5.6: The equations of motion
- Page ID
- 103450
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)When analysing dynamic systems we often want to determine the motion \(\overrightarrow{\boldsymbol{r}}(t)\) from a known acceleration \(\overrightarrow{\boldsymbol{a}}(t)\), velocity \(\overrightarrow{\boldsymbol{v}}(t)\) or a more complicated differential
equation that involves the position coordinates and their time-derivatives. Such an equation is called an equation of motion (EoM). Often multiple equations are needed to fully determine the time-dependent motion \(\overrightarrow{\boldsymbol{r}}_{i}(t)\) of every point in the system.
The equations of motion of a system are a set of differential equations of the position vectors/coordinates and their time-derivatives, that can be used to determine the motion \(\overrightarrow{\boldsymbol{r}}_{i}(t)\) of every point \(i\) in the system for any initial condition.
We also introduce the concepts initial condition and state:
Initial conditions fully describe the state of a system at a single initial time \(t_{0}\). The state of the system is defined as the combination of the position \(\overrightarrow{\boldsymbol{r}}_{i}\left(t_{0}\right)\) and velocity vectors \(\overrightarrow{\boldsymbol{v}}_{i}\left(t_{0}\right)\) of all points \(i\) in the system at a single time \(t_{0}\).
If the initial conditions, equations of motion and constraint equations are known, the future and history of a system can be calculated by kinematic techniques. As an example of an equation of motion let us consider the situation where we know that the velocity along a path curve is given by the function \(v(t)=c_{1} t^{2}\). Then from the definition of velocity and Equation 5.22 we find the following equation of motion:
\[\frac{\mathrm{d} s}{\mathrm{~d} t}=c_{1} t^{2} \tag{5.26} \label{5.26}\]
Such an equation of motion fixes the slope of the motion \(s(t)\) at every \((s, t)\) coordinate as is graphically shown by the arrows in Figure 5.6. A main challenge in kinematic analysis is to determine the motion \(s(t)\) from the EoMs and initial conditions as illustrated by the red lines in the figure. In this chapter we will discuss EoMs that can be solved by integration. In Ch. 13 on vibrations we will also deal with other solution methods.