63.1: Review of Newtonian Mechanics

We begin by reviewing Newtonian classical mechanics in one dimension. In this formulation, we begin by writing Newton's second law, which gives the force $F$ required to give an acceleration $a$ to a mass $m$ :

$$F=m a .$$

Generally the force is a function of $x$. Since the acceleration $a=d^{2} x / d t^{2}$, Eq. $\PageIndex{1}$ may be written

$$F(x)=m \frac{d^{2} x}{d t^{2}}$$

This is a second-order ordinary differential equation, which we solve for $x(t)$ to find the position $x$ at any time $t$. Solving a problem in Newtonian mechanics then consists of these steps:

• Write down Newton’s second law (Eq. $\PageIndex{2}$);
• Substitute for $F(x)$ the specific force present in the problem;
• Solve the resulting differential equation for $x(t)$.