page_title: Mechanics of planar particle motion
section_title: Fictitious forces in a local coordinate system
hierarchical_section_title: Mechanics of planar particle motion / Fictitious forces in a local coordinate system
caption_attribution_description: English: local unit vectors for planar motion on a curve
context_page_description: 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.
  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, and a co-rotating frame. 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.
context_section_description: In discussion of a particle moving in a circular orbit, in an inertial frame of reference one can identify the centripetal and tangential forces. It then seems to be no problem to switch hats, change perspective, and talk about the fictitious forces commonly called the centrifugal and Euler force. But what underlies this switch in vocabulary is a change of observational frame of reference from the inertial frame where we started, where centripetal and tangential forces make sense, to a rotating frame of reference where the particle appears motionless and fictitious centrifugal and Euler forces have to be brought into play. That switch is unconscious, but real.
  Suppose we sit on a particle in general planar motion (not just a circular orbit). What analysis underlies a switch of hats to introduce fictitious centrifugal and Euler forces?
  To explore that question, begin in an inertial frame of reference. By using a coordinate system commonly used in planar motion, the so-called local coordinate system, as shown in Figure 1, it becomes easy to identify formulas for the centripetal inward force normal to the trajectory (in direction opposite to uₙ in Figure 1), and the tangential force parallel to the trajectory (in direction uₜ), as shown next.
  To introduce the unit vectors of the local coordinate system shown in Figure 1, one approach is to begin in Cartesian coordinates in an inertial framework and describe the local coordinates in terms of these Cartesian coordinates. In Figure 1, the arc length s is the distance the particle has traveled along its path in time t.  The path r (t) with components x(t), y(t) in Cartesian coordinates is described using arc length s(t) as:
  One way to look at the use of s is to think of the path of the particle as sitting in space, like the trail left by a skywriter, independent of time. Any position on this path is described by stating its distance s from some starting point on the path. Then an incremental displacement along the path ds is described by:
  where primes are introduced to denote derivatives with respect to s. The magnitude of this displacement is ds, showing that:
      (Eq. 1)
  This displacement is necessarily tangent to the curve at s, showing that the unit vector tangent to the curve is:
  while the outward unit vector normal to the curve is 
  Orthogonality can be verified by showing the vector dot product is zero. The unit magnitude of these vectors is a consequence of Eq. 1.
  As an aside, notice that the use of unit vectors that are not aligned along the Cartesian xy-axes does not mean we are no longer in an inertial frame. All it means is that we are using unit vectors that vary with s to describe the path, but still observe the motion from the inertial frame.
  Using the tangent vector, the angle of the tangent to the curve, say θ, is given by:
     and   
  The radius of curvature is introduced completely formally (without need for geometric interpretation) as:
  The derivative of θ can be found from that for sin θ:
  Now:
    
  in which the denominator is unity according to Eq. 1. With this formula for the derivative of the sine, the radius of curvature becomes:
   
  where the equivalence of the forms stems from differentiation of Eq. 1:
  Having set up the description of any position on the path in terms of its associated value for s, and having found the properties of the path in terms of this description, motion of the particle is introduced by stating the particle position at any time t as the corresponding value s (t).
  Using the above results for the path properties in terms of s, the acceleration in the inertial reference frame as described in terms of the components normal and tangential to the path of the particle can be found in terms of the function s(t) and its various time derivatives (as before, primes indicate differentiation with respect to s):
    
  as can be verified by taking the dot product with the unit vectors uₜ(s) and  uₙ(s). This result for acceleration is the same as that for circular motion based on the radius ρ.  Using this coordinate system in the inertial frame, it is ea
caption: Figure 1: Local coordinate system for planar motion on a curve. Two different positions are shown for distances s and s + ds along the curve. At each position s, unit vector un points along the outward normal to the curve and unit vector ut is tangential to the path. The radius of curvature of the path  is ρ as found from the rate of rotation of the tangent to the curve with respect to arc length, and is the radius of the osculating circle at position s.  The unit circle on the left shows the rotation of the unit vectors with s.
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