Time as a Coordinate: A New Geometric Language for Classical Mechanics

In classical mechanics, the trajectory of a planar motion is conventionally described as (x(t), y(t)), where time t is treated as a parameter external to the coordinate space. This paper proposes a reinterpretation: by including the time axis t as an explicit coordinate component, every planar trajectory is redefined as the three-dimensional curve r(t) = (x(t), y(t), t). Applying the Frenet–Serret framework to this curve yields the curvature κ and torsion τ, whose ratio τ/κ emerges as a geometric descriptor of the system's dynamics in (x, y, t) space. For uniform circular motion, τ/κ = 1/(rω) = T/L is derived analytically, where T is the period and L the arc length. This ratio is interpreted as the proportion of time path to space path. The same geometric structure appears in the rhythmic gymnastics ribbon trajectory. For the double pendulum, τ/κ varies with time, revealing a non-uniform helix structure that geometrically distinguishes irregular motion from regular motion without recourse to phase space reconstruction, limiting procedures, or statistical estimation. The central contribution is the identification of a simple geometric descriptor: in the examined cases, τ/κ = const serves as a geometric indicator of regular motion, while τ/κ(t) varying in time indicates irregular motion. This descriptor follows analytically from the Frenet–Serret framework applied to the redefined trajectory. ​