Mechanics of planar particle motion

This article describes a particle in planar motion when observed from non-inertial reference frames. The most famous examples of planar motion are related to the motion of two spheres that are gravitationally attracted to one another, and the generalization of this problem to planetary motion. See centrifugal force, two-body problem, orbit and Kepler's laws of planetary motion. Those problems fall in the general field of analytical dynamics, the determination of orbits from given laws of force. This article is focused more on the kinematical issues surrounding planar motion, that is, determination of the forces necessary to result in a certain trajectory given the particle trajectory. General results presented in fictitious forces here are applied to observations of a moving particle as seen from several specific non-inertial frames, for example, a local frame (one tied to the moving particle so it appears stationary), and a co-rotating frame (one with an arbitrarily located but fixed axis and a rate of rotation that makes the particle appear to have only radial motion and zero azimuthal motion). The Lagrangian approach to fictitious forces is introduced.The equations of motion in a non-inertial system differ from the equations in an inertial system by additional terms called inertial forces. This allows us to detect experimentally the non-inertial nature of a system.An additional force due to nonuniform relative motion of two reference frames is called a pseudo-force.Treat the fictitious forces like real forces, and pretend you are in an inertial frame.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 R {displaystyle {mathfrak {R}}} , is said to move with the observer.… The spatial positions of particles are labelled relative to a frame R {displaystyle {mathfrak {R}}} by establishing a coordinate system R with origin O. The corresponding set of axes, sharing the rigid body motion of the frame R {displaystyle {mathfrak {R}}} , can be considered to give a physical realization of R {displaystyle {mathfrak {R}}} . In a frame R {displaystyle {mathfrak {R}}} , 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.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.In the above equations, there are three types of terms. The first involves the second derivative of the generalized co-ordinates. The second is quadratic in q ˙ {displaystyle mathbf {dot {q}} } where the coefficients may depend on q {displaystyle mathbf {q} } . These are further classified into two types. Terms involving a product of the type q ˙ i 2 {displaystyle {{dot {q}}_{i}}^{2}} are called centrifugal forces while those involving a product of the type q ˙ i q ˙ j {displaystyle {dot {q}}_{i}{dot {q}}_{j}} for i ≠ j are called Coriolis forces. The third type is functions of q {displaystyle mathbf {q} } only and are called gravitational forces. This article describes a particle in planar motion when observed from non-inertial reference frames. The most famous examples of planar motion are related to the motion of two spheres that are gravitationally attracted to one another, and the generalization of this problem to planetary motion. See centrifugal force, two-body problem, orbit and Kepler's laws of planetary motion. Those problems fall in the general field of analytical dynamics, the determination of orbits from given laws of force. This article is focused more on the kinematical issues surrounding planar motion, that is, determination of the forces necessary to result in a certain trajectory given the particle trajectory. General results presented in fictitious forces here are applied to observations of a moving particle as seen from several specific non-inertial frames, for example, a local frame (one tied to the moving particle so it appears stationary), and a co-rotating frame (one with an arbitrarily located but fixed axis and a rate of rotation that makes the particle appear to have only radial motion and zero azimuthal motion). The Lagrangian approach to fictitious forces is introduced. Unlike real forces such as electromagnetic forces, fictitious forces do not originate from physical interactions between objects. The appearance of fictitious forces normally is associated with use of a non-inertial frame of reference, and their absence with use of an inertial frame of reference. The connection between inertial frames and fictitious forces (also called inertial forces or pseudo-forces), is expressed, for example, by Arnol'd: A slightly different tack on the subject is provided by Iro: Fictitious forces do not appear in the equations of motion in an inertial frame of reference: in an inertial frame, the motion of an object is explained by the real impressed forces. In a non-inertial frame such as a rotating frame, however, Newton's first and second laws still can be used to make accurate physical predictions provided fictitious forces are included along with the real forces. For solving problems of mechanics in non-inertial reference frames, the advice given in textbooks is to treat the fictitious forces like real forces and to pretend you are in an inertial frame. It should be mentioned that 'treating the fictitious forces like real forces' means, in particular, that fictitious forces as seen in a particular non-inertial frame transform as vectors under coordinate transformations made within that frame, that is, like real forces.