10.1: Example- Equations of Motion in Classical Mechanics
- Page ID
- 34857
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The above standard formulation of the initial-value problem can be used to describe a very large class of time-dependent ODEs found in physics. For example, suppose we have a classical mechanical particle with position \(\vec{r}\), subject to an arbitrary external space-and-time-dependent force \(\vec{f}(\vec{r},t)\) and a friction force \(-\lambda d\vec{r}/dt\) (where \(\lambda\) is a damping coefficient). Newton's second law gives the following equation of motion:
\[m \frac{d^2 \vec{r}}{dt^2} = - \lambda \frac{d\vec{r}}{dt} + \vec{f}(\vec{r}, t).\]
This is a second-order ODE, whereas the standard initial-value problem involves a first-order ODE. However, we can turn it into a first-order ODE with the following trick. Define the velocity vector
\[\vec{v} = \frac{d\vec{r}}{dt},\]
and define the state vector by combining the position and velocity vectors:
\[\vec{y} = \begin{bmatrix}\vec{r} \\ \vec{v}\end{bmatrix}.\]
Then the equation of motion takes the form
\[\frac{d\vec{y}}{dt} = \frac{d}{dt}\begin{bmatrix}\vec{r} \\ \vec{v}\end{bmatrix} = \begin{bmatrix}\vec{v} \\ - (\lambda/m) \vec{v} + \vec{f}(\vec{r}, t)/m\end{bmatrix},\]
which is a first-order ODE, as desired. The quantity on the right-hand side is the derivative function \(\vec{F}(\vec{y},t)\) for the initial-value problem. Its dependence on \(\vec{r}\) and \(\vec{v}\) is simply regarded as a dependence on the upper and lower portions of the state vector \(\vec{y}\). In particular, note that the derivative function does not need to be linear, since \(\vec{f}\) can have any arbitrary nonlinear dependence on \(\vec{r}\), e.g. it could depend on the quantity \(|\vec{r}|\).
The "initial state", \(\vec{y}(t_0)\), requires us to specify both the initial position and velocity of the particle, which is consistent with the fact that the original equation of motion was a second-order equation, requiring two sets of initial values to fully specify a solution. In a similar manner, ODEs of higher order can be converted into first-order form, by defining the higher derivatives as state variables and increasing the size of the state vector.