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Hamilton's principle
Hamilton's principle/Applications/Central force and the Kepler problem
Hamilton's principle/Applications/Classical field theory
Hamilton's principle/Applications/Continuum systems: the vibrating string
Hamilton's principle/Applications/Fermat's principle
Hamilton's principle/Applications/Geodesics
Hamilton's principle/Applications/The Einstein-Hilbert action
Hamilton's principle/Applications/The Feynman path integral
Hamilton's principle/Applications/The brachistochrone
Hamilton's principle/Applications/The catenary
Hamilton's principle/Applications/The electromagnetic Lagrangian
Hamilton's principle/Applications/The optical-mechanical analogy
Hamilton's principle/Applications/The rigid body
Hamilton's principle/Chronology
Hamilton's principle/Equation index
Hamilton's principle/Explainers/Action from the endpoints
Hamilton's principle/Explainers/Comparing paths
Hamilton's principle/Explainers/Constraints
Hamilton's principle/Explainers/Electromagnetism from an action
Hamilton's principle/Explainers/Fields
Hamilton's principle/Explainers/Fixed length problems
Hamilton's principle/Explainers/Free ends
Hamilton's principle/Explainers/Gravity from an action
Hamilton's principle/Explainers/Is it really a minimum
Hamilton's principle/Explainers/Light and particles
Hamilton's principle/Explainers/More than one variable
Hamilton's principle/Explainers/Multipliers
Hamilton's principle/Explainers/Reading a curve two ways
Hamilton's principle/Explainers/Strong and weak
Hamilton's principle/Explainers/Sum over histories
Hamilton's principle/Explainers/Symmetry and conservation
Hamilton's principle/Explainers/The Hamiltonian
Hamilton's principle/Explainers/Two different actions
Hamilton's principle/Explainers/Waves and rays
Hamilton's principle/Explainers/What the Euler-Lagrange equation says
Hamilton's principle/Explainers/When symmetry is local
Hamilton's principle/Explainers/Where the minimum fails
Hamilton's principle/Figure gallery
Hamilton's principle/Glossary
Hamilton's principle/History/Euler and the Methodus inveniendi
Hamilton's principle/History/Feynman 1948
Hamilton's principle/History/Hamilton 1834 and 1835
Hamilton's principle/History/Jacobi
Hamilton's principle/History/Lagrange and the Mecanique analytique
Hamilton's principle/History/Least time before least action
Hamilton's principle/History/Maupertuis and the 1744 memoir
Hamilton's principle/History/Noether 1918
Hamilton's principle/History/The Maupertuis-Koenig affair
Hamilton's principle/History/d'Alembert and virtual work
Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton
Hamilton's principle/Mathematics/Action as a function of its endpoints
Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions
Hamilton's principle/Mathematics/Functionals and the space of paths
Hamilton's principle/Mathematics/Holonomic constraints
Hamilton's principle/Mathematics/Isoperimetric problems
Hamilton's principle/Mathematics/Lagrange multipliers
Hamilton's principle/Mathematics/Noether's first theorem
Hamilton's principle/Mathematics/Noether's second theorem
Hamilton's principle/Mathematics/Several dependent variables
Hamilton's principle/Mathematics/Several independent variables
Hamilton's principle/Mathematics/The Beltrami identity
Hamilton's principle/Mathematics/The Euler-Lagrange equation
Hamilton's principle/Mathematics/The Hamilton-Jacobi equation
Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations
Hamilton's principle/Mathematics/The Jacobi condition and conjugate points
Hamilton's principle/Mathematics/The Legendre condition
Hamilton's principle/Mathematics/The Legendre transform
Hamilton's principle/Mathematics/The Weierstrass excess function
Hamilton's principle/Mathematics/The first variation
Hamilton's principle/Mathematics/The fundamental lemma
Hamilton's principle/Mathematics/The second variation
Hamilton's principle/Notation
Hamilton's principle/Sources
Lean theorem prover
Lean theorem prover/Applications/Machine learning and theorem proving
Lean theorem prover/Applications/Physics in Lean
Lean theorem prover/Applications/Principia Machina
Lean theorem prover/Applications/Research mathematics
Lean theorem prover/Applications/Software verification
Lean theorem prover/Applications/Teaching with Lean
Lean theorem prover/Applications/The Fermat's Last Theorem project
Lean theorem prover/Applications/The equational theories project
Lean theorem prover/Applications/The polynomial Freiman-Ruzsa conjecture
Lean theorem prover/Bibliography
Lean theorem prover/Chronology
Lean theorem prover/Explainers/A library of mathematics
Lean theorem prover/Explainers/Automation and its limits
Lean theorem prover/Explainers/Building things from constructors
Lean theorem prover/Explainers/Checking a proof again later
Lean theorem prover/Explainers/Checking instead of convincing
Lean theorem prover/Explainers/Computation inside proofs
Lean theorem prover/Explainers/Functions that always finish
Lean theorem prover/Explainers/Holes and goals
Lean theorem prover/Explainers/Induction, as the machine sees it
Lean theorem prover/Explainers/Lean and physics
Lean theorem prover/Explainers/Programs and proofs in one language
Lean theorem prover/Explainers/Propositions as types
Lean theorem prover/Explainers/Stating the right thing
Lean theorem prover/Explainers/Tactics
Lean theorem prover/Explainers/The small trusted core
Lean theorem prover/Explainers/Two kinds of equality
Lean theorem prover/Explainers/Types as specifications
Lean theorem prover/Explainers/What a proof assistant does
Lean theorem prover/Explainers/What has to be trusted
Lean theorem prover/Explainers/What the axioms are
Lean theorem prover/Explainers/When Lean says no
Lean theorem prover/Explainers/Why there is no type of all types
Lean theorem prover/Foundations/Classical logic from choice
Lean theorem prover/Foundations/Consistency and the metatheory
Lean theorem prover/Foundations/Definitional equality
Lean theorem prover/Foundations/Dependent function types
Lean theorem prover/Foundations/Inductive types and recursors
Lean theorem prover/Foundations/Proof irrelevance and subsingleton elimination
Lean theorem prover/Foundations/Propositional equality
Lean theorem prover/Foundations/Quotient types
Lean theorem prover/Foundations/Structural and well-founded recursion
Lean theorem prover/Foundations/Terms, types and judgments
Lean theorem prover/Foundations/The three axioms
Lean theorem prover/Foundations/Universes and Prop
Lean theorem prover/Further reading
Lean theorem prover/Glossary
Lean theorem prover/History/Automath
Lean theorem prover/History/LCF and the tactic idea
Lean theorem prover/History/Lean 1 to Lean 3
Lean theorem prover/History/Lean 4
Lean theorem prover/History/Lean since 2023
Lean theorem prover/History/Martin-Löf type theory
Lean theorem prover/History/Mathlib and its community
Lean theorem prover/History/The Calculus of Constructions
Lean theorem prover/History/The Liquid Tensor Experiment
Lean theorem prover/Notation
Lean theorem prover/Sources
Lean theorem prover/System/Decision procedures
Lean theorem prover/System/Elaboration
Lean theorem prover/System/Lake and Lean projects
Lean theorem prover/System/Metaprogramming in Lean
Lean theorem prover/System/Metavariables and unification
Lean theorem prover/System/Monads and do-notation
Lean theorem prover/System/Proof export and independent checkers
Lean theorem prover/System/Syntax extension and hygienic macros
Lean theorem prover/System/The Lean kernel
Lean theorem prover/System/The compiler and runtime
Lean theorem prover/System/The tactic framework
Lean theorem prover/System/Type classes and instance resolution
Main Page
Noether's theorems
Noether's theorems/Applications/Conserved quantities in point mechanics
Noether's theorems/Applications/Continuum mechanics
Noether's theorems/Applications/Electromagnetism and charge conservation
Noether's theorems/Applications/General relativity and the Bianchi identity
Noether's theorems/Applications/General relativity and the energy question
Noether's theorems/Applications/Kelvin's circulation theorem
Noether's theorems/Applications/Photon energy in an expanding universe
Noether's theorems/Applications/The Runge-Lenz vector
Noether's theorems/Applications/The conserved currents of the Standard Model
Noether's theorems/Applications/Topological charge, which is not a Noether charge
Noether's theorems/Applications/Variational integrators and discrete symmetry
Noether's theorems/Applications/Yang-Mills theory
Noether's theorems/Bibliography
Noether's theorems/Chronology
Noether's theorems/Explainers/A conservation law that is only wearing the costume
Noether's theorems/Explainers/A conserved quantity Noether does not explain
Noether's theorems/Explainers/A flowing quantity and a standing total
Noether's theorems/Explainers/An identity hiding inside Einstein's equations
Noether's theorems/Explainers/Counting the equations a gauge theory really has
Noether's theorems/Explainers/Drifting centres and electric charge
Noether's theorems/Explainers/Equations that were never independent
Noether's theorems/Explainers/Groups of continuous motions
Noether's theorems/Explainers/Moving the bookkeeping without changing the total
Noether's theorems/Explainers/Nudging a path and seeing what changes
Noether's theorems/Explainers/Reading the second theorem backwards
Noether's theorems/Explainers/Reading the theorem backwards
Noether's theorems/Explainers/Symmetries of a curved space
Noether's theorems/Explainers/Symmetry and conservation, the first theorem
Noether's theorems/Explainers/Symmetry that can differ from place to place
Noether's theorems/Explainers/The conserved thing is the thing that does the moving
Noether's theorems/Explainers/The second theorem, and why it is not the first
Noether's theorems/Explainers/The symmetry behind the strong and weak forces
Noether's theorems/Explainers/The theorem after quantum mechanics
Noether's theorems/Explainers/The two theorems compared
Noether's theorems/Explainers/True everywhere against true for real motions
Noether's theorems/Explainers/Two ways a thing can stay the same
Noether's theorems/Explainers/What a continuous symmetry is
Noether's theorems/Explainers/What an action is
Noether's theorems/Explainers/What happens at the edges
Noether's theorems/Explainers/What the Standard Model conserves
Noether's theorems/Explainers/When a symmetry does not survive
Noether's theorems/Explainers/Where the light's energy goes
Noether's theorems/Explainers/Why Noether needed a third symbol
Noether's theorems/Explainers/Why energy and momentum are conserved
Noether's theorems/Explainers/Why gravity has no local energy
Noether's theorems/Explainers/Why speed has to come along for the ride
Noether's theorems/Explainers/Why the moment map resists plain language
Noether's theorems/Figure gallery
Noether's theorems/First theorem/Centre-of-mass motion and electric charge
Noether's theorems/First theorem/Energy, momentum and angular momentum
Noether's theorems/First theorem/Invariance of the action, strict and up to a divergence
Noether's theorems/First theorem/On-shell and off-shell
Noether's theorems/First theorem/One-parameter groups and their generators
Noether's theorems/First theorem/Prolongation to derivatives
Noether's theorems/First theorem/The Lagrange expressions and the first variation
Noether's theorems/First theorem/The Noether current and the Noether charge
Noether's theorems/First theorem/The action and the variational setup
Noether's theorems/First theorem/The boundary term
Noether's theorems/First theorem/The converse of the first theorem
Noether's theorems/First theorem/The first theorem stated and proved
Noether's theorems/Further reading
Noether's theorems/Glossary
Noether's theorems/History/Bessel-Hagen and the first uses
Noether's theorems/History/Gottingen and the energy problem
Noether's theorems/History/Hilbert and Klein
Noether's theorems/History/Noether's position and the writing of the paper
Noether's theorems/History/The 1918 paper read in its own order
Noether's theorems/History/The long neglect
Noether's theorems/History/The precursors
Noether's theorems/History/The recovery in field theory
Noether's theorems/History/What Noether's theorem now means
Noether's theorems/Index of symmetries
Noether's theorems/Mathematics/Anomalies
Noether's theorems/Mathematics/Jet space and prolongation
Noether's theorems/Mathematics/Killing vectors and conserved quantities along geodesics
Noether's theorems/Mathematics/Lie groups and Lie algebras
Noether's theorems/Mathematics/Poisson brackets and the generator identity
Noether's theorems/Mathematics/The Belinfante-Rosenfeld improvement
Noether's theorems/Mathematics/The Ward-Takahashi identity
Noether's theorems/Mathematics/The symplectic form of the theorem and the moment map
Noether's theorems/Notation
Noether's theorems/Second theorem/Fewer independent field equations than fields
Noether's theorems/Second theorem/Groups depending on arbitrary functions
Noether's theorems/Second theorem/Proper and improper conservation laws
Noether's theorems/Second theorem/The Noether identities
Noether's theorems/Second theorem/The converse of the second theorem
Noether's theorems/Second theorem/The second theorem stated and proved
Noether's theorems/Second theorem/The two theorems side by side
Noether's theorems/Sources
On Physical Lines of Force
On Physical Lines of Force/Chronology
On Physical Lines of Force/Explainers
On Physical Lines of Force/Explainers/A wrong model that produced right answers
On Physical Lines of Force/Explainers/How Maxwell found the speed of light
On Physical Lines of Force/Explainers/Magnetism twists light
On Physical Lines of Force/Explainers/The missing term
On Physical Lines of Force/Explainers/The sea of spinning cells
On Physical Lines of Force/Explainers/Two ways to measure electricity
On Physical Lines of Force/Explainers/What came after
On Physical Lines of Force/Explainers/What is a line of force
On Physical Lines of Force/Explainers/What is an electric current
On Physical Lines of Force/Explainers/Why Maxwell wanted a mechanism
On Physical Lines of Force/Explainers/Why a changing magnet makes a current
On Physical Lines of Force/Explainers/Why spinning makes a pull
On Physical Lines of Force/Explainers/Why the cells had to be springy
On Physical Lines of Force/Figures
On Physical Lines of Force/Glossary
On Physical Lines of Force/Index of numbered equations
On Physical Lines of Force/Introduction
On Physical Lines of Force/Modern derivations
On Physical Lines of Force/Modern derivations/Ampere's law and its correction
On Physical Lines of Force/Modern derivations/Faraday's law of induction
On Physical Lines of Force/Modern derivations/Faraday rotation today
On Physical Lines of Force/Modern derivations/From components to vectors
On Physical Lines of Force/Modern derivations/Gauss's law for electricity
On Physical Lines of Force/Modern derivations/No magnetic monopoles
On Physical Lines of Force/Modern derivations/The Lorentz force
On Physical Lines of Force/Modern derivations/The Maxwell stress tensor
On Physical Lines of Force/Modern derivations/The constitutive relations
On Physical Lines of Force/Modern derivations/The displacement current
On Physical Lines of Force/Modern derivations/The electromagnetic wave equation
On Physical Lines of Force/Modern derivations/The unit ratio
On Physical Lines of Force/Modern derivations/The vector potential
On Physical Lines of Force/Notation and concordance
On Physical Lines of Force/Part I
On Physical Lines of Force/Part I/Applications
On Physical Lines of Force/Part I/Prop. I
On Physical Lines of Force/Part I/Prop. II
On Physical Lines of Force/Part I/Prop. III
On Physical Lines of Force/Part I/The introductory discussion
On Physical Lines of Force/Part II
On Physical Lines of Force/Part II/Induced currents
On Physical Lines of Force/Part II/Prop. IV
On Physical Lines of Force/Part II/Prop. V
On Physical Lines of Force/Part II/Prop. VI
On Physical Lines of Force/Part II/Prop. VII
On Physical Lines of Force/Part II/Prop. VIII
On Physical Lines of Force/Part II/Prop. XI
On Physical Lines of Force/Part II/Props. IX and X
On Physical Lines of Force/Part II/The mechanism of idle wheels
On Physical Lines of Force/Part III
On Physical Lines of Force/Part III/Conduction and induction
On Physical Lines of Force/Part III/Prop. XII
On Physical Lines of Force/Part III/Prop. XIII
On Physical Lines of Force/Part III/Prop. XIV
On Physical Lines of Force/Part III/Prop. XV
On Physical Lines of Force/Part III/Prop. XVI
On Physical Lines of Force/Part III/Prop. XVII
On Physical Lines of Force/Part IV
On Physical Lines of Force/Part IV/The six results
On Physical Lines of Force/Part IV/What became of the vortices
On Physical Lines of Force/Part IV/Why magnetism must be rotatory
On Physical Lines of Force/Rendering test
On Physical Lines of Force/Sources and editorial method
Principia Machina/Lean transition
Principia Machina/Progress/Log
Principia Machina/Progress/Phase 0
Principia Machina/Progress/Phase 1
Principia Machina/Progress/Phase 2
Principia Machina/Progress/Phase 3
Principia Machina/Progress/Status
Principia Machina/Terminology
The Principia Machina
The Principia Machina/Agent/Bibliography
The Principia Machina/Agent/Contract
The Principia Machina/Agent/Falcons
The Principia Machina/Agent/Forms
The Principia Machina/Agent/Index
The Principia Machina/Agent/Inputs
The Principia Machina/Agent/Post-refusal
The Principia Machina/Agent/Reliability
The Principia Machina/Agent/Research toolkit
The Principia Machina/Agent/Tools
The Principia Machina/Agent/Top performers
The Principia Machina/Bootstrap archive/Build chain
The Principia Machina/Bootstrap archive/Declarations
The Principia Machina/Build chain
The Principia Machina/Build chain/Volume 1
The Principia Machina/Decisions
The Principia Machina/Declarations
The Principia Machina/Declarations/Volume 1
The Principia Machina/Diagrams/01 Five-layer map
The Principia Machina/Diagrams/02 Container lifecycle
The Principia Machina/Diagrams/03 Digest gate and recovery
The Principia Machina/Diagrams/04 CAC check flow
The Principia Machina/Diagrams/05 Copy-forward
The Principia Machina/Diagrams/06 Three-zone TFK
The Principia Machina/Diagrams/07 Strike-rule flow
The Principia Machina/Diagrams/08 Regime and contradiction tests
The Principia Machina/Diagrams/09 Lock and petition queue
The Principia Machina/Diagrams/10 Landing and post-refusal
The Principia Machina/Diagrams/11 Work-block lifecycle
The Principia Machina/Diagrams/12 Index of record
The Principia Machina/Diagrams/13 Canary and scorecard pipeline
The Principia Machina/Diagrams/C1 Kernel and register
The Principia Machina/Diagrams/C2 Recorder pages
The Principia Machina/Diagrams/C3 Lifecycle pipeline
The Principia Machina/Diagrams/C4 Landing pipeline
The Principia Machina/Diagrams/C5 Certification pipeline

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