Hamilton's principle/Glossary

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

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 Notation.

A

Abbreviated action. The integral of momentum along a path. Distinct from Hamilton's action, and the quantity Hamilton himself called $V$. See 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 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 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 the Beltrami identity.

Brachistochrone. The curve of fastest descent under gravity. A cycloid. See the brachistochrone.

C

Canonical equations. Hamilton's first-order equations for position and momentum. See 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 the catenary.

Characteristic function. Hamilton's name for the action evaluated on the true motion, treated as a function of the endpoints. See 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 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 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 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 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 Several independent variables.

E

Euler-Lagrange equation. The differential equation satisfied by every stationary path. Necessary, never sufficient. See 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 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 Fermat's principle.

Field. A quantity with a value at every point of space, or of spacetime. See 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 the first variation.

Functional. A rule assigning one number to a whole function. See 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 the fundamental lemma.

G

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

Geodesic. A stationary curve of arc length on a surface or in a curved space. See 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 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 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 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 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 the Legendre condition.

Legendre transform. Exchange of a variable for the slope of a function with respect to it. Requires convexity. See the Legendre transform.

M

Maslov index. The count of conjugate points a path crosses, contributing a phase in the semiclassical approximation. See 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 Lagrange multipliers.

N

Natural boundary condition. A condition at a free endpoint produced by the variational problem rather than imposed on it. See 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 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 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.