三阶相对论运动学与基于局部Darboux标架的Siacci冲击分解
Third-order relativistic kinematics and Siacci shock decompositions via localised Darboux frames
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中文总结 AI 辅助
本文利用局部Darboux标架建立闵可夫斯基3空间中三阶粒子运动学框架，推导相对论不变参数，推广Siacci定理以分解加速度与加加速度，揭示平面约束下的中心力动力学与维度紧化机制。
中文摘要 AI 辅助
受约束粒子沿低维世界片的轨迹构成了相对论运动学和子流形理论的基础领域。本文建立了闵可夫斯基3空间中受约束于类时曲面的三阶粒子动力学的综合几何与运动学框架。通过利用局部Darboux标架，我们显式推导了相对论不变参数，即测地曲率、法曲率和测地挠率，并分析了内蕴曲面拓扑与外在场时空几何之间的非线性耦合。主要焦点致力于表述相对论“加加速度”矢量，该矢量考虑了加速度的瞬时时间变化率并捕捉运动的高阶结构应力。此外，我们将Siacci定理推广到这一洛伦兹设定中，提供了加速度场和加加速度场相对于固定坐标原点的显式切向和径向分量的非垂直几何分解。我们的公式显式揭示了平面约束下隐藏的中心力动力学和维度紧化机制，表明当修正测地挠率被抑制时，面外扭转冲击恒等于零。这些结果为非欧几里得时空中受约束物理系统的结构稳定性和类守恒角动量量提供了新颖的分析见解。
英文摘要
Constrained particle trajectories along lower-dimensional world sheets constitute a foundational domain in relativistic kinematics and submanifold theory. This paper establishes a comprehensive geometric and kinematic framework for third-order particle dynamics constrained to timelike surfaces in Minkowski 3-space. By utilising the localised Darboux frame, we explicitly derive the relativistically invariant parameters, namely the geodesic curvature, normal curvature, and geodesic torsion and analyse the non-linear coupling between the intrinsic surface topology and the extrinsic spacetime geometry. Primary focus is dedicated to formulating the relativistic "jerk" vector, which accounts for the instantaneous time rate of change of acceleration and captures the high-order structural stresses of the motion. Furthermore, we generalise Siacci's theorem to this Lorentzian setting, providing a non-perpendicular geometric decomposition of both the acceleration and jerk fields into explicit tangential and radial components relative to a fixed coordinate origin. Our formulations explicitly unveil the hidden central-force dynamics and dimensional compactification mechanisms under planar constraints, showing that out-of-plane torsional shocks vanish identically when the modified geodesic torsion is suppressed. These results offer novel analytical insights for structural stability and conserved angular momentum-like quantities governing constrained physical systems in non-Euclidean spacetimes.
发表机构
- Batman University（巴特曼大学）
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