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The key data structure\n`DampedHarmonicOscillator` extends `HarmonicOscillator` (inheriting the positive\nmass `m`, spring constant `k`, and the undamped natural frequency / energy API)\nand adds a nonnegative damping coefficient `γ`. Position and velocity are modelled\nas `EuclideanSpace ℝ (Fin 1)` and trajectories as maps `Time → EuclideanSpace ℝ (Fin 1)`.\n\nThe `Basic` module defines the equation of motion, the force `- k x - γ ẋ` and its\nequivalence with Newton's second law, the rate of mechanical-energy dissipation\nalong solutions, the three damping regimes (underdamped, critically damped,\noverdamped) selected by the sign of the discriminant `γ² - 4 m k` together with\nthe decay rate and regime-selected angular frequency, the reduction to the\nundamped harmonic oscillator when `γ = 0`, and the Caldirola–Kanai lagrangian\n`exp (γ/m * t) * (T - V)`, whose variational gradient vanishes if and only if\nthe equation of motion holds. 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The key data structure\n`DampedHarmonicOscillator` extends `HarmonicOscillator` (inheriting the positive\nmass `m`, spring constant `k`, and the undamped natural frequency / energy API)\nand adds a nonnegative damping coefficient `γ`. Position and velocity are modelled\nas `EuclideanSpace ℝ (Fin 1)` and trajectories as maps `Time → EuclideanSpace ℝ (Fin 1)`.\n\nThe `Basic` module defines the equation of motion, the force `- k x - γ ẋ` and its\nequivalence with Newton's second law, the rate of mechanical-energy dissipation\nalong solutions, the three damping regimes (underdamped, critically damped,\noverdamped) selected by the sign of the discriminant `γ² - 4 m k` together with\nthe decay rate and regime-selected angular frequency, the reduction to the\nundamped harmonic oscillator when `γ = 0`, and the Caldirola–Kanai lagrangian\n`exp (γ/m * t) * (T - V)`, whose variational gradient vanishes if and only if\nthe equation of motion holds. 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The key data structure\n`DampedHarmonicOscillator` extends `HarmonicOscillator` (inheriting the positive\nmass `m`, spring constant `k`, and the undamped natural frequency / energy API)\nand adds a nonnegative damping coefficient `γ`. Position and velocity are modelled\nas `EuclideanSpace ℝ (Fin 1)` and trajectories as maps `Time → EuclideanSpace ℝ (Fin 1)`.\n\nThe `Basic` module defines the equation of motion, the force `- k x - γ ẋ` and its\nequivalence with Newton's second law, the rate of mechanical-energy dissipation\nalong solutions, the three damping regimes (underdamped, critically damped,\noverdamped) selected by the sign of the discriminant `γ² - 4 m k` together with\nthe decay rate and regime-selected angular frequency, the reduction to the\nundamped harmonic oscillator when `γ = 0`, and the Caldirola–Kanai lagrangian\n`exp (γ/m * t) * (T - V)`, whose variational gradient vanishes if and only if\nthe equation of motion holds. 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The key data structure\n`DampedHarmonicOscillator` extends `HarmonicOscillator` (inheriting the positive\nmass `m`, spring constant `k`, and the undamped natural frequency / energy API)\nand adds a nonnegative damping coefficient `γ`. Position and velocity are modelled\nas `EuclideanSpace ℝ (Fin 1)` and trajectories as maps `Time → EuclideanSpace ℝ (Fin 1)`.\n\nThe `Basic` module defines the equation of motion, the force `- k x - γ ẋ` and its\nequivalence with Newton's second law, the rate of mechanical-energy dissipation\nalong solutions, the three damping regimes (underdamped, critically damped,\noverdamped) selected by the sign of the discriminant `γ² - 4 m k` together with\nthe decay rate and regime-selected angular frequency, the reduction to the\nundamped harmonic oscillator when `γ = 0`, and the Caldirola–Kanai lagrangian\n`exp (γ/m * t) * (T - V)`, whose variational gradient vanishes if and only if\nthe equation of motion holds. 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The key data structure\n`DampedHarmonicOscillator` extends `HarmonicOscillator` (inheriting the positive\nmass `m`, spring constant `k`, and the undamped natural frequency / energy API)\nand adds a nonnegative damping coefficient `γ`. Position and velocity are modelled\nas `EuclideanSpace ℝ (Fin 1)` and trajectories as maps `Time → EuclideanSpace ℝ (Fin 1)`.\n\nThe `Basic` module defines the equation of motion, the force `- k x - γ ẋ` and its\nequivalence with Newton's second law, the rate of mechanical-energy dissipation\nalong solutions, the three damping regimes (underdamped, critically damped,\noverdamped) selected by the sign of the discriminant `γ² - 4 m k` together with\nthe decay rate and regime-selected angular frequency, the reduction to the\nundamped harmonic oscillator when `γ = 0`, and the Caldirola–Kanai lagrangian\n`exp (γ/m * t) * (T - V)`, whose variational gradient vanishes if and only if\nthe equation of motion holds. 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The key data structure\n`DampedHarmonicOscillator` extends `HarmonicOscillator` (inheriting the positive\nmass `m`, spring constant `k`, and the undamped natural frequency / energy API)\nand adds a nonnegative damping coefficient `γ`. Position and velocity are modelled\nas `EuclideanSpace ℝ (Fin 1)` and trajectories as maps `Time → EuclideanSpace ℝ (Fin 1)`.\n\nThe `Basic` module defines the equation of motion, the force `- k x - γ ẋ` and its\nequivalence with Newton's second law, the rate of mechanical-energy dissipation\nalong solutions, the three damping regimes (underdamped, critically damped,\noverdamped) selected by the sign of the discriminant `γ² - 4 m k` together with\nthe decay rate and regime-selected angular frequency, the reduction to the\nundamped harmonic oscillator when `γ = 0`, and the Caldirola–Kanai lagrangian\n`exp (γ/m * t) * (T - V)`, whose variational gradient vanishes if and only if\nthe equation of motion holds. 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The key data structure\n`DampedHarmonicOscillator` extends `HarmonicOscillator` (inheriting the positive\nmass `m`, spring constant `k`, and the undamped natural frequency / energy API)\nand adds a nonnegative damping coefficient `γ`. Position and velocity are modelled\nas `EuclideanSpace ℝ (Fin 1)` and trajectories as maps `Time → EuclideanSpace ℝ (Fin 1)`.\n\nThe `Basic` module defines the equation of motion, the force `- k x - γ ẋ` and its\nequivalence with Newton's second law, the rate of mechanical-energy dissipation\nalong solutions, the three damping regimes (underdamped, critically damped,\noverdamped) selected by the sign of the discriminant `γ² - 4 m k` together with\nthe decay rate and regime-selected angular frequency, the reduction to the\nundamped harmonic oscillator when `γ = 0`, and the Caldirola–Kanai lagrangian\n`exp (γ/m * t) * (T - V)`, whose variational gradient vanishes if and only if\nthe equation of motion holds. 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The key data structure\n`DampedHarmonicOscillator` extends `HarmonicOscillator` (inheriting the positive\nmass `m`, spring constant `k`, and the undamped natural frequency / energy API)\nand adds a nonnegative damping coefficient `γ`. Position and velocity are modelled\nas `EuclideanSpace ℝ (Fin 1)` and trajectories as maps `Time → EuclideanSpace ℝ (Fin 1)`.\n\nThe `Basic` module defines the equation of motion, the force `- k x - γ ẋ` and its\nequivalence with Newton's second law, the rate of mechanical-energy dissipation\nalong solutions, the three damping regimes (underdamped, critically damped,\noverdamped) selected by the sign of the discriminant `γ² - 4 m k` together with\nthe decay rate and regime-selected angular frequency, the reduction to the\nundamped harmonic oscillator when `γ = 0`, and the Caldirola–Kanai lagrangian\n`exp (γ/m * t) * (T - V)`, whose variational gradient vanishes if and only if\nthe equation of motion holds. 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The key data structure\n`DampedHarmonicOscillator` extends `HarmonicOscillator` (inheriting the positive\nmass `m`, spring constant `k`, and the undamped natural frequency / energy API)\nand adds a nonnegative damping coefficient `γ`. Position and velocity are modelled\nas `EuclideanSpace ℝ (Fin 1)` and trajectories as maps `Time → EuclideanSpace ℝ (Fin 1)`.\n\nThe `Basic` module defines the equation of motion, the force `- k x - γ ẋ` and its\nequivalence with Newton's second law, the rate of mechanical-energy dissipation\nalong solutions, the three damping regimes (underdamped, critically damped,\noverdamped) selected by the sign of the discriminant `γ² - 4 m k` together with\nthe decay rate and regime-selected angular frequency, the reduction to the\nundamped harmonic oscillator when `γ = 0`, and the Caldirola–Kanai lagrangian\n`exp (γ/m * t) * (T - V)`, whose variational gradient vanishes if and only if\nthe equation of motion holds. The `Solution` module defines the\ninitial conditions, the explicit closed-form `trajectory` selected from the\ndamping regime (trigonometric, polynomial, or hyperbolic, each times an\nexponential decay factor), and proves that the selected trajectory satisfies the\nequation of motion.","license_root":"sha256:c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4","lakefile_root":"sha256:9b6608c26fa30873fb8d823eadd7f66835992389a80c3e44b62e9803c8c65c01","ai_policy_root":"sha256:a639a2c51091911123600a376c7556b52d92ffd324b041df2e2a66ec7597ed49","api_map_schema":"v0.1","lean_toolchain":"leanprover/lean4:v4.32.0","api_map_guide_root":"sha256:0663b6828ab7635bd69a1bc5913d75c5f1b81640071f43f3db0e5e10f5d29ad1","lake_manifest_root":"sha256:a9531c3978f21e50cb72cc89c2a7644b296bee59a7f2ff92ca2408aac4a67311","native_null_fields":"[]","agent_guidance_root":"sha256:487a9754f996304b173dd871326a094f4552d27df7151d60c56d83374e7711ca","lean_toolchain_root":"sha256:cc346fd7850a83fb7af3902dfc378f74d8eb8c59c6d34322488269ebfe7615b1","requirement_ordinal":4,"source_declared_done":true,"source_declared_state":"implemented","review_guidelines_root":"sha256:e28f680f6fc0767840942aaa48da28bd9983cd48db8beec4ab77dafb4dbdab0b","source_declared_location":"Physlib/ClassicalMechanics/DampedHarmonicOscillator/Basic.lean (energy_dissipation_rate, energy_not_conserved)","declared_source_file_paths":"[\"Physlib/ClassicalMechanics/DampedHarmonicOscillator/Basic.lean\"]","declared_source_file_roots":"[\"sha256:f6119ad202abdd618a1a2fdbb75101c9634c603c8188ee95bf199f7eeb242d13\"]"},"metadata_root":"sha256:87ed19c0c33289bffe757b0df23effa80491957180264ef8aa8b1beaaec564f6","content_root":"sha256:b8d08e05c871667af9d708e8e33b4e6bd2bb80f91f67e87ad49e2e879270f5fb","availability":"reference_only","row_root":"sha256:fbf133ae1d5828c741ed0dc4fc0154b77ddc998e8f6c060abe5f95219a13c6eb"},{"schema":"vela.math-native-record.v1","source_id":"source:physlib","observation_root":"sha256:fce692e559477ec2cbcab6a5931c35bb0a903ff861f4100872eb141f3ed3e6f9","native_id":"api-map:Physlib/ClassicalMechanics/DampedHarmonicOscillator/API-map.yaml#requirement:5","native_kind":"api_requirement","native_revision":"e882411d1b6bcbdfdd336d4c509c6cc72e96842d","title":"The API classifies the underdamped, critically damped, and overdamped regimes via the discriminant `γ² - 4 m k`, and defines the decay rate and the regime-selected angular frequency.\n","summary":"Damped harmonic oscillator · source reports implemented","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/leanprover-community/physlib/blob/e882411d1b6bcbdfdd336d4c509c6cc72e96842d/Physlib/ClassicalMechanics/DampedHarmonicOscillator/API-map.yaml"},{"locator_id":"native-2","kind":"artifact","url":"https://github.com/leanprover-community/physlib/blob/e882411d1b6bcbdfdd336d4c509c6cc72e96842d/Physlib/ClassicalMechanics/DampedHarmonicOscillator/Basic.lean"}],"metadata":{"api_title":"Damped harmonic oscillator","references":"[\"Landau & Lifshitz, Mechanics, page 76, section 25.\",\"Goldstein, Classical Mechanics, Chapter 2.\",\"Caldirola, Nuovo Cimento 18 (1941) 393.\",\"Kanai, Progress of Theoretical Physics 3 (1948) 440.\"]","parent_apis":"[\"Harmonic oscillator (Physlib/ClassicalMechanics/HarmonicOscillator)\",\"Time (Physlib/SpaceAndTime/Time)\"]","api_map_path":"Physlib/ClassicalMechanics/DampedHarmonicOscillator/API-map.yaml","api_map_root":"sha256:17709dd8ce024919d0a3aafa16908e6771b59ba88df7637171aac6e0fead8a71","api_overview":"The damped harmonic oscillator is a classical mechanical system consisting of a\nmass `m` under a restoring force `- k x` and a damping force `- γ ẋ`, with\nequation of motion `m ẍ + γ ẋ + k x = 0`. The key data structure\n`DampedHarmonicOscillator` extends `HarmonicOscillator` (inheriting the positive\nmass `m`, spring constant `k`, and the undamped natural frequency / energy API)\nand adds a nonnegative damping coefficient `γ`. Position and velocity are modelled\nas `EuclideanSpace ℝ (Fin 1)` and trajectories as maps `Time → EuclideanSpace ℝ (Fin 1)`.\n\nThe `Basic` module defines the equation of motion, the force `- k x - γ ẋ` and its\nequivalence with Newton's second law, the rate of mechanical-energy dissipation\nalong solutions, the three damping regimes (underdamped, critically damped,\noverdamped) selected by the sign of the discriminant `γ² - 4 m k` together with\nthe decay rate and regime-selected angular frequency, the reduction to the\nundamped harmonic oscillator when `γ = 0`, and the Caldirola–Kanai lagrangian\n`exp (γ/m * t) * (T - V)`, whose variational gradient vanishes if and only if\nthe equation of motion holds. 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The key data structure\n`DampedHarmonicOscillator` extends `HarmonicOscillator` (inheriting the positive\nmass `m`, spring constant `k`, and the undamped natural frequency / energy API)\nand adds a nonnegative damping coefficient `γ`. Position and velocity are modelled\nas `EuclideanSpace ℝ (Fin 1)` and trajectories as maps `Time → EuclideanSpace ℝ (Fin 1)`.\n\nThe `Basic` module defines the equation of motion, the force `- k x - γ ẋ` and its\nequivalence with Newton's second law, the rate of mechanical-energy dissipation\nalong solutions, the three damping regimes (underdamped, critically damped,\noverdamped) selected by the sign of the discriminant `γ² - 4 m k` together with\nthe decay rate and regime-selected angular frequency, the reduction to the\nundamped harmonic oscillator when `γ = 0`, and the Caldirola–Kanai lagrangian\n`exp (γ/m * t) * (T - V)`, whose variational gradient vanishes if and only if\nthe equation of motion holds. 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The key data structure\n`DampedHarmonicOscillator` extends `HarmonicOscillator` (inheriting the positive\nmass `m`, spring constant `k`, and the undamped natural frequency / energy API)\nand adds a nonnegative damping coefficient `γ`. Position and velocity are modelled\nas `EuclideanSpace ℝ (Fin 1)` and trajectories as maps `Time → EuclideanSpace ℝ (Fin 1)`.\n\nThe `Basic` module defines the equation of motion, the force `- k x - γ ẋ` and its\nequivalence with Newton's second law, the rate of mechanical-energy dissipation\nalong solutions, the three damping regimes (underdamped, critically damped,\noverdamped) selected by the sign of the discriminant `γ² - 4 m k` together with\nthe decay rate and regime-selected angular frequency, the reduction to the\nundamped harmonic oscillator when `γ = 0`, and the Caldirola–Kanai lagrangian\n`exp (γ/m * t) * (T - V)`, whose variational gradient vanishes if and only if\nthe equation of motion holds. The `Solution` module defines the\ninitial conditions, the explicit closed-form `trajectory` selected from the\ndamping regime (trigonometric, polynomial, or hyperbolic, each times an\nexponential decay factor), and proves that the selected trajectory satisfies the\nequation of motion.","license_root":"sha256:c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4","lakefile_root":"sha256:9b6608c26fa30873fb8d823eadd7f66835992389a80c3e44b62e9803c8c65c01","ai_policy_root":"sha256:a639a2c51091911123600a376c7556b52d92ffd324b041df2e2a66ec7597ed49","api_map_schema":"v0.1","lean_toolchain":"leanprover/lean4:v4.32.0","api_map_guide_root":"sha256:0663b6828ab7635bd69a1bc5913d75c5f1b81640071f43f3db0e5e10f5d29ad1","lake_manifest_root":"sha256:a9531c3978f21e50cb72cc89c2a7644b296bee59a7f2ff92ca2408aac4a67311","native_null_fields":"[]","agent_guidance_root":"sha256:487a9754f996304b173dd871326a094f4552d27df7151d60c56d83374e7711ca","lean_toolchain_root":"sha256:cc346fd7850a83fb7af3902dfc378f74d8eb8c59c6d34322488269ebfe7615b1","requirement_ordinal":7,"source_declared_done":true,"source_declared_state":"implemented","review_guidelines_root":"sha256:e28f680f6fc0767840942aaa48da28bd9983cd48db8beec4ab77dafb4dbdab0b","source_declared_location":"Physlib/ClassicalMechanics/DampedHarmonicOscillator/Solution.lean (InitialConditions)","declared_source_file_paths":"[\"Physlib/ClassicalMechanics/DampedHarmonicOscillator/Solution.lean\"]","declared_source_file_roots":"[\"sha256:0d5a1d1650a59fa3380afc7dee858154c0081f6abd2e5aedcc1e65f4d73c62d3\"]"},"metadata_root":"sha256:3f4588bbc28068be9a80415ccde9b2310350b44b4d800607c232c26492fd0dc8","content_root":"sha256:4d9fc144dce6b05b97f2f9239da41f24bc6d0fa922e4c0df8310d3f0671cba92","availability":"reference_only","row_root":"sha256:21bf1f65c7792b89c14dcee8160890629c9a39aa7a57563a053c2966cc8178e8"},{"schema":"vela.math-native-record.v1","source_id":"source:physlib","observation_root":"sha256:fce692e559477ec2cbcab6a5931c35bb0a903ff861f4100872eb141f3ed3e6f9","native_id":"api-map:Physlib/ClassicalMechanics/DampedHarmonicOscillator/API-map.yaml#requirement:8","native_kind":"api_requirement","native_revision":"e882411d1b6bcbdfdd336d4c509c6cc72e96842d","title":"The API contains a definition of the trajectory of the damped harmonic oscillator, selecting the closed form appropriate to the damping regime.\n","summary":"Damped harmonic oscillator · source reports implemented","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/leanprover-community/physlib/blob/e882411d1b6bcbdfdd336d4c509c6cc72e96842d/Physlib/ClassicalMechanics/DampedHarmonicOscillator/API-map.yaml"},{"locator_id":"native-2","kind":"artifact","url":"https://github.com/leanprover-community/physlib/blob/e882411d1b6bcbdfdd336d4c509c6cc72e96842d/Physlib/ClassicalMechanics/DampedHarmonicOscillator/Solution.lean"}],"metadata":{"api_title":"Damped harmonic oscillator","references":"[\"Landau & Lifshitz, Mechanics, page 76, section 25.\",\"Goldstein, Classical Mechanics, Chapter 2.\",\"Caldirola, Nuovo Cimento 18 (1941) 393.\",\"Kanai, Progress of Theoretical Physics 3 (1948) 440.\"]","parent_apis":"[\"Harmonic oscillator (Physlib/ClassicalMechanics/HarmonicOscillator)\",\"Time (Physlib/SpaceAndTime/Time)\"]","api_map_path":"Physlib/ClassicalMechanics/DampedHarmonicOscillator/API-map.yaml","api_map_root":"sha256:17709dd8ce024919d0a3aafa16908e6771b59ba88df7637171aac6e0fead8a71","api_overview":"The damped harmonic oscillator is a classical mechanical system consisting of a\nmass `m` under a restoring force `- k x` and a damping force `- γ ẋ`, with\nequation of motion `m ẍ + γ ẋ + k x = 0`. The key data structure\n`DampedHarmonicOscillator` extends `HarmonicOscillator` (inheriting the positive\nmass `m`, spring constant `k`, and the undamped natural frequency / energy API)\nand adds a nonnegative damping coefficient `γ`. Position and velocity are modelled\nas `EuclideanSpace ℝ (Fin 1)` and trajectories as maps `Time → EuclideanSpace ℝ (Fin 1)`.\n\nThe `Basic` module defines the equation of motion, the force `- k x - γ ẋ` and its\nequivalence with Newton's second law, the rate of mechanical-energy dissipation\nalong solutions, the three damping regimes (underdamped, critically damped,\noverdamped) selected by the sign of the discriminant `γ² - 4 m k` together with\nthe decay rate and regime-selected angular frequency, the reduction to the\nundamped harmonic oscillator when `γ = 0`, and the Caldirola–Kanai lagrangian\n`exp (γ/m * t) * (T - V)`, whose variational gradient vanishes if and only if\nthe equation of motion holds. The `Solution` module defines the\ninitial conditions, the explicit closed-form `trajectory` selected from the\ndamping regime (trigonometric, polynomial, or hyperbolic, each times an\nexponential decay factor), and proves that the selected trajectory satisfies the\nequation of motion.","license_root":"sha256:c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4","lakefile_root":"sha256:9b6608c26fa30873fb8d823eadd7f66835992389a80c3e44b62e9803c8c65c01","ai_policy_root":"sha256:a639a2c51091911123600a376c7556b52d92ffd324b041df2e2a66ec7597ed49","api_map_schema":"v0.1","lean_toolchain":"leanprover/lean4:v4.32.0","api_map_guide_root":"sha256:0663b6828ab7635bd69a1bc5913d75c5f1b81640071f43f3db0e5e10f5d29ad1","lake_manifest_root":"sha256:a9531c3978f21e50cb72cc89c2a7644b296bee59a7f2ff92ca2408aac4a67311","native_null_fields":"[]","agent_guidance_root":"sha256:487a9754f996304b173dd871326a094f4552d27df7151d60c56d83374e7711ca","lean_toolchain_root":"sha256:cc346fd7850a83fb7af3902dfc378f74d8eb8c59c6d34322488269ebfe7615b1","requirement_ordinal":8,"source_declared_done":true,"source_declared_state":"implemented","review_guidelines_root":"sha256:e28f680f6fc0767840942aaa48da28bd9983cd48db8beec4ab77dafb4dbdab0b","source_declared_location":"Physlib/ClassicalMechanics/DampedHarmonicOscillator/Solution.lean (trajectory)","declared_source_file_paths":"[\"Physlib/ClassicalMechanics/DampedHarmonicOscillator/Solution.lean\"]","declared_source_file_roots":"[\"sha256:0d5a1d1650a59fa3380afc7dee858154c0081f6abd2e5aedcc1e65f4d73c62d3\"]"},"metadata_root":"sha256:f4f1460f52273ceb000a2ff890d17770eaf7cfe4b4aa0c77dacafb4a0a1311d1","content_root":"sha256:df80c0b7cca5fa95dc7eb82eb4923c6faa7137986e7629bbcbdb0a0d807259f1","availability":"reference_only","row_root":"sha256:30cd9d5b166e36cf125f9551cb925b1698a518c104d702a1a97bb96e39685c60"},{"schema":"vela.math-native-record.v1","source_id":"source:physlib","observation_root":"sha256:fce692e559477ec2cbcab6a5931c35bb0a903ff861f4100872eb141f3ed3e6f9","native_id":"api-map:Physlib/ClassicalMechanics/DampedHarmonicOscillator/API-map.yaml#requirement:9","native_kind":"api_requirement","native_revision":"e882411d1b6bcbdfdd336d4c509c6cc72e96842d","title":"The API proves that the selected trajectory satisfies the equation of motion.","summary":"Damped harmonic oscillator · source reports implemented","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/leanprover-community/physlib/blob/e882411d1b6bcbdfdd336d4c509c6cc72e96842d/Physlib/ClassicalMechanics/DampedHarmonicOscillator/API-map.yaml"},{"locator_id":"native-2","kind":"artifact","url":"https://github.com/leanprover-community/physlib/blob/e882411d1b6bcbdfdd336d4c509c6cc72e96842d/Physlib/ClassicalMechanics/DampedHarmonicOscillator/Solution.lean"}],"metadata":{"api_title":"Damped harmonic oscillator","references":"[\"Landau & Lifshitz, Mechanics, page 76, section 25.\",\"Goldstein, Classical Mechanics, Chapter 2.\",\"Caldirola, Nuovo Cimento 18 (1941) 393.\",\"Kanai, Progress of Theoretical Physics 3 (1948) 440.\"]","parent_apis":"[\"Harmonic oscillator (Physlib/ClassicalMechanics/HarmonicOscillator)\",\"Time (Physlib/SpaceAndTime/Time)\"]","api_map_path":"Physlib/ClassicalMechanics/DampedHarmonicOscillator/API-map.yaml","api_map_root":"sha256:17709dd8ce024919d0a3aafa16908e6771b59ba88df7637171aac6e0fead8a71","api_overview":"The damped harmonic oscillator is a classical mechanical system consisting of a\nmass `m` under a restoring force `- k x` and a damping force `- γ ẋ`, with\nequation of motion `m ẍ + γ ẋ + k x = 0`. The key data structure\n`DampedHarmonicOscillator` extends `HarmonicOscillator` (inheriting the positive\nmass `m`, spring constant `k`, and the undamped natural frequency / energy API)\nand adds a nonnegative damping coefficient `γ`. Position and velocity are modelled\nas `EuclideanSpace ℝ (Fin 1)` and trajectories as maps `Time → EuclideanSpace ℝ (Fin 1)`.\n\nThe `Basic` module defines the equation of motion, the force `- k x - γ ẋ` and its\nequivalence with Newton's second law, the rate of mechanical-energy dissipation\nalong solutions, the three damping regimes (underdamped, critically damped,\noverdamped) selected by the sign of the discriminant `γ² - 4 m k` together with\nthe decay rate and regime-selected angular frequency, the reduction to the\nundamped harmonic oscillator when `γ = 0`, and the Caldirola–Kanai lagrangian\n`exp (γ/m * t) * (T - V)`, whose variational gradient vanishes if and only if\nthe equation of motion holds. The `Solution` module defines the\ninitial conditions, the explicit closed-form `trajectory` selected from the\ndamping regime (trigonometric, polynomial, or hyperbolic, each times an\nexponential decay factor), and proves that the selected trajectory satisfies the\nequation of motion.","license_root":"sha256:c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4","lakefile_root":"sha256:9b6608c26fa30873fb8d823eadd7f66835992389a80c3e44b62e9803c8c65c01","ai_policy_root":"sha256:a639a2c51091911123600a376c7556b52d92ffd324b041df2e2a66ec7597ed49","api_map_schema":"v0.1","lean_toolchain":"leanprover/lean4:v4.32.0","api_map_guide_root":"sha256:0663b6828ab7635bd69a1bc5913d75c5f1b81640071f43f3db0e5e10f5d29ad1","lake_manifest_root":"sha256:a9531c3978f21e50cb72cc89c2a7644b296bee59a7f2ff92ca2408aac4a67311","native_null_fields":"[]","agent_guidance_root":"sha256:487a9754f996304b173dd871326a094f4552d27df7151d60c56d83374e7711ca","lean_toolchain_root":"sha256:cc346fd7850a83fb7af3902dfc378f74d8eb8c59c6d34322488269ebfe7615b1","requirement_ordinal":9,"source_declared_done":true,"source_declared_state":"implemented","review_guidelines_root":"sha256:e28f680f6fc0767840942aaa48da28bd9983cd48db8beec4ab77dafb4dbdab0b","source_declared_location":"Physlib/ClassicalMechanics/DampedHarmonicOscillator/Solution.lean (trajectory_equationOfMotion)","declared_source_file_paths":"[\"Physlib/ClassicalMechanics/DampedHarmonicOscillator/Solution.lean\"]","declared_source_file_roots":"[\"sha256:0d5a1d1650a59fa3380afc7dee858154c0081f6abd2e5aedcc1e65f4d73c62d3\"]"},"metadata_root":"sha256:898a3ec833738633d686cd70ec76aa850f748131b4deee088dcf25e0d6c7adc2","content_root":"sha256:c015f6e872bd4581b5e1ad83048439309c38717749b926d24af09f9e96665fe6","availability":"reference_only","row_root":"sha256:14b849d3e076feabdf2b10dc13e5e68ed1dfa1f18eec695c893ce747bcf2eb87"},{"schema":"vela.math-native-record.v1","source_id":"source:physlib","observation_root":"sha256:fce692e559477ec2cbcab6a5931c35bb0a903ff861f4100872eb141f3ed3e6f9","native_id":"api-map:Physlib/ClassicalMechanics/HarmonicOscillator/Geometric/API-map.yaml#requirement:1","native_kind":"api_requirement","native_revision":"e882411d1b6bcbdfdd336d4c509c6cc72e96842d","title":"The key data structure of the API, the configuration space of the classical harmonic oscillator, is defined as a structure carrying a single global coordinate (a one-dimensional Euclidean value), with extensionality in that coordinate and function-like coordinate access.\n","summary":"Configuration space of the harmonic oscillator · source reports implemented","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/leanprover-community/physlib/blob/e882411d1b6bcbdfdd336d4c509c6cc72e96842d/Physlib/ClassicalMechanics/HarmonicOscillator/Geometric/API-map.yaml"},{"locator_id":"native-2","kind":"artifact","url":"https://github.com/leanprover-community/physlib/blob/e882411d1b6bcbdfdd336d4c509c6cc72e96842d/Physlib/ClassicalMechanics/HarmonicOscillator/Geometric/Basic.lean"}],"metadata":{"api_title":"Configuration space of the harmonic oscillator","references":"[\"Ivo Terek, Introductory Variational Calculus on Manifolds, Section 1 (pages 1-2), Basic definitions and examples.\"]","parent_apis":"[\"Harmonic oscillator (Physlib/ClassicalMechanics/HarmonicOscillator)\",\"Space (Physlib/SpaceAndTime/Space)\",\"Time (Physlib/SpaceAndTime/Time)\"]","api_map_path":"Physlib/ClassicalMechanics/HarmonicOscillator/Geometric/API-map.yaml","api_map_root":"sha256:c13d44b23dea0fc97ea74ce7a1f18af6b6e31c25efbd968090fab8fe3cfaf84b","api_overview":"The one-dimensional harmonic oscillator is a mass on a spring: a single particle\nmoving on a line under a restoring force proportional to its displacement. Its\nconfiguration is one real coordinate, so its configuration space `Q` is a\none-dimensional smooth manifold; trajectories are curves from `Time` into `Q`,\nand the kinetic energy comes from the mass metric, the mass-scaled Euclidean\ninner product on tangent vectors.\n\nThe API formalises this geometrically. `ConfigurationSpace` wraps a global\ncoordinate valued in `EuclideanSpace ℝ (Fin 1)` and carries the induced\ntopology, a coordinate homeomorphism and analytic diffeomorphism, charted-space\nand manifold instances, tangent-coordinate infrastructure, and a map to physical\n`Space 1`. 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Its\nconfiguration is one real coordinate, so its configuration space `Q` is a\none-dimensional smooth manifold; trajectories are curves from `Time` into `Q`,\nand the kinetic energy comes from the mass metric, the mass-scaled Euclidean\ninner product on tangent vectors.\n\nThe API formalises this geometrically. `ConfigurationSpace` wraps a global\ncoordinate valued in `EuclideanSpace ℝ (Fin 1)` and carries the induced\ntopology, a coordinate homeomorphism and analytic diffeomorphism, charted-space\nand manifold instances, tangent-coordinate infrastructure, and a map to physical\n`Space 1`. Smoothness of a trajectory reduces to smoothness of its coordinate\ncurve, the velocity is a tangent vector recovered as the coordinate derivative,\nand the geometric kinetic energy recovers the standard mass-scaled expression.","license_root":"sha256:c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4","lakefile_root":"sha256:9b6608c26fa30873fb8d823eadd7f66835992389a80c3e44b62e9803c8c65c01","ai_policy_root":"sha256:a639a2c51091911123600a376c7556b52d92ffd324b041df2e2a66ec7597ed49","api_map_schema":"v0.1","lean_toolchain":"leanprover/lean4:v4.32.0","api_map_guide_root":"sha256:0663b6828ab7635bd69a1bc5913d75c5f1b81640071f43f3db0e5e10f5d29ad1","lake_manifest_root":"sha256:a9531c3978f21e50cb72cc89c2a7644b296bee59a7f2ff92ca2408aac4a67311","native_null_fields":"[]","agent_guidance_root":"sha256:487a9754f996304b173dd871326a094f4552d27df7151d60c56d83374e7711ca","lean_toolchain_root":"sha256:cc346fd7850a83fb7af3902dfc378f74d8eb8c59c6d34322488269ebfe7615b1","requirement_ordinal":9,"source_declared_done":true,"source_declared_state":"implemented","review_guidelines_root":"sha256:e28f680f6fc0767840942aaa48da28bd9983cd48db8beec4ab77dafb4dbdab0b","source_declared_location":"Physlib/ClassicalMechanics/HarmonicOscillator/Geometric/KineticEnergy.lean (geometricKineticEnergy, geometricKineticEnergy_eq, geometricKineticEnergy_massMetric_eq)","declared_source_file_paths":"[\"Physlib/ClassicalMechanics/HarmonicOscillator/Geometric/KineticEnergy.lean\"]","declared_source_file_roots":"[\"sha256:3b21b4243bce257670296ded0f7dcdca973593ae0c729a7184dc0d43340275e0\"]"},"metadata_root":"sha256:51a4da628235e1e74e4c4cc993bb62adef7b1b53e21167f90d27865109a6f7d3","content_root":"sha256:8ec5e3962a7d4c50853c50caeb7b33d90e13a6cc6b9b9ec49d828aac3859dc2f","availability":"reference_only","row_root":"sha256:2eb61ebb953827a55a13094116a02171c4311d45d44c3c7aea584a5cd0c421a2"},{"schema":"vela.math-native-record.v1","source_id":"source:physlib","observation_root":"sha256:fce692e559477ec2cbcab6a5931c35bb0a903ff861f4100872eb141f3ed3e6f9","native_id":"api-map:Physlib/ClassicalMechanics/Lagrangian/API-map.yaml#requirement:1","native_kind":"api_requirement","native_revision":"e882411d1b6bcbdfdd336d4c509c6cc72e96842d","title":"A general, system-agnostic `ConfigurationSpace` construction (a configuration manifold abstracting over the choice of mechanical system) usable across the Lagrangian API.\n","summary":"Configuration Space · source reports planned","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/leanprover-community/physlib/blob/e882411d1b6bcbdfdd336d4c509c6cc72e96842d/Physlib/ClassicalMechanics/Lagrangian/API-map.yaml"}],"metadata":{"api_title":"Configuration Space","references":"[\"Ivo Terek, Introductory Variational Calculus on Manifolds, Section 1\",\"Landau and Lifshitz, Mechanics, 3rd ed., Chapter 1 (generalised coordinates, the principle of least action, the Lagrangian)\"]","parent_apis":"[\"Space (Physlib/SpaceAndTime/Space)\"]","api_map_path":"Physlib/ClassicalMechanics/Lagrangian/API-map.yaml","api_map_root":"sha256:08708c0d067e9b7e91e9c783683425cef96a2af23a3b1ebbf1451bac174051ba","api_overview":"The configuration space `Q` of a mechanical system is the manifold of its\npossible instantaneous configurations, coordinatised by generalized coordinates.\nIt is the base space of the Lagrangian formulation of mechanics: a Lagrangian is\na function on the tangent bundle `TQ` of positions and velocities, and the\nEuler-Lagrange equations describe physical motions as curves in `Q`.\n\nThe library does not yet have a general, system-agnostic configuration-space\nabstraction, nor a Lagrangian built over one; those are the requirements below.\nConcrete per-system configuration spaces exist and are tracked in their own API\nmaps (the geometric harmonic oscillator and the pendulum).","license_root":"sha256:c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4","lakefile_root":"sha256:9b6608c26fa30873fb8d823eadd7f66835992389a80c3e44b62e9803c8c65c01","ai_policy_root":"sha256:a639a2c51091911123600a376c7556b52d92ffd324b041df2e2a66ec7597ed49","api_map_schema":"v0.1","lean_toolchain":"leanprover/lean4:v4.32.0","api_map_guide_root":"sha256:0663b6828ab7635bd69a1bc5913d75c5f1b81640071f43f3db0e5e10f5d29ad1","lake_manifest_root":"sha256:a9531c3978f21e50cb72cc89c2a7644b296bee59a7f2ff92ca2408aac4a67311","native_null_fields":"[]","agent_guidance_root":"sha256:487a9754f996304b173dd871326a094f4552d27df7151d60c56d83374e7711ca","lean_toolchain_root":"sha256:cc346fd7850a83fb7af3902dfc378f74d8eb8c59c6d34322488269ebfe7615b1","requirement_ordinal":1,"source_declared_done":false,"source_declared_state":"planned","review_guidelines_root":"sha256:e28f680f6fc0767840942aaa48da28bd9983cd48db8beec4ab77dafb4dbdab0b","source_declared_location":"N/A","declared_source_file_paths":"[]","declared_source_file_roots":"[]"},"metadata_root":"sha256:263e7250f17ce4cd8219a3c3417854e966f431aaf7798377adc1ddbf0a68427b","content_root":"sha256:e4b79700366fc5bf1613e922d77cf41e16790b35f64731b283be570fd3a775b8","availability":"reference_only","row_root":"sha256:02722dc5112cd6f97a40f4093743374e32b283b2746a59cc6fb523acf7b589f0"},{"schema":"vela.math-native-record.v1","source_id":"source:physlib","observation_root":"sha256:fce692e559477ec2cbcab6a5931c35bb0a903ff861f4100872eb141f3ed3e6f9","native_id":"api-map:Physlib/ClassicalMechanics/Lagrangian/API-map.yaml#requirement:2","native_kind":"api_requirement","native_revision":"e882411d1b6bcbdfdd336d4c509c6cc72e96842d","title":"A general treatment of generalized coordinates and the velocity phase space (tangent bundle `TQ`) built over a shared configuration-space abstraction, on which a Lagrangian is defined for the Euler-Lagrange API.\n","summary":"Configuration Space · source reports planned","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/leanprover-community/physlib/blob/e882411d1b6bcbdfdd336d4c509c6cc72e96842d/Physlib/ClassicalMechanics/Lagrangian/API-map.yaml"}],"metadata":{"api_title":"Configuration Space","references":"[\"Ivo Terek, Introductory Variational Calculus on Manifolds, Section 1\",\"Landau and Lifshitz, Mechanics, 3rd ed., Chapter 1 (generalised coordinates, the principle of least action, the Lagrangian)\"]","parent_apis":"[\"Space (Physlib/SpaceAndTime/Space)\"]","api_map_path":"Physlib/ClassicalMechanics/Lagrangian/API-map.yaml","api_map_root":"sha256:08708c0d067e9b7e91e9c783683425cef96a2af23a3b1ebbf1451bac174051ba","api_overview":"The configuration space `Q` of a mechanical system is the manifold of its\npossible instantaneous configurations, coordinatised by generalized coordinates.\nIt is the base space of the Lagrangian formulation of mechanics: a Lagrangian is\na function on the tangent bundle `TQ` of positions and velocities, and the\nEuler-Lagrange equations describe physical motions as curves in `Q`.\n\nThe library does not yet have a general, system-agnostic configuration-space\nabstraction, nor a Lagrangian built over one; those are the requirements below.\nConcrete per-system configuration spaces exist and are tracked in their own API\nmaps (the geometric harmonic oscillator and the pendulum).","license_root":"sha256:c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4","lakefile_root":"sha256:9b6608c26fa30873fb8d823eadd7f66835992389a80c3e44b62e9803c8c65c01","ai_policy_root":"sha256:a639a2c51091911123600a376c7556b52d92ffd324b041df2e2a66ec7597ed49","api_map_schema":"v0.1","lean_toolchain":"leanprover/lean4:v4.32.0","api_map_guide_root":"sha256:0663b6828ab7635bd69a1bc5913d75c5f1b81640071f43f3db0e5e10f5d29ad1","lake_manifest_root":"sha256:a9531c3978f21e50cb72cc89c2a7644b296bee59a7f2ff92ca2408aac4a67311","native_null_fields":"[]","agent_guidance_root":"sha256:487a9754f996304b173dd871326a094f4552d27df7151d60c56d83374e7711ca","lean_toolchain_root":"sha256:cc346fd7850a83fb7af3902dfc378f74d8eb8c59c6d34322488269ebfe7615b1","requirement_ordinal":2,"source_declared_done":false,"source_declared_state":"planned","review_guidelines_root":"sha256:e28f680f6fc0767840942aaa48da28bd9983cd48db8beec4ab77dafb4dbdab0b","source_declared_location":"N/A","declared_source_file_paths":"[]","declared_source_file_roots":"[]"},"metadata_root":"sha256:63735f2b6f456c1fa174d1da746d6d023f5a90d1af0f1c89b29ebdc373d2ad5a","content_root":"sha256:9b9d9e0a5887cc13344175853ec394f0da73c059464b04967c3494abf1c0c54f","availability":"reference_only","row_root":"sha256:10773ff770414f518f5a2ed8f48b25e3c69eadb3d59231114278b9e6fe9e45b5"},{"schema":"vela.math-native-record.v1","source_id":"source:physlib","observation_root":"sha256:fce692e559477ec2cbcab6a5931c35bb0a903ff861f4100872eb141f3ed3e6f9","native_id":"api-map:Physlib/ClassicalMechanics/Pendulum/API-map.yaml#requirement:1","native_kind":"api_requirement","native_revision":"e882411d1b6bcbdfdd336d4c509c6cc72e96842d","title":"The key data structure, the configuration space of the pendulum, is defined (chosen here as generalized coordinates, the support position and the string angle for the sliding pendulum).\n","summary":"Configuration space for pendulum · source reports implemented","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/leanprover-community/physlib/blob/e882411d1b6bcbdfdd336d4c509c6cc72e96842d/Physlib/ClassicalMechanics/Pendulum/API-map.yaml"},{"locator_id":"native-2","kind":"artifact","url":"https://github.com/leanprover-community/physlib/blob/e882411d1b6bcbdfdd336d4c509c6cc72e96842d/Physlib/ClassicalMechanics/Pendulum/SlidingPendulum.lean"}],"metadata":{"api_title":"Configuration space for pendulum","references":"[\"Landau & Lifshitz, Mechanics, 3rd Edition, Chapter 1 (The Equations of motion), Section 5 (The Lagrangian for a system of particles).\"]","parent_apis":"[\"Classical mechanics Lagrangian (Physlib/ClassicalMechanics/Lagrangian)\",\"Space (Physlib/SpaceAndTime/Space)\"]","api_map_path":"Physlib/ClassicalMechanics/Pendulum/API-map.yaml","api_map_root":"sha256:0ba7979724c2109f4c7c79d3fe0776d0a8da50435d757716dd1aebfe58e2cb9d","api_overview":"A pendulum is a mass swinging under gravity on a string or rod; its\nconfiguration is set by angles and, for a moving support, the support position,\nso its configuration space is naturally a product of circles and lines. This API\ncovers the pendulum problems of Landau and Lifshitz, Mechanics, 3rd ed.,\nChapter 1, Section 5.\n\nAt present only the sliding pendulum has a defined configuration space, with\nthe horizontal support position and the string angle as its generalized\ncoordinates. The coplanar double pendulum's configuration space is declared but\nnot yet defined, and the miscellaneous pivot-motion problems have documentation\nonly. The remaining requirements, a manifold structure on the configuration\nspace, a map into real space, trajectories, and the lagrangian, are open and\nrecorded below with location N/A.","license_root":"sha256:c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4","lakefile_root":"sha256:9b6608c26fa30873fb8d823eadd7f66835992389a80c3e44b62e9803c8c65c01","ai_policy_root":"sha256:a639a2c51091911123600a376c7556b52d92ffd324b041df2e2a66ec7597ed49","api_map_schema":"v0.1","lean_toolchain":"leanprover/lean4:v4.32.0","api_map_guide_root":"sha256:0663b6828ab7635bd69a1bc5913d75c5f1b81640071f43f3db0e5e10f5d29ad1","lake_manifest_root":"sha256:a9531c3978f21e50cb72cc89c2a7644b296bee59a7f2ff92ca2408aac4a67311","native_null_fields":"[]","agent_guidance_root":"sha256:487a9754f996304b173dd871326a094f4552d27df7151d60c56d83374e7711ca","lean_toolchain_root":"sha256:cc346fd7850a83fb7af3902dfc378f74d8eb8c59c6d34322488269ebfe7615b1","requirement_ordinal":1,"source_declared_done":true,"source_declared_state":"implemented","review_guidelines_root":"sha256:e28f680f6fc0767840942aaa48da28bd9983cd48db8beec4ab77dafb4dbdab0b","source_declared_location":"Physlib/ClassicalMechanics/Pendulum/SlidingPendulum.lean (ConfigurationSpace)","declared_source_file_paths":"[\"Physlib/ClassicalMechanics/Pendulum/SlidingPendulum.lean\"]","declared_source_file_roots":"[\"sha256:5a225cb730e1498e89625c7517ad74ec97457d970ae118ee0970a798a855d6d5\"]"},"metadata_root":"sha256:05e026e54b2b80bb00be034ef3b90c396147d117a36edee79f734e763691eb50","content_root":"sha256:e6c8506073a8ac5cfceddca38b092a23d27700777bb2d694318926b038a4e5d1","availability":"reference_only","row_root":"sha256:2c0e8f16dbe5ceb4c4b5bb44644c4a38d39123f9b722b731b70922edbba0d5b0"},{"schema":"vela.math-native-record.v1","source_id":"source:physlib","observation_root":"sha256:fce692e559477ec2cbcab6a5931c35bb0a903ff861f4100872eb141f3ed3e6f9","native_id":"api-map:Physlib/ClassicalMechanics/Pendulum/API-map.yaml#requirement:2","native_kind":"api_requirement","native_revision":"e882411d1b6bcbdfdd336d4c509c6cc72e96842d","title":"The API shall contain the structure of a manifold on the configuration space.","summary":"Configuration space for pendulum · source reports planned","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/leanprover-community/physlib/blob/e882411d1b6bcbdfdd336d4c509c6cc72e96842d/Physlib/ClassicalMechanics/Pendulum/API-map.yaml"}],"metadata":{"api_title":"Configuration space for pendulum","references":"[\"Landau & Lifshitz, Mechanics, 3rd Edition, Chapter 1 (The Equations of motion), Section 5 (The Lagrangian for a system of particles).\"]","parent_apis":"[\"Classical mechanics Lagrangian (Physlib/ClassicalMechanics/Lagrangian)\",\"Space (Physlib/SpaceAndTime/Space)\"]","api_map_path":"Physlib/ClassicalMechanics/Pendulum/API-map.yaml","api_map_root":"sha256:0ba7979724c2109f4c7c79d3fe0776d0a8da50435d757716dd1aebfe58e2cb9d","api_overview":"A pendulum is a mass swinging under gravity on a string or rod; its\nconfiguration is set by angles and, for a moving support, the support position,\nso its configuration space is naturally a product of circles and lines. This API\ncovers the pendulum problems of Landau and Lifshitz, Mechanics, 3rd ed.,\nChapter 1, Section 5.\n\nAt present only the sliding pendulum has a defined configuration space, with\nthe horizontal support position and the string angle as its generalized\ncoordinates. The coplanar double pendulum's configuration space is declared but\nnot yet defined, and the miscellaneous pivot-motion problems have documentation\nonly. The remaining requirements, a manifold structure on the configuration\nspace, a map into real space, trajectories, and the lagrangian, are open and\nrecorded below with location N/A.","license_root":"sha256:c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4","lakefile_root":"sha256:9b6608c26fa30873fb8d823eadd7f66835992389a80c3e44b62e9803c8c65c01","ai_policy_root":"sha256:a639a2c51091911123600a376c7556b52d92ffd324b041df2e2a66ec7597ed49","api_map_schema":"v0.1","lean_toolchain":"leanprover/lean4:v4.32.0","api_map_guide_root":"sha256:0663b6828ab7635bd69a1bc5913d75c5f1b81640071f43f3db0e5e10f5d29ad1","lake_manifest_root":"sha256:a9531c3978f21e50cb72cc89c2a7644b296bee59a7f2ff92ca2408aac4a67311","native_null_fields":"[]","agent_guidance_root":"sha256:487a9754f996304b173dd871326a094f4552d27df7151d60c56d83374e7711ca","lean_toolchain_root":"sha256:cc346fd7850a83fb7af3902dfc378f74d8eb8c59c6d34322488269ebfe7615b1","requirement_ordinal":2,"source_declared_done":false,"source_declared_state":"planned","review_guidelines_root":"sha256:e28f680f6fc0767840942aaa48da28bd9983cd48db8beec4ab77dafb4dbdab0b","source_declared_location":"N/A","declared_source_file_paths":"[]","declared_source_file_roots":"[]"},"metadata_root":"sha256:ddd5a3c66388b2e1962d2fc5eed4ccad776b11557b95da1946ce9ce2676288e4","content_root":"sha256:299a75993a385c47312b318a5cbb3fdd2e68a29cbcaab002e1439ab660cb50d4","availability":"reference_only","row_root":"sha256:17e01cdccb8e162fd0651c37ac0984dde3032f6c7505b1a039016fd40990699f"},{"schema":"vela.math-native-record.v1","source_id":"source:physlib","observation_root":"sha256:fce692e559477ec2cbcab6a5931c35bb0a903ff861f4100872eb141f3ed3e6f9","native_id":"api-map:Physlib/ClassicalMechanics/Pendulum/API-map.yaml#requirement:3","native_kind":"api_requirement","native_revision":"e882411d1b6bcbdfdd336d4c509c6cc72e96842d","title":"The API shall contain a map from the configuration space to `Space`, giving the position of the pendulum in real space.\n","summary":"Configuration space for pendulum · source reports planned","locators":[{"locator_id":"native-1","kind":"artifact","url":"https://github.com/leanprover-community/physlib/blob/e882411d1b6bcbdfdd336d4c509c6cc72e96842d/Physlib/ClassicalMechanics/Pendulum/API-map.yaml"}],"metadata":{"api_title":"Configuration space for pendulum","references":"[\"Landau & Lifshitz, Mechanics, 3rd Edition, Chapter 1 (The Equations of motion), Section 5 (The Lagrangian for a system of particles).\"]","parent_apis":"[\"Classical mechanics Lagrangian (Physlib/ClassicalMechanics/Lagrangian)\",\"Space (Physlib/SpaceAndTime/Space)\"]","api_map_path":"Physlib/ClassicalMechanics/Pendulum/API-map.yaml","api_map_root":"sha256:0ba7979724c2109f4c7c79d3fe0776d0a8da50435d757716dd1aebfe58e2cb9d","api_overview":"A pendulum is a mass swinging under gravity on a string or rod; its\nconfiguration is set by angles and, for a moving support, the support position,\nso its configuration space is naturally a product of circles and lines. 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D. Jackson, Classical Electrodynamics, 3rd ed., Sections 6.2-6.3 (vector and scalar potentials; gauge transformations)\"]","parent_apis":"[\"SpaceTime (Physlib/SpaceAndTime/SpaceTime)\",\"LorentzGroup (Physlib/Relativity/LorentzGroup)\"]","api_map_path":"Physlib/Electromagnetism/Kinematics/API-map.yaml","api_map_root":"sha256:7cd1855ac6332a7d1f87e0c0aee23badd503df2d6e0ab43c84d0365b64c7943b","api_overview":"The electromagnetic potential is the primitive kinematic object of the electromagnetism\nAPI. 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