Newton's Laws

All of you have previously encountered Newton's laws of motion. These laws form the basis of the part of mechanics which is known as dynamics. They are as follows:

1. A body does not change its state of rest or uniform straight-line motion unless it is compelled by some force to change that state.
2. The change of motion is proportional to the force and takes place in the same direction as the force.
3. Action is always contrary and equal to reaction.

The first and second laws of motion are usually combined in a single vector equation:

$${\bf F}=m{\bf a}=m\frac{d{\bf v}}{dt},$$ (1)

where ${\bf F}$ is the force, the scalar $m$ is the mass of the body, and ${\bf a}$ its acceleration. This simplified version assumes tacitly that the mass is a constant. Often, in our daily life experiences, this is a valid assumption or at least a good approximation. But even, when you press the accelerator in a car the mass of the car changes due to the burning of fuel and rubber. However, this change is insignificant compared with the mass of the vehicle and can be neglected.

Accelerating rockets is always accompanied by huge changes of the mass of the rocket itself. Furthermore, since space travel is often done at very high velocities, relativity effects can play a role, albeit a very minor one in the context of this class. To account for this we use the following form of Newton's law:

$${\bf F}=\frac{d{m\bf v}}{dt},$$ (2)

where ${\bf v}$ is the velocity of the body. It is easy to see that (1.2) implies (1.1) for a constant mass.

Newton also published his law of gravity

$${\bf F}=\frac{GMm}{\vert{\bf r}\vert^3}{\bf r},$$ (3)

where ${\bf F}$ is the attracting force between the two bodies of mass $M$ and mass $m$ and ${\bf r}$ is a vector pointing from one body to the other. The constant $G$ is the Gravitational constant which is measured at

$$G=6.672\times 10^{-11} {\rm m}^3/{\rm kg} {\rm s}^2.$$

If the bigger of the two bodies were fixed at the origin of a coordinate system, and the mass $m$ of the smaller body were fixed, we could describe the motion of the smaller body by

$$m\frac{d{\bf v}}{dt} = -\frac{GMm}{\vert{\bf r}\vert^3}{\bf r}$$ (4)

$$\frac{d{\bf r}}{dt} = {\bf v}$$ (5)

a coupled system of two first order ordinary differential equations. Since ${\bf r}$ and ${\bf v}$ are vectors, it is actually a coupled system of six ordinary differential equations. Given initial conditions ${\bf r_0}$ and ${\bf v_0}$ we can describe all future positions of the body of mass $m$. This tells us that we usually need seven variables to describe the motion completely, time, three spatial components and three velocity components.