Hamilton's principle/Figure gallery

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.

Whether a path is best

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

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

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

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

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.