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Mathematics for Physics

104 Inertial Frame

A reference frame in which Newton’s first law holds that is used to state the laws of mechanics without fictitious forces from acceleration of the axes.

Note: Also called an inertial reference frame.

definition [d] (Inertial Frame = Inertial Reference Frame) From Knight: we define an inertial reference frame as a reference frame in which Newton’s first law is valid.

where

  • a reference frame is a coordinate system for position and time measurements.
  • Newton’s first law is the law of inertia for free particles.

definition [d] (Inertial Frame) From Griffiths: to avoid this trap, we define an inertial frame formally as one in which Newton’s first law holds.

where

  • the frame is inertial precisely when free particles move with constant velocity in that frame.

definition [d] (Inertial Frame of Reference) From Emam: coordinate axes moving with constant velocities are called inertial frames of reference. Another way of defining inertial frames of reference is that they are those frames that preserve the form of Newton’s law.

where

  • constant-velocity motion of the axes is relative to other inertial frames.
  • Newton’s second law keeps the same form in every such frame.

104.1 Elementary Example

104.1.1 Simple

A frame fixed to the ground, treating Earth as non-accelerating for a short lab experiment, is treated as inertial: a puck on ice moves at constant velocity when no horizontal force acts.

\[ \mathbf{F} = \mathbf{0} \implies \mathbf{v} = \text{constant} \]

where

  • \(\mathbf{v}\) is the velocity measured in that frame.

104.1.2 General

Two frames related by a constant relative velocity \(\mathbf{u}\) are both inertial if one is.

\[ \mathbf{x}' = \mathbf{x} - \mathbf{u}t,\quad t' = t \]

\[ \mathbf{a}' = \mathbf{a} \]

where

  • \(\mathbf{a}\) is acceleration, unchanged by the constant boost.
  • Newton’s second law keeps the same form in both frames.

104.2 References

  1. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — source for the heading explanation.
  2. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — inertial reference frame as one where Newton’s first law is valid.
  3. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — inertial frame as one in which Newton’s first law holds.
  4. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — inertial frames as constant-velocity axes preserving Newton’s law.