Dynamics · Inertia and Momentum

01Premises and quantities · Starting Assumptions

This page retains Chapter 2’s chain from core structure to momentum. A1–A4 are website premise indices. Upstream LVE keeps its v1.1 conditional mathematical review; using it in dynamics does not make it unconditional.

02Near-core mass and effective volume · Mass and Effective Volume

It follows that $m_0=\rho_0V_{\mathrm{eff}}=4\pi\rho_0R_{\mathrm{eff}}^3/3$. The normalization in this equality must remain attached to the momentum result.

Chapter 2’s far-field $\Phi$ is a line-density coefficient, whereas $q_\rho$ is a near-core volume-density ratio. They are not interchangeable. Inserting an external $1/r$ line field directly into an all-space volume-deficit integral is not the construction of Eq. (2.8); extending the domain requires a separate convergence argument.

03LVE and axial gain · Directional Scale

Set $\cos\phi=+1,-1$ along the forward and backward axes:

The intermediate algebra is $K_+K_-=\gamma^2(1-\beta^2)=1$. Hence $K_+^{-1}=K_-$ and $K_-^{-1}=K_+$, whose average is $\gamma$. Up to this point, the result concerns the algebra of two axial factors.

04Inertial angular distribution · Constitutive Closure

$\rho_I$ has density dimensions and is an inertial weight density, not the actual volume density at every point. In general directions, $K(\phi)^{-1}\ne\gamma(1+\beta\cos\phi)$. At $\phi=\pi/2$, for example, these are $1/\gamma$ and $\gamma$. Equation (2.11) therefore cannot silently be derived by identifying density with inverse length in every direction.

05Total momentum and intermediate steps · Momentum Integration

In the chapter’s effective construction, $\rho_I$ retains only angular dependence. Integrating term by term:

Consequently,

Multiplying and applying A2 yields:

The odd directional term contributes zero to this integral. The mean gain $\gamma$ contributes to the total momentum. These steps retain the algebra needed between Eqs. (2.12) and (2.14).

06Structural meaning · Physical Interpretation

ASD interprets $\gamma$ as the inertial gain of the moving spatial structure. Changing the motion of a core and its accompanying structure changes the momentum of the joint system. The matching mathematical form is not interpreted as a change of rest mass caused by a time-flow rate.

Reconfiguration response is not dissipative drag during uniform motion. Maintaining free uniform motion costs no continuous work in this construction; work is associated with changing the state.

07Validity and consistency · Domain and Checks

• Rest limit: As $\beta\to0$, $\rho_I\to\rho_0$, $\gamma\to1$ and $p\to m_0v$.
• Dimensions and positivity: $\rho_I$ is mass per volume; the integral has mass-times-speed dimensions. For $\beta<1$, $1+\beta\cos\phi>0$.
• Symmetry: The angular mean of the directional bias vanishes, while $\mathbf v$ still determines the direction of total motion.
• Domain: Do not cross $v=c$ or apply the fixed-state $m_0$ to radiation or internal excitation without extending the model.

A common mean gain does not uniquely fix an angular distribution. Higher terms with zero angular mean can leave this integral unchanged; spatial dynamics must select the actual profile. This check does not replace the manuscript’s lowest-order constitutive choice.

08Mathematical review and inputs · Review and Dependencies

1. LVE input: Retain the upstream review of A5’s pairing and mapping conditions.
2. Constitutive origin: Derive the gain and angular distribution of Eq. (2.11) from closed spatial dynamics and a stable core solution, beyond the axial identity.
3. Core matching: Relate $q_\rho$, the effective-core integral and a real nonuniform structure, and establish a finite mass integral.
4. Common normalization: A single structural solution must test whether $C_m=1$, $E_0=m_0c^2$ and the universal coupling of Eq. (2.24c) hold simultaneously.

OP-DYN-01 (Chinese) · Chinese original collects these tasks. The angular integration is closed under A1–A4. Mathematical review remains CONDITIONAL to preserve the physical inputs.

09Sources and continuation · Sources

This page corresponds to Chapter 2 §III, final manuscript §III. Continue to Force and Work for $d\mathbf p/dt$, or Energy for $\int\mathbf v\cdot d\mathbf p$.