--- { "id": "ASD-DOC-EN-THEORY-DYNAMICS-FORCE-AND-WORK", "path": "/en/theory/dynamics/force-and-work", "title": "Dynamics · Force and Work", "title_en": "Dynamics · Force and Work", "category": "Foundations & Dynamics", "summary": "Differentiate momentum to obtain directional force response, low-speed limits and circular motion in a common calibration.", "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.3—4 (2.15)—(2.23)", "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/dynamics/momentum", "/en/theory/time/measurement-and-process" ], "related": [ "/en/theory/dynamics/energy", "/en/theory/dynamics/tests" ], "open_problems": [ "/open-problems/inertial-closure" ], "equations": [ "dyn-force-definition", "dyn-force-expansion", "dyn-gamma-dot", "dyn-force-tensor", "dyn-parallel", "dyn-newton-limit", "dyn-transverse", "dyn-circular", "dyn-force-inverse" ], "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 reference parameter · Starting Assumptions", "Differentiating total momentum · Momentum Derivative", "The response by direction · Directional Response", "Longitudinal, transverse and circular motion · Special Cases", "From work to kinetic energy · Work and Energy", "Physical meaning and domain · Interpretation and Validity", "Consistency and sources · Checks and Sources" ], "seo_title": "Dynamics · Force and Work | Anisotropic Space Dynamics", "canonical_url": "https://asd-physics.org/en/theory/dynamics/force-and-work", "translation_path": "/theory/dynamics/force-and-work", "equation": "dyn-force-definition", "translation_of": "/theory/dynamics/force-and-work", "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.3—4 (2.15)—(2.23)", "locators": [ "p0637", "p0643", "p0647", "p0649", "p0650", "p0658", "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 Retain the core–accompanying-field construction of [Inertia and Momentum](/en/theory/dynamics/momentum), with fixed $m_0$, a uniform background and no net radiative loss. $t$ is Chapter 2’s common reference counting parameter. Define force as the rate of change of total momentum with respect to the reference parameter. Mechanical work is $dW=\mathbf F\cdot d\mathbf x$, with $d\mathbf x=\mathbf v\,dt$. Path, velocity, force and counting use one calibration. Under these conditions $dW=\mathbf v\cdot d\mathbf p$. Internal excitation, radiation or a changing background requires its own momentum flux and energy exchange rather than only the fixed-$m_0$ translational term. ## Differentiating total momentum · Momentum Derivative Apply the product rule to $\mathbf p=m_0\gamma\mathbf v$: Apply the chain rule to $\gamma=(1-\mathbf v\cdot\mathbf v/c^2)^{-1/2}$: $$ \frac{d\gamma}{dt} =\frac12\left(1-\frac{v^2}{c^2}\right)^{-3/2} \frac{2\mathbf v\cdot\mathbf a}{c^2}. $$ This differentiates the scale factor along the motion process. No time-dilation premise is inserted alongside the product rule. ## The response by direction · Directional Response Substitute $\dot\gamma$ into the force expression: Resolve $\mathbf a=\mathbf a_\parallel+\mathbf a_\perp$, with $\mathbf v\cdot\mathbf a_\perp=0$. The coefficient along velocity is $$ \gamma+\gamma^3\frac{v^2}{c^2} =\gamma(1+\gamma^2\beta^2)=\gamma^3, $$ whereas the perpendicular coefficient is $\gamma$. Both coefficients arise from the derivative of one momentum with respect to velocity, not from two rest masses. At $v=0$, the directional decomposition is arbitrary and the total response approaches $m_0\mathbf a$ continuously. ### Inverting the force response The longitudinal and transverse eigenvalues of the response tensor are $m_0\gamma^3$ and $m_0\gamma$. Both are positive for $m_0>0$ and $v This is final-manuscript equation (2.20). It gives the acceleration produced by a specified total external force, while (2.18) gives the force required for a specified acceleration. At zero speed it continuously becomes $\mathbf a=\mathbf F/m_0$. The separately retained low-speed Newtonian expansion is a website check of (2.18), not the content of final equation (2.20). ## Longitudinal, transverse and circular motion · Special Cases For uniform circular motion in one local calibration, $a_\perp=v^2/R$, so Here $v$ and $R$ use the same local length and counting reference. Since $\gamma$ already enters momentum, replacing $R$ by $\gamma R$ without an additional measurement map would count a scale effect again. Comparing a different length realization requires that separate map first. Chapter 2 explicitly preserves this condition after Eq. (2.21). ## From work to kinetic energy · Work and Energy $$ dW=\mathbf F\cdot d\mathbf x =\frac{d\mathbf p}{dt}\cdot\mathbf v\,dt =\mathbf v\cdot d\mathbf p. $$ When only speed changes, $dp/dv=m_0\gamma^3$, allowing integration from rest to $v$. A change of momentum perpendicular to velocity contributes zero to the dot product even when the direction changes. See [Energy and Action](/en/theory/dynamics/energy) for the complete integral and total-energy normalization. A work-derived energy difference does not already determine rest energy. ## Physical meaning and domain · Interpretation and Validity ASD interprets the directional force response as reconfiguration of a moving spatial structure. Longitudinal acceleration changes speed and mean gain; transverse acceleration changes direction. The instantaneous force in uniform circular motion is perpendicular to velocity and does not change kinetic energy in this model. If a charged structure radiates, that radiation must be added. The expressions apply at fixed mass, in a uniform background, with differentiable velocity and $v