On Concept of Mechanical System

Abstract: The paper gives a screw axiomatics of rational mechanics, namely: 1. introduces the main measures of mechanics: the mass measure, the scalar and (vector) screw measures of motion, the (vector) screw measure of impressed action, the increment velocity of the vector measure of motion, the (vector) screw measure of constraint action; 2. postulates the (stronger) local integral form of conservation law for the vector measuare of motion (fundamental principle of dynamics), and 3. defines the central concept of rational mechanics {mechanical system being realized in the form of all classical mechanical systems (mass-points, rigid bodies, continua, point-bodies, etc.). The presentation is based on new notions of vector calculus -- homogeneous and inhomogeneous vector and tensor slider-functions and screw measures.

• The paper rigorously constructs a unified axiomatic framework that integrates measure theory and screw calculus for classical mechanics.
• It introduces homogeneous and inhomogeneous sliders along with screw measures to generalize the conventional descriptions of force and momentum.
• The framework recovers established models for mass-points, rigid bodies, continua, and multiphase systems via a local integral form of conservation laws.

Formal Axiomatic Foundations for the Mechanical System

The paper "On Concept of Mechanical System" (1302.7092) offers a rigorous axiomatization of rational mechanics anchored in advanced measure theory, screw calculus, and the systematic abstraction of physical systems. This formulation reconstructs the foundational notions of mass, momentum, forces, and constraints on a mathematically explicit basis, integrating points, continua, multibody, and constrained systems under a unified formalism. The work includes the introduction of homogeneous and inhomogeneous sliders, screw measures, and group-theoretic representations, leading to a local integral form of dynamics applicable to all classical mechanical models.

Measure-Theoretic Formalism and Screw Calculus

At the heart of this framework is the abstraction of physical bodies as measurable sets in affine space, equipped with Borel measures that may include absolutely continuous and pure point (atomic) parts. The author defines the fundamental kinematic and dynamic descriptors—mass, kinetic energy, momentum—not as naive quantities but as scalar or screw-valued measures (Lebesgue–Stieltjes integrals over the sets representing the bodies).

The principal innovation involves generalizing the classical notion of force and momentum to screw measures, built upon vector and tensor sliders. A slider, in this context, is a field associating to each point a pair of vectors (the fundamental pair), with screw properties under affine transport. Screw measures aggregate these into integral objects invariant and reducible with respect to changes of reference points, naturally encompassing translation and rotation, and thus the description of both mass-points and extended bodies.

Local Integral Axiomatics of Mechanics

A central axiom is the local integral form of the conservation law for the vector measure of motion (the screw measure of momentum):

$$\frac{d}{dt} \mathcal{P}(\Lambda_t) = \mathcal{F}(\Lambda_t),$$

where $\mathcal{P}$ is the screw measure of motion, and $\mathcal{F}$ is the screw measure of impressed action over the (possibly time-dependent) measurable set $\Lambda_t$ occupied by the system at time $t$.

This form is strictly more general than the conventional differential laws—it is valid even when the divergence theorem does not hold (e.g., for concentrated or non-smooth distributions of mass, or in presence of atomic components). It immediately specializes to classical forms (Newton's second law for a point, Newton–Euler equations for rigid bodies, balance laws in continua) under respective choices of $\Lambda_t$ and slider densities.

Hierarchies and Generalizations of Mechanical Systems

The axiomatics is shown to instantiate conventional mechanical models:

• Mass-Points: Obtained for pure atomic measures; yields Newton's law.
• Rigid Bodies: Recovered by integrating the field of twists over a closed, connected set with ideal constraints, yielding the Newton–Euler equations.
• Consecutive and Tree-Like Multibody Systems: The formalism naturally expresses the kinematic and dynamic coupling of rigid components, as illustrated by system graphs (see below).

  Figure 1: Multibody system graphs encode the hierarchical and branching structure of consecutively and tree-connected rigid bodies along with their relative motions.

• Continua: By employing absolutely continuous measures and introducing stress tensors as densities of internal constraint actions, the field equations of continuum mechanics are rigorously derived as special cases.
• Point-Bodies and Multiphase Media: Addressed via inhomogeneous sliders and non-symmetric stress measures, with the same measure-theoretic machinery extending to generalized bodies and subsystems.

Parameterizations, Group Structure, and Symmetry

The geometric aspect is systematically resolved using group-theoretic apparatus: the motion of frames and their velocities (translations, rotations, twists) are encoded as actions of groups of screw transformations. The rotation group is parameterized by Euler angles, Fedorov vector-parameterization (Gibbs–Rodrigues vectors), and unit quaternions (Euler–Rodrigues parameters), and kinematic relations are systematically derived in these representations. The approach preserves generality and computational tractability (as for the kinematic chain composition rules), and the explicit form of all transformation matrices is given.

Isotropic Constitutive Theory and Correct Continua

The framework incorporates a general, invariant description of isotropic linear constitutive relations (including the stress–strain mappings for 2D and 3D cases), with necessary and sufficient conditions for invertibility (“correct” continuum). The stress tensor is shown to be symmetric in physical cases and is expressed as an isotropic function of the strain or strain-rate tensor and, when possible, pressure terms. The constitutive mappings are fully parametrized in terms of scalar rheological invariants, with explicit formulae for their inverses, enabling treatment of elastic materials, viscous fluids, and more general rheological media in a uniform measure-theoretic way.

Discussion, Claims, and Implications

The author asserts that all classical mechanical systems—mass points, rigid bodies (with or without constraints), continua, multiphase systems—are mathematically realizations of the mechanical system structure as defined by this screw–measure axiomatics and local integral dynamics. The local integral form is claimed to be strictly stronger than the conventional differential formulations and enables rigorous treatment of configurations inaccessible to classical methods, such as those involving atomic measures or non-smooth sets, and provides a uniform language for the transition from the particle model to the field model.

Further, the approach dismantles classical ambiguities regarding the assignment of forces, inertia, and momentum to “shapes” or parts of bodies rather than to bodies themselves, and resolves the inclusion problem of point particles within continuum dynamics. These advances have strong implications for the theoretical completeness of mechanical theory, clarity in the definition of mechanical systems, and the rigorous formulation of constraint and interaction forces.

Concluding Remarks

The measure-theoretic, screw-based axiomatics presented in "On Concept of Mechanical System" delivers a complete and unified mathematical foundation for rational mechanics. This structure seamlessly encompasses all major classical system types and reconciles measure-based and geometric approaches. The implications are significant for both foundations (resolving old theoretical discrepancies and ambiguities) and computational approaches (enabling hierarchical, group-theoretic, and measure-theoretic methods in modeling complex, multi-component, or singular systems). Future developments may exploit this formalism to extend mechanical theory to even more general physical models, refined treatments of constraints, or hybrid discrete–continuum systems.