2.2: Newton's Laws of motion

Newton defined a vector quantity called linear momentum \(\mathbf{p}\) which is the product of mass and velocity.

\begin{equation}\label{eq:2.1}
    \mathbf{p} = m\dot{\mathbf{r}}
\end{equation}

Since the mass m is a scalar quantity, then the velocity vector \(\dot{r}\) and the linear momentum vector \(\mathbf{p}\) are colinear.

Newton’s laws, expressed in terms of linear momentum, are:

1. Law of inertia: A body remains at rest or in uniform motion unless acted upon by a force.
2. Equation of motion: A body acted upon by a force moves in such a manner that the time rate of change of momentum equals the force.\begin{equation}\label{eq:2.2}\mathbf{F} = \frac{d\mathbf{p}}{dt}\end{equation}
3. Action and reaction: If two bodies exert forces on each other these forces are equal in magnitude and opposite in direction.

Newton’s second law contains the essential physics relating the force \(\mathbf{F}\) and the rate of change of linear momentum \(\mathbf{p}\).

Newton’s first law, the law of inertia, is a special case of Newton’s second law in that if

\begin{equation}\label{eq:2.3}\tag{2.3}\mathbf{F}=\frac{d\mathbf{p}}{dt}=0\end{equation}

then \(\mathbf{p}\) is a constant of motion. Newton’s third law also can be interpreted as a statement of the conservation of momentum, that is, for a two particle system with no external forces acting,

\begin{equation}\label{eq:2.4} F_{12} = -F_{21}\end{equation}

If the forces acting on two bodies are their mutual action and reaction, then Equation \ref{eq:2.4} simplifies to

\begin{equation}\label{eq:2.5}
F_{12}=-F_{21}=
\frac{d\mathbf{p_1}}{dt}+
\frac{d\mathbf{p_2}}{dt} = \frac{d}{dt}(\mathbf{p_1+p_2})
\end{equation}

This implies that the total linear momentum \(\mathbf{P = p_1 + p_2}\) is a constant of motion. Combining Equations \ref{eq:2.1} and \ref{eq:2.2} leads to a second-order differential equation

\begin{equation}\label{2.6}
\mathbf{F}=\frac{d\mathbf{p}}{dt}=m\frac{d^2\mathbf{r}}{dt^2}=m\mathbf{\ddot{r}}
\end{equation}

Note that the force on a body \(\mathbf{F}\), and the resultant acceleration \(a = \ddot{r}\) are colinear. Appendix C2 gives explicit expressions for the acceleration a in cartesian and curvilinear coordinate systems. The definition of force depends on the definition of the mass m. Newton’s laws of motion are obeyed to a high precision for velocities much less than the velocity of light. For example, recent experiments have shown they are obeyed with an error in the acceleration of \(\Delta a \leq 5 \times 10^{-14}\mathit{m/s^2}\)