Newtons Laws of Motion

I

In an intertial frame of reference (i.e. a frame of reference undergoing zero acceleration), an object at rest stays at rest (or, similarly, keeps its initial velocity) unless acted upon by an outside force.

$$\begin{equation} \sum \mathbf{F} = 0 \Leftrightarrow \frac{d\mathbf{v}}{dt} = 0 \end{equation}$$

II

In the simplest of terms:

$$\begin{equation} \mathbf{F} = m\mathbf{a} \end{equation}$$

which states that

The vector sum of the forces on an object is equal to that objects mass times the acceleration

where $m$ is assumed as a constant.

In terms of infinitesimal change, the same law is seen as

$$\begin{equation} \mathbf{F} = m\frac{d\mathbf{v}}{dt} = m\mathbf{a} \end{equation}$$

III

When one body exerts a force on another body, the second body exerts an equal force in the opposing direction

$$\begin{equation} F_{a} = -F_{b} \end{equation}$$