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New page

Every figure in the volume, with the pages that carry it. This page is generated
from the figure manifest, so it cannot fall out of step with what is shipped.

'''58 figures, 131 placements.'''

All figures are original to this volume. They are drawn as SVG and shipped as PNG;
the reasons are given at the foot of this page.

== The method ==

{| class="wikitable"
! figure !! shows !! carried by
|-
| F-01 || A true path and several comparison paths between fixed endpoints || [[Hamilton's principle/Mathematics/Functionals and the space of paths]] · [[Hamilton's principle/Mathematics/The first variation]] · [[Hamilton's principle]] · [[Hamilton's principle/Explainers/Comparing paths]]
|-
| F-02 || A one-parameter family of paths through the true path || [[Hamilton's principle/Mathematics/The first variation]] · [[Hamilton's principle/Explainers/Comparing paths]]
|-
| F-03 || A localised test function || [[Hamilton's principle/Mathematics/The fundamental lemma]] · [[Hamilton's principle/Explainers/What the Euler-Lagrange equation says]]
|-
| F-04 || A free endpoint and its natural boundary condition || [[Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions]] · [[Hamilton's principle/Explainers/Free ends]]
|-
| F-16 || A needle variation || [[Hamilton's principle/Mathematics/The Legendre condition]] · [[Hamilton's principle/Explainers/Is it really a minimum]]
|-
| F-39 || The two rates the Euler-Lagrange equation balances || [[Hamilton's principle/Mathematics/The Euler-Lagrange equation]] · [[Hamilton's principle/Explainers/What the Euler-Lagrange equation says]]
|-
| F-40 || The boundary term left by integration by parts || [[Hamilton's principle/Mathematics/The first variation]] · [[Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions]] · [[Hamilton's principle/Explainers/Free ends]]
|-
| F-41 || An endpoint free to lie on a curve || [[Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions]]
|-
| F-42 || A corner in an extremal || [[Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions]] · [[Hamilton's principle/Mathematics/The Weierstrass excess function]]
|-
| F-43 || A quantity conserved along the whole path || [[Hamilton's principle/Mathematics/The Beltrami identity]] · [[Hamilton's principle/Explainers/Free ends]]
|-
| F-44 || Varying one degree of freedom at a time || [[Hamilton's principle/Mathematics/Several dependent variables]] · [[Hamilton's principle/Mathematics/Holonomic constraints]] · [[Hamilton's principle/Explainers/More than one variable]] · [[Hamilton's principle/Explainers/Constraints]]
|-
| F-56 || From one independent variable to several || [[Hamilton's principle/Mathematics/Several independent variables]] · [[Hamilton's principle/Explainers/More than one variable]]
|}

== Whether a path is best ==

{| class="wikitable"
! figure !! shows !! carried by
|-
| F-05 || Geodesics from a pole refocusing at the antipode || [[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]] · [[Hamilton's principle/Applications/Geodesics]] · [[Hamilton's principle/Explainers/Where the minimum fails]]
|-
| F-07 || The second variation changing sign as the interval lengthens || [[Hamilton's principle/Mathematics/The second variation]] · [[Hamilton's principle/Explainers/Is it really a minimum]]
|-
| F-13 || A sawtooth competitor close in position and far in slope || [[Hamilton's principle/Mathematics/The Weierstrass excess function]] · [[Hamilton's principle/Explainers/Strong and weak]]
|-
| F-14 || Two stationary light paths around a mass || [[Hamilton's principle/Applications/The Einstein-Hilbert action]] · [[Hamilton's principle/Explainers/Where the minimum fails]] · [[Hamilton's principle/Explainers/Gravity from an action]]
|-
| F-17 || Local penalty against accumulated reward || [[Hamilton's principle/Mathematics/The second variation]] · [[Hamilton's principle/Explainers/Is it really a minimum]]
|-
| F-22 || Geodesic behaviour on flat, spherical and saddle surfaces || [[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]] · [[Hamilton's principle/Applications/Geodesics]]
|-
| F-34 || The Weierstrass excess function as a gap above the tangent || [[Hamilton's principle/Mathematics/The Weierstrass excess function]] · [[Hamilton's principle/Explainers/Strong and weak]]
|-
| F-49 || The four conditions of the sufficiency theory || [[Hamilton's principle/Mathematics/The Legendre condition]] · [[Hamilton's principle/Mathematics/The Weierstrass excess function]] · [[Hamilton's principle/Explainers/Strong and weak]]
|-
| F-50 || Rays crossing a focus || [[Hamilton's principle/Applications/The Feynman path integral]] · [[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]] · [[Hamilton's principle/Explainers/Waves and rays]]
|}

== Structure ==

{| class="wikitable"
! figure !! shows !! carried by
|-
| F-10 || The Legendre transform as a tangent-line description || [[Hamilton's principle/Mathematics/The Legendre transform]] · [[Hamilton's principle/Explainers/Reading a curve two ways]]
|-
| F-11 || Phase portrait of a pendulum || [[Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations]] · [[Hamilton's principle/Explainers/The Hamiltonian]]
|-
| F-12 || Wavefronts and the rays orthogonal to them || [[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation]] · [[Hamilton's principle/Applications/The optical-mechanical analogy]] · [[Hamilton's principle/Explainers/Waves and rays]]
|-
| F-24 || Lagrangian and Hamiltonian descriptions compared || [[Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations]] · [[Hamilton's principle/Explainers/The Hamiltonian]]
|-
| F-25 || Action on the true motion as a function of its endpoints || [[Hamilton's principle/Mathematics/Action as a function of its endpoints]] · [[Hamilton's principle/Explainers/Action from the endpoints]]
|-
| F-32 || From n second-order equations to one partial differential equation || [[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation]] · [[Hamilton's principle/Explainers/Waves and rays]]
|-
| F-36 || What each action principle holds fixed || [[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton]] · [[Hamilton's principle/Explainers/Two different actions]]
|-
| F-38 || The optical-mechanical dictionary || [[Hamilton's principle/Applications/The optical-mechanical analogy]] · [[Hamilton's principle/Explainers/Light and particles]]
|-
| F-48 || The empty corner of the optical-mechanical analogy || [[Hamilton's principle/Applications/The optical-mechanical analogy]] · [[Hamilton's principle/Explainers/Light and particles]] · [[Hamilton's principle/History/Hamilton 1834 and 1835]]
|}

== Symmetry ==

{| class="wikitable"
! figure !! shows !! carried by
|-
| F-09 || Symmetry to conservation, four examples || [[Hamilton's principle/Mathematics/Noether's first theorem]] · [[Hamilton's principle/Explainers/Symmetry and conservation]]
|-
| F-26 || Counting physical polarisations from gauge redundancy || [[Hamilton's principle/Applications/The electromagnetic Lagrangian]] · [[Hamilton's principle/Explainers/Electromagnetism from an action]]
|-
| F-37 || Identities among field equations from a local symmetry || [[Hamilton's principle/Mathematics/Noether's second theorem]] · [[Hamilton's principle/Applications/The electromagnetic Lagrangian]] · [[Hamilton's principle/Explainers/When symmetry is local]]
|-
| F-51 || Locality: influence passing from neighbour to neighbour || [[Hamilton's principle/Applications/Classical field theory]] · [[Hamilton's principle/Explainers/Fields]]
|}

== Worked problems ==

{| class="wikitable"
! figure !! shows !! carried by
|-
| F-06 || The brachistochrone as a cycloid || [[Hamilton's principle/Applications/The brachistochrone]]
|-
| F-08 || Snell's law from stationary time || [[Hamilton's principle/Applications/Fermat's principle]] · [[Hamilton's principle/Explainers/Light and particles]]
|-
| F-15 || Phases adding near the stationary path and cancelling elsewhere || [[Hamilton's principle/Applications/The Feynman path integral]] · [[Hamilton's principle/Explainers/Sum over histories]]
|-
| F-18 || A constraint force normal to the permitted motion || [[Hamilton's principle/Mathematics/Holonomic constraints]] · [[Hamilton's principle/History/d'Alembert and virtual work]] · [[Hamilton's principle/Explainers/Constraints]]
|-
| F-19 || Tension in a swinging load || [[Hamilton's principle/Mathematics/Lagrange multipliers]] · [[Hamilton's principle/Explainers/Multipliers]]
|-
| F-20 || Fixed perimeter, varying area || [[Hamilton's principle/Mathematics/Isoperimetric problems]] · [[Hamilton's principle/Explainers/Fixed length problems]]
|-
| F-23 || A catenary and the directrix from which its tension is measured || [[Hamilton's principle/Applications/The catenary]] · [[Hamilton's principle/Explainers/Free ends]]
|-
| F-27 || Stability of rotation about the three principal axes || [[Hamilton's principle/Applications/The rigid body]]
|-
| F-28 || Fundamental modes with both ends fixed and with one end free || [[Hamilton's principle/Applications/Continuum systems: the vibrating string]] · [[Hamilton's principle/Explainers/Free ends]]
|-
| F-29 || Local conservation as a flow across a boundary || [[Hamilton's principle/Applications/Classical field theory]] · [[Hamilton's principle/Explainers/Fields]]
|-
| F-30 || Lambert's theorem: chord and sum of radii || [[Hamilton's principle/Applications/Central force and the Kepler problem]] · [[Hamilton's principle/Explainers/Action from the endpoints]]
|-
| F-31 || A bead chain approaching a continuous string || [[Hamilton's principle/Applications/Continuum systems: the vibrating string]] · [[Hamilton's principle/Explainers/More than one variable]]
|-
| F-33 || D'Alembert's principle: dynamics rewritten as statics || [[Hamilton's principle/History/d'Alembert and virtual work]] · [[Hamilton's principle/Explainers/Constraints]]
|-
| F-35 || A two-parameter family for an isoperimetric problem || [[Hamilton's principle/Mathematics/Isoperimetric problems]] · [[Hamilton's principle/Explainers/Fixed length problems]]
|-
| F-45 || Free fall as a straight path in a curved geometry || [[Hamilton's principle/Applications/The Einstein-Hilbert action]] · [[Hamilton's principle/Explainers/Gravity from an action]]
|-
| F-46 || Maxwell's equations split into dynamics and identities || [[Hamilton's principle/Applications/The electromagnetic Lagrangian]] · [[Hamilton's principle/Explainers/Electromagnetism from an action]]
|-
| F-47 || The effective potential of a central-force problem || [[Hamilton's principle/Applications/Central force and the Kepler problem]] · [[Hamilton's principle/Applications/The rigid body]]
|-
| F-52 || The boundary term in one dimension and in a field || [[Hamilton's principle/Mathematics/Several independent variables]] · [[Hamilton's principle/Applications/The Einstein-Hilbert action]] · [[Hamilton's principle/Explainers/Gravity from an action]]
|}

== History and method ==

{| class="wikitable"
! figure !! shows !! carried by
|-
| F-21 || Chronology of the action principle || [[Hamilton's principle/Chronology]] · [[Hamilton's principle]]
|-
| F-53 || The three provenance classes, and the empty one || [[Hamilton's principle/Sources]]
|-
| F-54 || Four states an electronic source can be in || [[Hamilton's principle/Sources]] · [[Hamilton's principle/History/Least time before least action]]
|-
| F-55 || Three modernised statements attributed to their authors || [[Hamilton's principle/Sources]] · [[Hamilton's principle/History/Least time before least action]] · [[Hamilton's principle/History/Maupertuis and the 1744 memoir]] · [[Hamilton's principle/History/Euler and the Methodus inveniendi]]
|-
| F-57 || Transcribed against corroborated || [[Hamilton's principle/Sources]] · [[Hamilton's principle/History/Maupertuis and the 1744 memoir]]
|-
| F-58 || The two long gaps in the subject's history || [[Hamilton's principle/Chronology]] · [[Hamilton's principle/History/Lagrange and the Mecanique analytique]] · [[Hamilton's principle/History/Jacobi]] · [[Hamilton's principle/History/Feynman 1948]] · [[Hamilton's principle]]
|}

== Note on format ==

The volume ships raster figures rather than vector ones. A MediaWiki install
accepts PNG with no configuration change, while SVG needs the upload extension
list altered and the rasteriser bypassed. The masters are vector and the PNGs
are rendered from them at twice the display size.

Every figure carries an opaque background, so the set reads consistently on a
light skin and sits as a light panel on a dark one. That trade was chosen, not
defaulted.

Hamilton's principle/Figure gallery - Revision history

PMA Publisher: Published from the Principia Machina source

BFP Rauch: Pipe internal links so cross-references read as titles, not paths

BFP Rauch: Add DISPLAYTITLE: title the page by its section rather than its path

BFP Rauch: 1 revision imported

hamiltons-principle>HPVBuild: Initial import of R-04