$$\require{cancel}$$

# 10.1: Example- Equations of Motion in Classical Mechanics

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

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.

10.1: Example- Equations of Motion in Classical Mechanics is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Y. D. Chong via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.