Hamilton's principle/Figure gallery

Hamilton's principle: map
Reading the volume How to read this volume Notation Glossary Equation index Figure gallery
Reading by level Start here: Comparing paths The mathematics: Functionals and the space of paths Stationary is not least: The Jacobi condition and conjugate points The applications: The brachistochrone The history: Hamilton 1834 and 1835
Sources Sources Chronology
Related Noether's theorems On Physical Lines of Force

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

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

Whether a path is best

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

Structure

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

Symmetry

figure	shows	carried by
F-09	Symmetry to conservation, four examples	Noether's first theorem · Symmetry and conservation
F-26	Counting physical polarisations from gauge redundancy	the electromagnetic Lagrangian · Electromagnetism from an action
F-37	Identities among field equations from a local symmetry	Noether's second theorem · the electromagnetic Lagrangian · When symmetry is local
F-51	Locality: influence passing from neighbour to neighbour	Classical field theory · Fields

Worked problems

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

History and method

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