Consider a particle m α i within the group. Its force is given by Equation (5) as follows:

On the other hand, the total external force on the entire group is given by Equation (13). Thus, from Equations (13) and (16), we have the following:

Since ∑ β = 1 k M β = ∑ j = 1 n m j , the above equation can be transformed into:

Because

where

Substituting the above results into Equation (17) gives:

Substituting V α with V α = ∑ j m α j v α j ∑ j m α j in Equation (18) gives:

Thus, we obtain:

The above equation indicates that the resultant force F α i acting on particle m α i can be clearly decomposed into two parts:

1) The external force component F α i ( e x t ) = m α i M α F α is outside the group. This

term is proportional to the share of the particle in the total mass of the group. It represents how the external influence on the entire group is “distributed” to each internal particle according to “mass allocation”.

2) The internal force F α i ( i n t ) = m α i ∑ j m α j ∑ j m α j d ( v α i − v α j ) d t from within the

group. This term is entirely determined by the interaction forces between the particle and other particles within the group.

Therefore, the resultant force acting on any particle within a particle group is the vector sum of its external and internal forces. This decomposition method conforms to the principle of force superposition and provides a clear physical basis for distinguishing between internal and external action mechanisms within the system.

According to the superposition of momenta, the momentum of a particle can also be naturally decomposed. Analogous to the decomposition of external and internal

forces, its momentum expression P α i = m α i ∑ j = 1 n m j ∑ j = 1 n m j ( v α i − v j ) can be written as follows:

Thus, we define the following:

1) External momentum P α i ( e x t ) = m α i M α P α represents the momentum share

carried by particle m α i because it “belongs to the group” and moves with the center of mass as a whole.

2) Internal momentum P α i ( i n t ) = m α i ∑ j m α j ∑ j m α j ( v α i − v α j ) : the momentum obtained from the relative motion between particle m α i and other particles within the group.

Total momentum P α i = P α i ( e x t ) + P α i ( i n t ) . In summary, the total momentum of the group is P α = ∑ i P α i ( e x t ) (the sum of the internal momenta is zero).

Compared with vectorial forces and momentum, kinetic energy, as a scalar quantity, offers richer connotations in its decomposition. When the relative motion and interactions among particles within a particle group are neglected and the entire group is abstracted as an equivalent particle, the kinetic energy possessed by this equivalent particle during interactions with other particles or groups is defined as the external kinetic energy of the particle group. From Equation (15), the external kinetic energy of the system after particle grouping is given by:

Evidently, the system’s external kinetic energy expressed in Equation (21) differs from the total kinetic energy in Equation (11), indicating that a portion of the kinetic energy remains confined within each particle group. To address this, the relationship between the two quantities requires further formalization.

Starting from the differential of total kinetic energy d T = ∑ i = 1 n F i v i d t , we derive the following:

where

is the differential of the kinetic energy generated by the external forces acting on the particle group, and its corresponding external kinetic energy is given by

Equation (21). d T ( i n t ) = ∑ α = 1 k ∑ i F α i ( i n t ) v α i d t is the sum of the differentials of the

kinetic energy generated by the internal forces acting within each particle group. We call this the internal kinetic energy of the system, which is the sum of the

internal kinetic energies of each particle group T ( i n t ) = ∑ α = 1 k T α ( i n t ) ; thus, d T α ( i n t ) = ∑ i F α i ( i n t ) v α i d t . Substituting the formula for internal forces gives:

Substituting ∑ i ∑ j m α i m α j v α i d ( v α i − v α j ) = 1 2 ∑ i ∑ j m α i m α j ( v α i − v α j ) d ( v α i − v α j ) for ∑ i ∑ j m α i m α j v α i d ( v α i − v α j ) in the above equation yields:

After integration, we obtain

Considering that T α ( i n t ) = 0 holds when the particles within the α th particle group are relatively stationary, C α = 0 can be derived. Therefore, the expression for the internal kinetic energy of the α th particle group is:

The total kinetic energy of a closed particle group is equal to the sum of the internal kinetic energy of its subsystems and the external kinetic energy of the system as a whole; that is,

Equation (25) is a profound physical statement: the total kinetic energy of a closed system is the superposition of the “overall translational kinetic energy” (external kinetic energy) of each substructure (i.e., “particle group”) and the “relative motion kinetic energy” (internal kinetic energy) of all the particles within these substructures.

The theory of internal and external kinetic energy of an object reveals the hierarchical structure of energy: when an object participates in external interactions as a whole, its macroscopic motion manifests as external kinetic energy, whereas as a composite system composed of microscopic particles, its internal motion is expressed as internal kinetic energy. External kinetic energy is caused mainly by work carried out by external forces, whereas internal kinetic energy stems from the action of internal forces and internal relative motion. When an external force is applied to a nonrigid object, it often simultaneously triggers adjustments in its internal structure, thereby leading to changes in internal kinetic energy. The divisibility of matter determines that energy has a multilevel organizational form. Many experiments have shown that between the internal and external kinetic energy of an object, a reduction in one form of energy is often accompanied by an increase in the other form of energy, and vice versa; this finding indicates that energy at different levels can be mutually converted and maintain the conservation of total energy under specific conditions.

The “layered” view of energy provides a unified theoretical framework for understanding energy storage and transfer methods from macroscopic celestial bodies and nonrigid mechanical structures to microscopic multiparticle systems. This finding reveals that the internal energy and macroscopic kinetic energy of an object are not isolated but exist in a hierarchical structure that can be mutually converted.

Equation (5) can then be transformed into:

For m k ≫ ∑ j ≠ k m j , λ = ∑ j ≠ k m j / m k → 0 , so

Thus, there is a limit:

This is the same as the standard form of Newton’s second law in an inertial frame.

Taking the limit of Equation (7) with respect to λ = ∑ j ≠ k m j / m k → 0 , similar to the derivation of the limit of force, we obtain the following:

This limit form is also the same as the expression of momentum for a particle in an inertial frame.

Transforming the kinetic energy expression (11) into the form of

Because ∑ j = 1 n m j ( v k − v j ) 2 = ∑ i = 1 n m i ( v i − v k ) 2 , the above equation becomes:

Take the limit of λ = ∑ j ≠ k m j / m k → 0 in the above equation, where

(where A = max { | v i − v j | } )

So,

This limit also has the same form as the expression of the kinetic energy of a particle in an inertial frame. Through the above research, our main conclusion is that under the condition of “there being a central body with an absolutely dominant mass”, the mechanical equations of other particles in a general reference frame relative to this central body naturally converge to the standard equations in the inertial frame of Newtonian classical mechanics in the limit form. This indicates that a celestial body with a sufficiently large mass (such as the Sun in relation to its planetary system or the Earth in relation to macroscopic objects near its surface) provides a local and approximately ideal inertial reference environment in its vicinity. The classical theory and experiments of inertial frames can be regarded as special cases established under the “infinite mass background approximation” rather than the theoretical starting point. This also explains at a deeper level why classical mechanics has extremely high accuracy at the scale of the solar system, the Earth’s surface, etc.

From v ′ i = v i − V C M , we obtain v i − v j = v ′ i − v ′ j , and substituting it into Equation (5) gives:

Because ∑ j = 1 n m j v ′ j = ∑ i = 1 n m i v ′ i = 0 , therefore,

The above equation indicates that the force in the center-of-mass coordinate system has the same form as that in the inertial system; in other words, dynamically, the center-of-mass coordinate system is equivalent to the inertial system.

Following the same principle as the derivation of the force equation, substituting

v i − v j = v ′ i − v ′ j and ∑ j = 1 n m j v ′ j = 0 into Equation (7) yields

The momentum equation also demonstrates the equivalence of the center-of-mass coordinate system and the inertial system.

Substituting v i − v j = v ′ i − v ′ j into Equation (11) gives

Among them

Substituting ∑ j = 1 n m j v ′ i 2 ∑ i = 1 n m i = ∑ i = 1 n m i v ′ i 2 ∑ j = 1 n m j and ∑ i = 1 n m i v ′ i = 0 into the above equation, we obtain

So,

The above equation indicates that T is the sum of n independent kinetic energy terms. On this basis, the kinetic energy of the i th particle can be defined as follows:

The kinetic energy of a particle is completely consistent with the definition of kinetic energy in an inertial frame. This shows that in the center-of-mass coordinate system, the kinetic energy of a particle system can be decomposed into the kinetic energy of individual particles and has physical significance.

In the center-of-mass system, the mechanical equations can automatically be present in the standard form of a classical inertial frame and are unaffected by whether the original system is accelerating as a whole. This finding indicates that an inertial frame is not the theoretical foundation but rather a dynamic manifestation in a specific reference frame. For an isolated system, the center-of-mass system is a convenient description framework that can be defined by the internal mass distribution of the system and has no inertial forces. It is among the physical sources of an ideal inertial frame and is a special normative choice among general reference frames. It is also a powerful tool for analysing isolated or nearly isolated systems.

StructuralLimitations of Classical Non-Inertial System Theory: This theory takes the idealized and unoperationalizable “inertial frame” as an unverifiable a priori postulate and relies on introducing “inertial forces” without corresponding physical sources as technical intermediaries to achieve mapping and solving within the inertial frame framework. This approach has two unavoidable theoretical tensions: First, it gives the inertial frame ontological priority, leading to a lack of theoretical self-sufficiency; second, inertial forces have neither independent dynamical origins nor direct observable carriers, making it impossible to obtain a unified and strict expression basis for the conservation of momentum, energy, and Newton’s third law in systems containing such forces, exposing the incompleteness of its physical foundation.Construction Logic of General Reference Frame Theory:This paper establishes the relative motion between all pairs of particles within the system as the primary, directly measurable fundamental physical quantity. Although the derivation process temporarily borrows the language of inertial frames to ensure logical consistency, the core equations ultimately obtained—the fundamental equation of force (5), the momentum Equation (7), the expression for the total kinetic energy of the system (11), and the strictly derived decomposition rules for internal and external forces and the hierarchical structure of internal and external energy (25)—have completely eliminated the dependence on specific parameters such as the position and velocity of the original inertial frame and do not contain any fictitious forces or virtual displacement terms. Thus, a logically self-consistent, physically self-sufficient, and universal dynamical system that does not require the assumption of an absolute rest reference frame is established.

On the basis of the general reference frame perspective for the systematic reconstruction of mechanical principles, this paper achieves ontological redefinitions of several fundamental physical concepts:

1)Ontological Reinterpretation ofForce: Force is clearly defined as a function of the relative acceleration between pairs of particles within the system, an intrinsic physical quantity determined solely by the relative configuration and its temporal evolution between particles. This definition eliminates the dependence on external absolute space or the state of the observer, providing solid and universal theoretical support for its objective reality and reference frame independence.

2)The Relativistic Nature of Momentum: Momentum is no longer interpreted as an attribute of an object relative to an abstract “absolute space” but as the weighted integral of the relative motion between all pairs of particles within the system (Equation (7)). This definition not only naturally satisfies the conservation law in any translational inertial frame but also strictly guarantees its generalization consistency in any accelerating reference frame, thereby eliminating the conceptual ambiguity caused by reference frame selection in classical formulations.

3)Hierarchical Structure of Kinetic Energy: By strictly distinguishing between internal kinetic energy (representing the energy carried by the relative motion within subsystems) and external kinetic energy (representing the energy carried by the motion of the system’s center of mass) (Equation (25)), this paper reveals the modular topological structure of energy storage and dissipation in composite systems. This structure provides a theoretical tool with clear physical meaning and computability for quantitatively analysing complex dynamic processes such as nonrigid body deformation dissipation, energy redistribution in flexible robots, multiscale cascades in turbulence, and angular momentum transport in gravitational multibody systems.

1)Strict Inclusion of Classical Mechanics: Through two strictly implementable physical limit processes—the “infinite mass limit” and the “centre of mass reference frame”—classical Newtonian mechanics is perfectly reproduced as a special case within the new framework. This finding indicates that the new framework is not a replacement for existing theories but rather achieves a logical unification and fundamental reconstruction under a broader mathematical structure and more solid physical premises, fundamentally eliminating the absolute dependence of classical theories on an inertial background.

2)Modelling advantages for complex systems:The mechanics of particle groups have transcended the numerical strategy level of traditional multibody problems and have been elevated to a system cognition paradigm with profound physical connotations. It systematically decouples high-dimensional strongly coupled dynamical problems into multilevel subsystem interaction problems with clear physical boundaries, thereby providing a universal analytical paradigm with both physical fidelity and computational feasibility for astrophysics (such as predicting the relaxation timescale of star clusters and simulating angular momentum transfer during galaxy mergers), molecular dynamics (such as the competition mechanism between internal and external kinetic energy in protein folding pathways and quantifying conformational entropy changes induced by ligand binding), and even the modelling of emergent dynamics in complex network systems.

3)Application potential in frontier cross-disciplinary fields

“Mechanics of particle groups in general reference frames” not only constitutes a logically closed theoretical system but also, owing to the ontological universality of its conceptual basis and the structural advantages of its analytical tools, clearly enables paths in multiple frontier technological fields:

a) Precision space dynamics and deep space exploration

High-precision orbit modelling in complex gravitational fields: In missions such as Earth-Moon transfer, long-term residence at Lagrange points, or interplanetary cruising under the perturbation of multiple small celestial bodies, the actual environment of spacecraft significantly deviates from the ideal two-body field. Traditional perturbation methods rely on numerous empirical correction terms and frequent coordinate transformations, which can easily introduce systematic modelling errors. This theory supports modelling the main gravitational body, perturbation sources (including the moon, planets, asteroid groups, and even the equivalent particle of solar radiation pressure), and the spacecraft itself as an independent particle group; analysing the energy‒momentum coupling mechanism between the intergroup external dynamics (orbital evolution) and intragroup internal dynamics (attitude manoeuvre, propulsion dissipation, thermal deformation), submeter orbit prediction accuracy, probabilistic assessment of collision risks, and global energy-optimal solutions for multipulse manoeuvres can be achieved.Cooperativecontrolof spacecraft clusters: The entire formation is regarded as the “main group,” with its center-of-mass motion dominated by the orbital constraints of the mission (external kinetic energy dominant); each satellite constitutes a functional subgroup within the main group, and issues such as maintaining relative configurations, autonomous obstacle avoidance, and dynamic fuel allocation can all be strictly expressed as the process of subgroups regulating internal momentum and internal kinetic energy through internal force work. This modelling approach provides verifiable theoretical constraints and performance boundaries for designing distributed controllers with physical consistency, anti-interference robustness, and online adaptability.

b) Advanced robotics and intelligent bionic systems

Dynamic modelling of flexible multibody systems: For systems such as snake-like robots, continuum manipulators, and soft actuators that have distributed elasticity, viscous dissipation, and active drive coupling, traditional rigid body models cannot describe the real-time redistribution of energy between macroscopic motion and microscopic vibration modes. This theory supports the natural division of the system into several physical subgroups (such as joint segments, fluid chambers, and flexible segments) on the basis of structural function and the strength of dynamical coupling; the dynamics of each subgroup are not only constrained by the motion of the upper level (external force input) but also determined by its internal elastic potential energy, damping dissipation, and electromechanical conversion (internal force action). On the basis of the hierarchical analysis of internal and external kinetic energy, the dominant flow paths of energy between rigid motion, elastic energy storage, viscous dissipation, and electromechanical feedback channels can be quantitatively identified, thereby providing a quantifiable theoretical basis for the design of compliant control laws, wideband vibration suppression strategies, and cyclic energy recovery architectures.Modelling of emergent behaviors in multiagent clusters: Systems such as drone swarms, autonomous vehicle fleets, or collections of molecular motors can all be formalized as multiparty dynamical objects. By applying the framework of group mechanics, it is possible to strictly model how local interaction rules among individuals (internal force model) can collaboratively guide the overall system toward an energy-optimal state (minimization of the sum of external and internal kinetic energy) under macroscopic task constraints (such as formation geometry maintenance and maximization of area coverage efficiency) or spontaneously form collective motion patterns such as vortices, travelling waves, and synchronous oscillations. This approach not only reproduces the experimental observations of biological swarms but also provides computable theoretical criteria for the robust design and failure-tolerant mechanisms of artificial swarm systems.

c) Cross-scale physical simulation and complex system science

Astrophysical and cosmological simulations:From star clusters and galactic disks to large-scale cosmic structures, the distribution of matter exhibits strict hierarchical nesting features. The recursive grouping strategy of this theory is highly consistent with this: for example, in galactic-scale simulations, the entire galaxy can be set as the first-level group, the spiral arms and halo as the second-level subgroups, and giant molecular clouds or globular clusters as the third-level subgroups. This structure not only reduces the computational complexity of N-body simulations from O ( N 2 ) to approximately O ( N ) but also strictly tracks the cross-scale transport fluxes of angular momentum, kinetic energy, and gravitational potential energy between different scale structures (such as the interface between the stellar disk and the dark matter halo), providing falsifiable diagnostic variables for understanding key evolutionary issues such as the excitation threshold of bar structures and the feedback efficiency of galaxy mergers.Mesoscopic-scale biophysical modelling: Although Newtonian mechanics fails at the quantum scale, its ideas of “hierarchical interaction” and “internal/external energy partitioning” have profound heuristic value. In molecular dynamics simulations, the entire protein can be regarded as the main group of the system, with its structural domains (α-helices, β-sheets, loop regions) as physical subgroups; when a ligand (external group) binds, in addition to analysing the overall binding free energy change (external kinetic energy level), the amplitude of internal conformational fluctuations, the rate of hydrogen bond network breakage/reformation, and the activation intensity of low-frequency collective modes (redistribution of internal kinetic energy) in specific structural domains can also be quantitatively calculated, thereby establishing a causal link from molecular binding events to biological functional outputs at the mechanical level.