# Newton’s Laws of Motion

1. First law: There exists at least one reference frame in which the velocity of a point particle remains constant unless the particle is acted upon by an external force;
2. Second law: The acceleration $\vec{a}$ of a body is parallel and directly proportional to the net force $\sum \vec{F}_i=\vec{F}_{tot}$ and inversely proportional to the mass m, i.e., $\vec{F}_{tot}=m\vec{a}$;
3. Third law: The mutual forces of action and reaction between two bodies are equal, opposite and collinear.

The three laws share some features. For instance:

• The first law can be seen as a special case of the second if the acceleration is null: $\vec{a}=0 \Rightarrow \vec{v}=const$
• The third law for point particles is $\vec{F}_{12} = -\vec{F}_{21}$
• Non-instantaneous acceleration is given by a force integrated in time: $F\Delta t = m\Delta v$ (impulse)

Newton’s laws are valid in inertial frames only, defined by the First Law. The simplest inertial frame is the one given by the fixed stars (which, btw, are not fixed after all!). In non-inertial frames the force is given by a much more complicated expression:

• $\vec{F}' = \vec{F} - 2m\vec{\Omega}\times\vec{v}-m\vec{\Omega}\times\left ( \vec{\Omega}\times\vec{r} \right ) - m\frac{d\vec{\Omega}}{dt}\times \vec{r}$

where $\vec{\Omega}$ is the angular velocity, $\vec{r}$ is the position of the body, $\vec{v}$ is its velocity. The “fictitious” force adds extra terms to the “newtonian” force:

• The Coriolis force $- 2m\vec{\Omega}\times\vec{v}$ ;
• The centrifugal force: $-m\vec{\Omega}\times\left ( \vec{\Omega}\times\vec{r} \right )$
• The Euler force: $- m\frac{d\vec{\Omega}}{dt}\times \vec{r}$