Newtonian mechanics and equations of motion

Dealing with second order derivatives

The apparent problem with the equations of motion

\[ \begin{eqnarray*} m \ddot{x}&=&F_x(x,\,y,\,v_x,\,v_y,\,t) \\ m \ddot{y}&=&F_y(x,\,y,\,v_x,\,v_y,\,t) \end{eqnarray*} \] is that they involve second order derivatives which makes them unsuitable for the Euler method discussed in the previous lecture.

There is, however, a simple trick to deal with this difficulty. This trick essentially consists of using not only the positions x and y as relevant variables, but to include also the corresponding velocities $v_x$ and $v_y$, respectively. This amounts to replacing the two differential equations of second order above with four first-order differential equations:

\[ \begin{eqnarray*} \begin{array}{lclcl} \displaystyle \dot{x}\, &=&\, \frac{\mathrm{d}x}{\mathrm{d}t}\, &=&\,v_x \\ \dot{y}\, &=&\, \frac{\mathrm{d}y}{\mathrm{d}t}\, &=&\,v_y \\ \dot{v}_x\,&=&\, \frac{\mathrm{d}v_x}{\mathrm{d}t}\,&=& \,\frac{F_x}{m} \\ \dot{v}_y\,&=&\, \frac{\mathrm{d}v_y}{\mathrm{d}t}\,&=& \,\frac{F_y}{m}\,. \end{array}\end{eqnarray*} \] Ultimately, this allows us to apply Euler's method to each of the four equations, which are then coupled by using the positions and velocities at time steps $t_i$ - i.e. $x_i$, $v_{x,i}$ etc. - to calculate new positions and velocities at times $t_{i+1}$, namely $x_{i+1}$, $v_{x,i+1}$ etc..

Frank Krauss and Daniel Maitre

Last modified: Tue Oct 3 14:43:58 BST 2017