PlanetPhysics/Reference Frames in Newtonian Physics

Newton's First and Second Laws gain meaning only when they are cast in a coordinate system. The coordinate system in which they are valid is referred to as an inertial reference frame. Suppose a particle is set into motion at some time $$t_{0}$$, by some force, $$\mathbf{F}_{0}$$, and at all later times, $$t>t_{0}$$, the net force acting on that particle is zero. According to Newton's First Law, the particle will remain in motion, and have constant velocity at all subsequent times. Any coordinate system reflecting this property is an inertial reference frame. That is, in an inertial reference frame, motion can result only from forces acting upon the body, and not through the system of coordinates that have been chosen to characterise the system.

At first glance, the inertial reference frame seems intuitively obvious, and it seems easier to imagine such a coordinate system, then to form a counter example. The counter example of an inertial reference frame is the non-inertial reference frame. In such a coordinate system, acceleration of the body arises due to acceleration of the coordinate system, relative to that body. Suppose some particle is moving in a straight line, with some constant velocity, $$v$$. An observer of that body is moving in the same straight line, but at a velocity, $$u = u(t)$$ i.e. a time dependent velocity. Relative to the observer, the body moves with velocity $$v' = v - u(t)$$. Recall that force is proportional to acceleration. Relative to the observer, the body has acceleration $$a = \frac{d}{dt}(v - u(t)) = \frac{du}{dt}\neq 0.$$ Thus, since the acceleration of the body relative to the observer is non-zero, so too is the net force acting upon that body in this coordinate system. This directly violates Newton's First Law. That is, Newton's First Law states that an object undergoing constant velocity has no net force acting on it. However, in the accelerated coordinate system, the body is preceived to have acceleration, and hence a non-zero net force.

By illustrating the concept of a non-inertial reference frame, the inertial reference frame can be given more specific meaning. An inertial reference frame is one that moves with constant velocity relative too all objects that are not subject to a net force. If the coordinate system accelerates, then fictitous forces result, which contradict Newton's laws.

Associated with the inertial reference frame is the idea of the homogeneity of time. Suppose a body moves with constant velocity in an inertial reference frame. Let $$\Delta t$$ denote some time interval. An observer in the reference frame measures that on the time interval, the body moves a total distance of $$\Delta x$$. If time is homogenous, then at all subsequent times, if the distance travelled by the body is measured in the interval $$\Delta t$$, then the result will always be $$\Delta x$$. If time is homogeneous, then Newton's Laws are valid. Once again, this principle seems intuitive, yet must be clearly stated in order to make use of Newton's Laws, and determine in what situations they apply.

A final point related to inertial reference frames is that of Galilean Invariance. Suppose an inertial reference frame has been established. Another coordinate system is then constructed, This second coordinate system is specified to be in constant velocity relative to the first. That is, if one thinks of both coordinate systems as observers, then neither experiences acceleration relative to one another. Subject to the observations made above regarding the inertial reference frame, it follows that on any interval of time, both coordinate systems will measure the same displacement of the body. This is the statement that if Newton's Laws are valid in one reference frame, then they will also be valid in any reference frame that is in uniform motion relative to the first.