--- { "id": "ASD-DOC-EN-THEORY-DYNAMICS-MOMENTUM", "path": "/en/theory/dynamics/momentum", "title": "Dynamics · Inertia and Momentum", "title_en": "Dynamics · Inertia and Momentum", "category": "Foundations & Dynamics", "summary": "Trace the near-core density deficit, axial LVE gain, inertial constitutive closure and angular integration of total momentum.", "template": "theory", "version": "1.0.0", "updated": "2026-10-24", "canonical": true, "source_tier": "canonical", "theory_status": "DERIVED", "evidence_status": "UNTESTED", "locale": "en", "module_key": "dynamics", "module_navigation": [ { "path": "/en/theory/dynamics", "label": "Overview", "en": "Overview" }, { "path": "/en/theory/dynamics/momentum", "label": "Inertia & Momentum", "en": "Inertia & Momentum" }, { "path": "/en/theory/dynamics/energy", "label": "Energy & Action", "en": "Energy & Action" }, { "path": "/en/theory/dynamics/force-and-work", "label": "Force & Work", "en": "Force & Work" }, { "path": "/en/theory/dynamics/tests", "label": "Tests", "en": "Tests" } ], "canonical_scope": "Preface · Chapters 1–2", "source_chapter": "Chapter 2", "source_reference": "§III.1—2 (2.8)—(2.14)", "source_version": "ASD-BOOK-ddbade997b45", "source_sha256": "dace8ff491b4e4ae8a59b2b98e2a6c9c10e689079021ce0bc117985baba68e86", "source_kind": "manifest", "source_records": [ { "id": "ASD-BOOK-ddbade997b45", "title": "各向异性空间动力学（最终合订稿）", "filename": "各向异性空间动力学.docx", "format": "docx", "size_bytes": 503564, "sha256": "ddbade997b45913248e588953b1d27ccb9c7c206c3c8d992d3aef2364a64836e" } ], "last_reviewed": "2026-09-14", "depends_on": [ "/en/theory/lve/derivation", "/en/theory/time" ], "related": [ "/en/theory/dynamics", "/en/theory/dynamics/force-and-work", "/en/theory/dynamics/energy" ], "open_problems": [ "/open-problems/inertial-closure" ], "equations": [ "dyn-mass-deficit", "dyn-effective-volume", "dyn-lve-input", "dyn-axial-gain", "dyn-inertial-density", "dyn-momentum-integral", "dyn-momentum" ], "sources": [ { "label": "Final combined manuscript — adopted scope documented on this page", "kind": "final-combined-manuscript", "url": "/downloads/ASD_Final_Manuscript_Source_Manifest.json" }, { "label": "Previous source: PREF-125df9543371 · 序章.pdf", "kind": "historical-source" }, { "label": "Previous source: CH1-46d43c8c39bf · 第一章(2).pdf", "kind": "historical-source" }, { "label": "Previous source: CH2-f9598318d123 · 第二章-ASD理论-修订书稿 (2).pdf", "kind": "historical-source" } ], "outline": [ "Premises and quantities · Starting Assumptions", "Near-core mass and effective volume · Mass and Effective Volume", "LVE and axial gain · Directional Scale", "Inertial angular distribution · Constitutive Closure", "Total momentum and intermediate steps · Momentum Integration", "Structural meaning · Physical Interpretation", "Validity and consistency · Domain and Checks", "Mathematical review and inputs · Review and Dependencies", "Sources and continuation · Sources" ], "seo_title": "Dynamics · Inertia and Momentum | Anisotropic Space Dynamics", "canonical_url": "https://asd-physics.org/en/theory/dynamics/momentum", "translation_path": "/theory/dynamics/momentum", "equation": "dyn-mass-deficit", "translation_of": "/theory/dynamics/momentum", "translation_version": "1.0.0", "mathematical_review": "CONDITIONAL", "review_path": "/en/theory/dynamics/momentum#review-notes", "review_note": "Momentum follows after the inertial constitutive closure; total energy additionally uses E₀ = m₀c². These inputs remain explicit and require microscopic matching.", "source_manifest": "final-manuscript-source-manifest.json", "source_manifest_url": "/downloads/ASD_Final_Manuscript_Source_Manifest.json", "final_source": { "source_id": "ASD-BOOK-ddbade997b45", "chapter": "Chapter 2", "reference": "§III.1—2 (2.8)—(2.14)", "locators": [ "p0610", "p0613", "p0615", "p0620", "p0624", "p0628", "p0630", "p0936" ], "adopted_scope": "Core–accompanying-field inertial constitution, momentum integration, inverse force response and energy; retain correct existing derivations.", "conditions": [ "Uniform background, fixed internal state, 0≤v Consider a localized material structure in a uniform background, with fixed internal state and common translational velocity $\mathbf v$. $\rho_0$ is the calibrated background volume density; $q_\rho$ is the near-core volume-density ratio. The effective-core deficit integral must be finite and give positive $m_0$. Source: Chapter 2, §III, final manuscript §III. Define $R_{\mathrm{eff}}$ from the equal-deficit volume. Adopt $m_0=C_m m_{\mathrm{def}}$ with $C_m=1$. $R_{\mathrm{eff}}$ is an effective integration radius, not automatically a measured geometric particle radius. Source: Eqs. (2.8)–(2.9) and adjacent text. Use Chapter 1’s $K(\phi)=\gamma(1-\beta\cos\phi)$, with $0\leq\beta<1$. $\phi$ is measured relative to velocity, and $K$ describes ASD true-length relations. Source: Eq. (2.10); [LVE derivation and review](/en/theory/lve/derivation#review-notes). Use the symmetric gain of inverse axial scales as the overall inertial gain and the lowest angular harmonic for the forward/backward bias. Set $\rho_I=\rho_0\gamma(1+\beta\cos\phi)$ and momentum density $\boldsymbol\pi=\rho_I\mathbf v$. Chapter 2 explicitly introduces this constitutive relation; it is not a geometric identity of LVE. Source: Eqs. (2.11)–(2.12). ## Near-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. ## LVE 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. ## Inertial angular distribution · Constitutive Closure Interpreting the average as the overall inertial gain, and adopting the lowest angular harmonic $1+\beta\cos\phi$ for the bias, gives the manuscript’s 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. ## Total momentum and intermediate steps · Momentum Integration In the chapter’s effective construction, $\rho_I$ retains only angular dependence. Integrating term by term: $$ \int_0^\pi\sin\phi\,d\phi=2, \qquad \int_0^\pi\cos\phi\sin\phi\,d\phi=0. $$ Consequently, $$ \int_0^{2\pi}d\alpha\int_0^\pi\rho_I\sin\phi\,d\phi =4\pi\rho_0\gamma, \qquad \int_0^{R_{\mathrm{eff}}}r^2dr=\frac{R_{\mathrm{eff}}^3}{3}. $$ 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). ## Structural 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. ## Validity 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. ## Mathematical 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)](/open-problems/inertial-closure) collects these tasks. The angular integration is closed under A1–A4. Mathematical review remains CONDITIONAL to preserve the physical inputs. ## Sources and continuation · Sources This page corresponds to Chapter 2 §III, final manuscript §III. Continue to [Force and Work](/en/theory/dynamics/force-and-work) for $d\mathbf p/dt$, or [Energy](/en/theory/dynamics/energy) for $\int\mathbf v\cdot d\mathbf p$.