Table of Contents
3.3 Newton's axioms and equations of motion (EOM)
In Section 3.1 we referred to a train compartment to point out that physical observations will be the same — irrespective of the velocity of its motion, as long as it is constant. A setting where we perform an experiment is denoted as reference frame, and reference frames that move with constant velocity are called inertial systems.
Definition 3.2 Reference Frames and Inertial Systems
A reference frame $(\mathbf Q, \{\hat{\boldsymbol e}_i(t), \; i=1\cdots D\})$ is an agreement about the, in general time dependent, position of the origin $\mathbf Q(t)$ of the coordinate system and a set of orthonormal basis vectors $\{\hat{\boldsymbol e}_i(t), \; i=1\cdots D\}$, that are adopted to indicate the positions of particles in a physical model. The reference frame refers to an inertial system when it does not rotate and when it moves with a constant velocity, i.e., if and only if $\ddot{\mathbf Q} = \mathbf 0$ and $\dot{\hat{\boldsymbol e}}_i = \mathbf 0$ for all $i\in\{1\cdots D\}$.
Remark 3.1. The requirement $\dot{\hat{\boldsymbol e}}_i = \mathbf 0$ implies that the orientation of the basis vectors ${\hat{\boldsymbol e}}_i$ does not change, i.e. the reference frame does not rotate.
Remark 3.2. The rest frame for a particle is a reference frame where the particle velocity takes the constant velocity $\mathbf 0$.
Remark 3.3.
Let $\mathbf q = (q_1, \dots q_D)$ be the coordinates of a particles,
as specified in in the inertial frame $(\mathbf Q, \{ \hat{\boldsymbol e}_i \})$,
and $\mathbf x = (x_1, \dots x_D)$ its position
given in the inertial frame $(\mathbf X, \{ \hat{\boldsymbol n}_i \})$.
Then
\begin{align*}
\mathbf q = \mathbf Q + \sum_{i=1}^D q_i \, \hat{\boldsymbol e}_i
= \mathbf X + \sum_{i=1}^D x_i \, \hat{\boldsymbol n}_i \, .
\end{align*}
Figure 3.1: Graphical illustration of the description of a position from the perspective of two different reference frames,
$\mathbf q = \mathbf Q + 3 \, \hat{\boldsymbol e}_1 + 3 \, \hat{\boldsymbol e}_2$
$= \mathbf X + 4 \, \hat{\boldsymbol n}_1 - \hat{\boldsymbol n}_2$
with the notations of Remark 3.3.
3.3.1 1st Law
As long as a reference frame moves with a constant velocity, it feels like at rest. Physical measurements can only detect acceleration. This is expressed by
Axiom 3.1: Newton's 1st law
The velocity of a particle moving in an inertial system is constant, unless a (net) force is acting on the particle, \begin{align*} \forall t \geq t_0 : \mathbf F (t) = \mathbf 0 &\quad \Leftrightarrow \quad \dot{\mathbf q}(t) = \mathbf v = \text{const} \\ &\quad \Leftrightarrow \quad \mathbf q (t) = \mathbf q_0 + \mathbf v \: ( t - t_0 ) \\ \end{align*} as sketched in the margin.
The particle moves then in a straight line with a constant speed. Indeed, when a particle moves with the constant velocity $\mathbf v = \dot{\mathbf q}(t)$ in the reference frame $(\mathbf Q_1, \{\hat{\boldsymbol e}_i(t), i=1\cdots D\}$ then it is at rest in the alternative reference frame $(\mathbf Q_2, \{\hat{\boldsymbol e}_i(t), i=1\cdots D\}$ where $\mathbf Q_2 = \mathbf Q_1 + \mathbf v \: t$. Therefore, in the latter coordinate system the particle is at rest, and it will remain at rest when it is not perturbed by a net external force. After all, \begin{align*} \mathbf q = \mathbf Q_1 + \mathbf v \, t = \mathbf Q_2 + \mathbf 0 \, . \end{align*}
3.3.2 2nd Law
Newton's second law spells out how the velocity of the particle changes when there is a force.
Axiom 3.2: Newton's 2nd law
The change, $\ddot{\mathbf q}(t)$, of the velocity of a particle, $\dot{\mathbf q}(t)$, at position, $\mathbf q (t)$, is proportional to the sum of the forces $\mathbf F_\alpha$ acting on the particle, and the proportionality factor is the particle mass $m$, \begin{align*} m \; \ddot{\mathbf q}(t) = \sum_\alpha \mathbf F_\alpha( t ) \, . \end{align*}
Remark 3.4.
In general the time dependence of the forces can be decomposed into three contributions
a) An implicit time dependence, $\mathbf F( \mathbf q(t) )$,
when the force depends on the position, $\mathbf q(t)$ of the particle.
For instance, for a Hookian spring with spring constant $k$ one has,
$\mathbf F( \mathbf q ) = - k \; \mathbf q$. 1)
b) An implicit time dependence, $\mathbf F( \dot{\mathbf q} (t) )$,
when the force depends on the velocity, $\dot{\mathbf q} (t)$ of the particle. For instance, the sliding friction for a particle with mass $m$ and friction coefficient $\gamma$
is, $\mathbf F( \dot{\mathbf q} ) = - m \, \gamma \; \dot{\mathbf q}$.
c) An explicit time dependence when the force is changing in time.
For instance, when pushing a child sitting on a swing one will only push when the swing is moving in forward direction.
Typically, one explicitly sorts out these dependencies and writes
\begin{align*}
m \, \ddot{\mathbf q}(t) = \sum_\alpha \mathbf F_\alpha( \mathbf q(t), \dot{\mathbf q} (t), t )
\end{align*}
The resulting relation between the acceleration and the force is called equation of motion of the particle.
Definition 3.3 Equation of Motion (E.O.M.)
Newton's second law establishes a relation between
the position $\mathbf q(t)$ of a particle of mass $m$,
its velocity $\dot{\mathbf q}(t)$, and
acceleration $\ddot{\mathbf q}(t)$,
\begin{align*}
m \, \ddot{\mathbf q}(t) = \mathbf F( \dot{\mathbf q}(t), \mathbf q(t), t )
\\
\end{align*}
that is referred to as the equation of motion (E.O.M.) of the particle.
The motion of $N$ particles
residing at the positions $\mathbf q_1(t), \dots, \mathbf q_N(t) \in \mathbb{R}^D$
and interacting with each other
amounts to $N\,D$ coupled equations
\begin{align*}
m \, \ddot{\mathbf q}_1 (t)
&= \mathbf F_1\bigl( \dot{\mathbf q}_1(t),\dots,\dot{\mathbf q}_N(t), \mathbf q_1(t), \dots, \mathbf q_N(t), t \bigr)
\\
\vdots \qquad & \quad \vdots
\\
m \, \ddot{\mathbf q}_N (t)
&= \mathbf F_N\bigl( \dot{\mathbf q}_1(t),\dots,\dot{\mathbf q}_N(t), \mathbf q_1(t), \dots, \mathbf q_N(t), t \bigr)
\\
\end{align*}
The primary aim of Theoretical Mechanics is to determine the solution of the E.O.M. for given initial conditions (cf. Definition 1.6), \begin{align*} \Gamma_0 = \Bigl( \mathbf q_1(t_0), \dots, \mathbf q_N(t_0), \dot{\mathbf q}_1(t_0),\dots,\dot{\mathbf q}_N(t_0) \Bigr) \end{align*} for the positions and velocities of the particles at time $t_0$. Bundles of phase-space trajectories characterize the motion of sets of trajectories, and they can be analyzed to determine how the behavior of a system changes upon varying the parameters of the setup.
Example 3.1: Particle moving in the gravitational field
The gravitational field induces a constant force $m\, \mathbf g$ on a particle with mass $m$. Let it have velocity $\mathbf v_0$ at time $t_0$ when it is taking off from the position $\mathbf q_0$. Then Newton's 2nd law states that $\ddot{\mathbf q} (t) = \mathbf g$, and this equation must be solved subject to the initial conditions $\mathbf q( t_0 ) = \mathbf q_0$ and $\dot{\mathbf q}(t_0) = \mathbf v$. By working out the derivatives one readily checks that this is given for \begin{align*} \mathbf q (t) = \mathbf q_0 + \mathbf v \; ( t - t_0) + \frac{1}{2} \; \mathbf g \; ( t - t_0)^2 \end{align*}
Example 3.2 Particle moving in a circle
Let a particle of mass $m$ move with constant speed in a circle of radius $R$
such that its position can be written as
\begin{align*}
\mathbf q (t) =
\begin{pmatrix}
R \; \cos( \omega t ) \\
R \: \sin( \omega t )
\end{pmatrix}
\end{align*}
with a constant angular velocity $\omega$.
Then its velocity and acceleration take the form
\begin{align*}
\dot{\mathbf q} (t) &=
\begin{pmatrix}
-\omega \, R \; \sin( \omega t ) \\
\omega \, R \: \cos( \omega t )
\end{pmatrix}
\\
\text{and}
\quad
\ddot{\mathbf q} (t) &=
\begin{pmatrix}
-\omega^2 \, R \; \cos( \omega t ) \\
-\omega^2 \, R \: \sin( \omega t )
\end{pmatrix}
= -\omega^2 \: \mathbf q(t)
\end{align*}
The speed is constant, taking the value $\sqrt{ \dot{\mathbf q} \cdot \dot{\mathbf q} } = \omega \, R$. The force is antiparallel to $\mathbf q$ with magnitude $m\, \omega^2 \, R$. Moreover, $\dot{\mathbf q} \cdot \mathbf F = 0$ at all times. Hence, the force only changes the direction of motion, and not the speed.
3.3.3 3rd Law
Newton's third law states that the reference frame does not matter for the description of the evolution of two particles, even when they interact with each other — i.e. when they exert forces on each other. Consider for instance the motion of two particles of the same mass $m$ that reside at the positions $\mathbf q_1(t)$ and $\mathbf q_2(t)$. We decide to observe them from a position right in the middle between the two particles $\mathbf Q = \bigl( \mathbf q_1(t) + \mathbf q_2(t) \bigr)/2$. In the absence of external forces this is an inertial frame, such that $\ddot{\mathbf Q} = \mathbf 0$ according to Newton's first law. However, Newton's second law implies that also \begin{align*} \mathbf 0 = 2m \ddot Q = m \, \ddot{\mathbf q}_1 + m \, \ddot{\mathbf q}_2 = \mathbf F_1 + \mathbf F_2 \end{align*} where $\mathbf F_1 = m\, \ddot{\mathbf q}_1$ and $\mathbf F_2 = m\, \ddot{\mathbf q}_2$ are the forces acting on particle $1$ and $2$, respectively. Up to a change of sign the forces are the same, $\mathbf F_1 = - \mathbf F_2$. This action-reaction principle is stipulated by
Axiom 3.3: Newton's 3rd law
Forces act in pairs:
- actio when a body $A$ is pushing a body $B$ with force $\mathbf F_{A \rightarrow B}$
- reactio then $B$ is pushing $A$ with force $\mathbf F_{B \rightarrow A} = - \mathbf F_{A \rightarrow B}$
- and these forces are always balanced, $\mathbf F_{A \rightarrow B} + \mathbf F_{B \rightarrow A} = \mathbf 0$.