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	{{DISPLAYTITLE:Hamilton's principle — Glossary}}		{{DISPLAYTITLE:Hamilton's principle — Glossary}}
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	Terms used in this volume, with the page where each is introduced. Definitions		Terms used in this volume, with the page where each is introduced. Definitions
	here are short; the pages carry the working ones.		here are short; the pages carry the working ones.

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	Terms used in this volume, with the page where each is introduced. Definitions		Terms used in this volume, with the page where each is introduced. Definitions
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	'''Weak minimum.''' Better than every competitor close in both position and		'''Weak minimum.''' Better than every competitor close in both position and
	slope. The weaker claim.		slope. The weaker claim.

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Terms used in this volume, with the page where each is introduced. Definitions
here are short; the pages carry the working ones.

Symbols, and the places where this volume's notation collides with the primary
sources, are at [[Hamilton's principle/Notation]].

== A ==

'''Abbreviated action.''' The integral of momentum along a path. Distinct from
Hamilton's action, and the quantity Hamilton himself called <math>V</math>. See
[[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton]].

'''Accessory equation.''' The equation whose solutions govern the second
variation, and whose zeros are the conjugate points. Also called the Jacobi
equation. See
[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]].

'''Action.''' Ambiguous without qualification. Two different quantities carry the
name. See the entry for abbreviated action.

'''Admissible.''' Satisfying whatever the problem requires of a competitor: the
right endpoints, the required smoothness, and any constraint. See
[[Hamilton's principle/Mathematics/Functionals and the space of paths]].

== B ==

'''Beltrami identity.''' The first integral available when the integrand carries
no explicit independent variable. In mechanics it is conservation of energy. See
[[Hamilton's principle/Mathematics/The Beltrami identity]].

'''Brachistochrone.''' The curve of fastest descent under gravity. A cycloid. See
[[Hamilton's principle/Applications/The brachistochrone]].

== C ==

'''Canonical equations.''' Hamilton's first-order equations for position and
momentum. See
[[Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations]].

'''Catenary.''' The curve of a uniform chain hanging under its own weight. A
hyperbolic cosine, not a parabola. See
[[Hamilton's principle/Applications/The catenary]].

'''Characteristic function.''' Hamilton's name for the action evaluated on the
true motion, treated as a function of the endpoints. See
[[Hamilton's principle/Mathematics/Action as a function of its endpoints]].

'''Conjugate momentum.''' The derivative of the integrand with respect to a
velocity. Conserved when its coordinate is absent from the integrand. See
[[Hamilton's principle/Mathematics/The Euler-Lagrange equation]].

'''Conjugate point.''' A point where neighbouring extremals from a common start
refocus. Beyond the first one, an extremal is not a minimum. Identical with an
optical focus. See
[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]].

'''Constraint.''' A restriction on the configurations or motions a system may
have. Holonomic if it restricts configurations; non-holonomic otherwise. See
[[Hamilton's principle/Mathematics/Holonomic constraints]].

'''Cyclic coordinate.''' One absent from the integrand, whose conjugate momentum
is therefore conserved. Also called ignorable.

== D ==

'''Degenerate.''' Said of a problem whose Hessian in the velocities is singular,
so that momenta cannot be solved for velocities. Characteristic of gauge
theories. See [[Hamilton's principle/Mathematics/The Legendre transform]].

'''Divergence.''' The sum of the derivatives of a vector's components. Noether's
theorems are statements about when combinations of field equations become
divergences. See
[[Hamilton's principle/Mathematics/Several independent variables]].

== E ==

'''Euler-Lagrange equation.''' The differential equation satisfied by every
stationary path. Necessary, never sufficient. See
[[Hamilton's principle/Mathematics/The Euler-Lagrange equation]].

'''Excess function.''' Weierstrass's measure of how much the integrand exceeds
its own tangent line in the velocity. Non-negativity is required for a strong
minimum. See [[Hamilton's principle/Mathematics/The Weierstrass excess function]].

'''Extremal.''' A solution of the Euler-Lagrange equation. The word records that
the equation is satisfied and claims nothing about minimality.

== F ==

'''Fermat's principle.''' Light follows a path of stationary optical length. See
[[Hamilton's principle/Applications/Fermat's principle]].

'''Field.''' A quantity with a value at every point of space, or of spacetime.
See [[Hamilton's principle/Applications/Classical field theory]].

'''First variation.''' The derivative of a functional along a one-parameter
family of paths, at the parameter value giving the path of interest. See
[[Hamilton's principle/Mathematics/The first variation]].

'''Functional.''' A rule assigning one number to a whole function. See
[[Hamilton's principle/Mathematics/Functionals and the space of paths]].

'''Fundamental lemma.''' A continuous function whose integral against every
admissible test function vanishes is itself zero. The step converting an integral
statement into a pointwise one. See
[[Hamilton's principle/Mathematics/The fundamental lemma]].

== G ==

'''Gauge freedom.''' Redundancy in a description: several descriptions
representing one physical situation. Produces identities among the field
equations. See [[Hamilton's principle/Mathematics/Noether's second theorem]].

'''Geodesic.''' A stationary curve of arc length on a surface or in a curved
space. See [[Hamilton's principle/Applications/Geodesics]].

== H ==

'''Hamiltonian.''' The Legendre transform of the Lagrangian in the velocities.
Conserved when the Lagrangian carries no explicit time. Equal to the energy only
under stated conditions. See
[[Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations]].

'''Hamilton-Jacobi equation.''' A single partial differential equation whose
complete solution integrates the whole mechanical problem. The eikonal equation
of mechanics. See
[[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation]].

'''Hamilton's principle.''' The action is stationary among paths with fixed
endpoints and fixed elapsed time.

== I ==

'''Improper conservation law.''' Noether's term for a divergence relation whose
conserved quantity is built from the field equations, so that it is an identity
rather than a dynamical statement. See
[[Hamilton's principle/Mathematics/Noether's second theorem]].

'''Isoperimetric.''' Said of a problem where a second integral is held at a
prescribed value. Its multiplier is a constant, not a function. See
[[Hamilton's principle/Mathematics/Isoperimetric problems]].

== J ==

'''Jacobi condition.''' No conjugate point in the open interval. Necessary for a
minimum, and with the other conditions strengthened, sufficient for a weak one.

== L ==

'''Lagrangian.''' The integrand of the action. For ordinary mechanical systems,
kinetic minus potential energy.

'''Legendre condition.''' The coefficient of the squared velocity variation in
the second variation must not be negative. Local, necessary, and decisively
insufficient. See [[Hamilton's principle/Mathematics/The Legendre condition]].

'''Legendre transform.''' Exchange of a variable for the slope of a function with
respect to it. Requires convexity. See
[[Hamilton's principle/Mathematics/The Legendre transform]].

== M ==

'''Maslov index.''' The count of conjugate points a path crosses, contributing a
phase in the semiclassical approximation. See
[[Hamilton's principle/Applications/The Feynman path integral]].

'''Maupertuis' principle.''' The abbreviated action is stationary among paths of
fixed energy joining fixed endpoints. Elapsed time is free.

'''Multiplier.''' The unknown introduced to carry a constraint. Equals the
constraint force. A function for pointwise constraints, a constant for
isoperimetric ones. See
[[Hamilton's principle/Mathematics/Lagrange multipliers]].

== N ==

'''Natural boundary condition.''' A condition at a free endpoint produced by the
variational problem rather than imposed on it. See
[[Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions]].

'''Noether's first theorem.''' A continuous symmetry with finitely many
parameters yields that many conserved quantities.

'''Noether's second theorem.''' A symmetry depending on arbitrary functions
yields identities among the field equations.

== P ==

'''Phase space.''' The space of positions and momenta, in which a single point
determines the entire future and trajectories never cross. See
[[Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations]].

'''Plebeian track.''' This volume's name for its plain-language pages, each
paired with a technical counterpart.

== S ==

'''Second variation.''' The second derivative of the functional along a family of
paths. Its sign begins to decide whether an extremal is a minimum. See
[[Hamilton's principle/Mathematics/The second variation]].

'''Stationary.''' The first variation vanishes. Weaker than minimal, and the
reason this volume avoids the word ''least''.

'''Strong minimum.''' Better than every competitor close in position, whatever
its slopes. The stronger claim, despite the weaker-sounding name of its
counterpart.

== T ==

'''Transversality condition.''' The condition at an endpoint constrained to a
curve rather than a point. For arc length it gives a right angle; in general it
is not about angles.

== V ==

'''Variation.''' An admissible direction in which a path may be nudged, vanishing
wherever the endpoints are fixed.

== W ==

'''Weak minimum.''' Better than every competitor close in both position and
slope. The weaker claim.

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