Frame of Reference and Coordinate System
The term frame of reference is used often in a very broad sense, but for the present discussion its meaning is restricted to refer to an observer's state of motion, that is, to either an inertial frame of reference or a non-inertial frame of reference.
The term coordinate system is used to differentiate between different possible choices for a set of variables to describe motion, choices available to any observer, regardless of their state of motion. Examples are Cartesian coordinates, polar coordinates and (more generally) curvilinear coordinates.
Here are two quotes relating "state of motion" and "coordinate system":
We first introduce the notion of reference frame, itself related to the idea of observer: the reference frame is, in some sense, the "Euclidean space carried by the observer". Let us give a more mathematical definition:… the reference frame is... the set of all points in the Euclidean space with the rigid body motion of the observer. The frame, denoted, is said to move with the observer.… The spatial positions of particles are labelled relative to a frame by establishing a coordinate system R with origin O. The corresponding set of axes, sharing the rigid body motion of the frame, can be considered to give a physical realization of . In a frame, coordinates are changed from R to R' by carrying out, at each instant of time, the same coordinate transformation on the components of intrinsic objects (vectors and tensors) introduced to represent physical quantities in this frame.
— Jean Salençon, Stephen Lyle. (2001). Handbook of Continuum Mechanics: General Concepts, Thermoelasticity p. 9
In traditional developments of special and general relativity it has been customary not to distinguish between two quite distinct ideas. The first is the notion of a coordinate system, understood simply as the smooth, invertible assignment of four numbers to events in spacetime neighborhoods. The second, the frame of reference, refers to an idealized system used to assign such numbers … To avoid unnecessary restrictions, we can divorce this arrangement from metrical notions. … Of special importance for our purposes is that each frame of reference has a definite state of motion at each event of spacetime.…Within the context of special relativity and as long as we restrict ourselves to frames of reference in inertial motion, then little of importance depends on the difference between an inertial frame of reference and the inertial coordinate system it induces. This comfortable circumstance ceases immediately once we begin to consider frames of reference in nonuniform motion even within special relativity.…the notion of frame of reference has reappeared as a structure distinct from a coordinate system.
— John D. Norton: General Covariance and the Foundations of General Relativity: eight decades of dispute, Rep. Prog. Phys., 56, pp. 835-7.
Read more about this topic: Mechanics Of Planar Particle Motion, Moving Objects and Observational Frames of Reference
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