--- { "id": "ASD-DOC-EN-THEORY-DYNAMICS-ENERGY", "path": "/en/theory/dynamics/energy", "title": "Dynamics · Energy and Action", "title_en": "Dynamics · Energy and Action", "category": "Foundations & Dynamics", "summary": "Retain the work integral and substitution, distinguish energy differences from rest energy, and construct and check the free Lagrangian.", "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.4 (2.22)—(2.24b)", "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/dynamics/force-and-work" ], "related": [ "/en/theory/dynamics/tests" ], "open_problems": [ "/open-problems/inertial-closure" ], "equations": [ "dyn-work-integral", "dyn-kinetic-energy", "dyn-rest-energy", "dyn-total-energy", "dyn-energy-identity", "dyn-lagrangian" ], "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 energy reference · Starting Assumptions", "From momentum to work · Work Integral", "Essential integration steps · Intermediate Steps", "Rest energy and total energy · Rest and Total Energy", "Energy–momentum consistency · Algebraic Consistency", "Free-motion action · Free Lagrangian", "Physical meaning, scope and review · Interpretation and Review", "Sources and experimental interface · Sources" ], "seo_title": "Dynamics · Energy and Action | Anisotropic Space Dynamics", "canonical_url": "https://asd-physics.org/en/theory/dynamics/energy", "translation_path": "/theory/dynamics/energy", "equation": "dyn-work-integral", "translation_of": "/theory/dynamics/energy", "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.4 (2.22)—(2.24b)", "locators": [ "p0654", "p0656", "p0658", "p0660", "p0661", "p0663", "p0666", "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 Use $\mathbf p=m_0\gamma\mathbf v$ and $dE_k=\mathbf v\cdot d\mathbf p$. The background is uniform, internal state fixed, and net radiative loss absent. Set kinetic energy to zero at $v=0$. Source: Eqs. (2.14) and (2.22). For total energy, Chapter 2 adopts $E_0=m_0c^2$. This is not needed to integrate the kinetic-energy difference. It is a normalization input linking static structural energy to the inertia parameter. Source: text between Eqs. (2.23) and (2.24). ## From momentum to work · Work Integral Differentiate $p=m_0\gamma v$: $$ \frac{dp}{dv}=m_0\left(\gamma+\gamma^3\frac{v^2}{c^2}\right)=m_0\gamma^3. $$ Kinetic energy from rest to speed $v$ is therefore: $w$ is the integration speed. For a general change of direction, $\mathbf v\cdot d\mathbf p$ retains only the speed-changing component. Under the stated conditions, kinetic energy is a function of speed. ## Essential integration steps · Intermediate Steps Set $z=1-w^2/c^2$, giving $w\,dw=-c^2dz/2$. The lower limit $w=0$ gives $z=1$, and the upper limit $w=v$ gives $z=1-v^2/c^2$: $$ \begin{aligned} E_k&=-\frac{m_0c^2}{2}\int_1^{1-v^2/c^2}z^{-3/2}\,dz\\ &=-\frac{m_0c^2}{2}\left[-2z^{-1/2}\right]_1^{1-v^2/c^2}\\ &=m_0c^2\left[\left(1-\frac{v^2}{c^2}\right)^{-1/2}-1\right]. \end{aligned} $$ This is the unnumbered calculation following manuscript Eq. (2.22); it does not work backwards from a standard energy formula. Using $\gamma=1+\beta^2/2+3\beta^4/8+O(\beta^6)$ recovers Newtonian kinetic energy and its next correction. Both coefficients come from the expansion of one function. ## Rest energy and total energy · Rest and Total Energy ASD treats a structure at rest as carrying strain and structural energy. To connect this energy to its inertia parameter, the chapter adopts: The work integral determines energy differences and cannot fix $E_0$ by itself. With this normalization: Thus $E=m_0\gamma c^2$ combines the kinetic-energy result and A2. A microscopic theory must compute $E_0$ and $m_0$ from the same stable structural solution to test the normalization; it is not a fact established by this integral alone. ## Energy–momentum consistency · Algebraic Consistency $$ E^2-c^2p^2 =m_0^2c^4\gamma^2\left(1-\frac{v^2}{c^2}\right) =m_0^2c^4. $$ This is an algebraic consistency relation after eliminating $v$. In this chapter it follows $p$ and $E$; a presumed Minkowski invariant does not replace ASD’s constitutive derivation. ## Free-motion action · Free Lagrangian Require the free translational Lagrangian to satisfy $\partial\mathcal L_0/\partial\mathbf v=\mathbf p$. Integrate over speed: $$ \mathcal L_0(v)-\mathcal L_0(0) =\int_0^v\frac{m_0w}{\sqrt{1-w^2/c^2}}\,dw =m_0c^2\left(1-\sqrt{1-v^2/c^2}\right). $$ Fix the constant with $\mathcal L_0(0)=-E_0=-m_0c^2$, giving: Direct differentiation recovers $m_0\gamma\mathbf v$. The Legendre transform gives $$ \mathbf p\cdot\mathbf v-\mathcal L_0 =m_0c^2\left(\gamma\beta^2+\frac1\gamma\right) =m_0c^2\gamma. $$ The website uses $\mathcal L_0$ to distinguish the Lagrangian from length $L$; it corresponds to $L_0$ in manuscript Eq. (2.24b). Its low-speed expansion is $$ \mathcal L_0=-m_0c^2+\frac{m_0v^2}{2}+\frac{m_0v^4}{8c^2}+O\!\left(\frac{m_0v^6}{c^4}\right). $$ Chapter 2’s later orbital calculation must retain the $v^4$ term alongside gravitational terms of the same order. This module supplies that interface without developing the orbital approximation here. ## Physical meaning, scope and review · Interpretation and Review Kinetic energy describes reconfiguration of the moving structure; rest energy describes the static core and accompanying structure. Matching mathematical forms do not remove these distinct physical roles. For $0\leq v