscillations></wrap> <wrap #eq_EOMharmonicOscillator_dampedOscillations></wrap>
book/chap4/4.5_linear_odes.1638899561.txt.gz · Last modified: 2021/12/07 18:52 by abril