三分量速度本体统一推导 ——圆周、摆线与开普勒椭圆 Unified Derivation under the Three‑Component‑Velocity Ontology — Circle, Cycloid and Keplerian Elliptical Orbits

摘要 本报告彻底摒弃牛顿力学中“力”的概念,全面回归T₃理论的“速度分解”本体论,建立了一套三分量速度本体统一推导框架。依托T₃三维数代数 及其指数映射(类欧拉公式),通过将任意轨道运动分解为两套质点内源正交速度(切向本征速度 与回旋本征速度 )与一套外源场矢量()的矢量叠加,本报告成功统一了正圆、摆线(最速降线)与开普勒椭圆轨道。 本报告严格区分了“法向回旋动力学速度 ”与“运动学法向加速度 ”,并引入外源场与内源双分量的矢量交叉耦合项。在此基础上,完成开普勒椭圆轨道的极坐标参数形式推导,给出以内源速度分量表达的总速率平方与瞬时曲率半径显式表达式,并通过近星点/远星点定性校验与圆轨道极限回归验证其自洽性。利用T₃指数映射,将圆周、摆线、椭圆位置、速度改写为T₃三维数欧拉指数形式。 Abstract Abandoning the Newtonian primitive concept of “force”, this report adopts the velocity‑decomposition ontology of T₃ theory and establishes a unified derivation framework built upon three‑component‑velocity ontology. Based on the T₃ three‑dimensional‑number algebra together with its exponential mapping (Euler‑like formula), orbital motion is decomposed into two intrinsic orthogonal velocity components of the particle — the tangential intrinsic velocity and the cyclotron intrinsic velocity — plus one external‑field vector . This formulation unifies circular orbit, cycloid (brachistochrone) and Keplerian elliptical orbit. This report strictly distinguishes the intrinsic cyclotron dynamical velocity from the kinematic normal acceleration , and introduces vector cross‑coupling terms between the external field and the dual intrinsic velocity components. Parameter‑form derivation in polar coordinates for the Keplerian ellipse is carried out; explicit expressions for total speed‑squared and instantaneous radius‑of‑curvature in terms of intrinsic velocity components are given. Self‑consistency is verified by qualitative periapsis‑apoapsis checks and the circular‑orbit limit. Using T₃ exponential mapping, position and velocity of circle, cycloid and ellipse are rewritten in T₃ three‑dimensional‑number Euler‑exponential form.