**Previous section: ****General Keplerian Motion****.**

__Next section__**: ****The Zeroth Law of Physics****.**

A **reference system **assigns each particle a definite **position **with respect to a given rigid body (or **reference frame**). The set of all possible **position vectors **is a 3D Euclidean vector space called the **position space **of the frame.

**Time dependent rigid displacement of one reference frame with respect to another is **** completely specified** by a time dependent

Accordingly, a particle path **x **= **x**(*t*) in the ** unprimed frame** is mapped into a particle path

**x**′(

*t*) = R(t)

**x**(

*t*) +

**a**(

*t*) in the

**. For this formula, we envision**

*primed frame***,which is being**

*the unprimed frame represents the frame attached to the rigid body***. Also implicit in this expression is the idea that both frames agree on a**

*displaced with respect to the primed frame***parameter.**

*common time*

When it proves convenient, we can often ** choose time t=0 as the time when the two frames coincide** (that is, we imagine a snapshot at this time and define the primed frame to match this snapshot copy). Thus, we can take the viewpoint that

**as the rigid body displaces past in its travels. In such a case, we would have**

*the primed frame is our fixed snapshot copy of the rigid body frame**θ*(0)=0,

*R*(0)=1 and

**a**(0)=0. We are considering this all taking place in a single plane, so both frames agree on the same unit bivector,

**i**. Any particle (attached or unattached to either frame) will be seen to have a path designated by

**x**(

*t*) with respect to the rigid body and

**x**′(

*t*) with respect to the snapshot copy frame.

**Problem: **Suppressing the time argument and writing

derive the following equations relating velocities and accelerations in the two frames aligned at time *t*=0:

(Click* *__here__ for solution and example.)

Newton’s 1^{st} **Law **(implicitly) defines an **inertial frame **as a rigid body with respect to which every free particle has constant velocity.

**Principle of Relativity **requires that the laws of physics are the same in all inertial systems.

[First formulated by Galileo and incorporated by Newton as a corollary in his theory.]

**Problem **(see link below for a solution)

**:**Apply this to Newton’s 2

^{nd}Law to prove that any two inertial frames are related by

a

**Galilean transformation**

** x **→ **x**′ = *R***x **+ **c **+ **u***t** *where *R*, **c **and **u **are constant**.**

Derive therefrom the Galilean **velocity addition theorem:**** ****v**′ = **v **+ **u .**

**Exercise **(see link below for a solution)

**:**Inside a cable car climbing a slope with constant velocity

**v**

_{0}, an object is dropped

from rest. Derive equations for the trajectory within the car and with respect to the earth outside.

**Problem: **Discuss invariance of Newton’s first and second laws with respect to ** Galilean time translation and scaling**: *t** *→ *t*′ = *α**t *+ *β*** **where *α* and *β* are constant.

(Click* *__here__ for solutions for these Galilean transformation problems and exercises.)

__Previous section__**: ****General Keplerian Motion****.**

**Next section: ****The Zeroth Law of Physics****.**