Foundations of Dynamics within the Multi-Origin High-Dimensional Geometry Framework

Abstract: Classical dynamics relies on single-origin Cartesian coordinate systems to describe motion and forces. Based on the framework of Multi-Origin High-Dimensional Geometry (MOC), this paper reconstructs the core concepts of dynamics. Within a multi-origin geometry with variable curvature, physical quantities including mass, force, momentum and energy are formulated as structural attributes of origin curvature. This paper proposes eight geometric correspondence principles for dynamical elements, providing a unified endogenous geometric interpretation for mechanical evolution.

I. Geometric Reinterpretation of Core Dynamical Concepts

In Multi-Origin High-Dimensional Geometry, motion is not defined by translation within an external coordinate system, but determined by high-dimensional curvature projections and multi-origin coupling. The correspondences for core dynamical quantities under the MOC framework are as follows:

1. Mass and inertia: the fundamental curvature stiffness of each independent origin, i.e., the geometric resistance of a domain to deformation.

2. Force and torque: coupled differences of curvature gradients between distinct origins; geometric potential differences drive the evolution of the system.

3. Displacement, velocity and acceleration: dynamic offset rates arising when high-dimensional curvature is projected onto lower dimensions.

4. Momentum: the first-order flux of curvature within space.

5. Angular momentum: the second-order circulatory quantity of space arising from rotational partitioning generated by multi-origin coupling.

6. Kinetic energy and potential energy: hierarchical differences of curvature energy between systems of different origins.

7. Temporal evolution: the natural sequence formed by iterative advancement of curvature states of multiple origins.

8. Constraint conditions: inherent geometric topological relations among origins.

II. Theoretical Significance and Scope

1. From coordinate description to curvature generation: Classical dynamics describes motion against coordinate positions, while MOC dynamics directly generates dynamical quantities from the curvature distribution of multiple origins.

2. Compatibility and generalisability: When the multi-origin setting reduces to a single origin and the spatial curvature tends to zero, this framework naturally degenerates into Newtonian mechanics in classical flat spacetime. The introduction of multi-origin structures offers new geometric interpretive pathways for strongly coupled fields, complex rotational systems and high-dimensional phase transitions.

3. Future work: This paper establishes geometric correspondences for dynamical elements. Exact formulations showing how curvature gradients yield Newton’s second law, as well as variational equations for multi-origin coupling, will be developed in subsequent dedicated dynamical models.

Closing Remark

These eight correspondences form the core of MOC physics; the entire body of dynamics is encapsulated within geometry.