Hamilton's principle/Notation

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

This page fixes the symbols used across the volume and records where they collide with the notation of the primary sources. It exists because three collisions in this subject are severe enough to make a source unreadable if they are not declared, and because a reader moving between this volume and Hamilton's essays will meet all three within a page.

Conventions of this volume

symbol	meaning	introduced at
$L$	integrand of a functional; Lagrangian	Functionals and the space of paths
$ℒ$	Lagrangian density, per unit volume	Classical field theory
$J[y]$	a general functional	Functionals and the space of paths
$S$	Hamilton's action, $\int Ldt$	Abbreviated action: Maupertuis versus Hamilton
$W$	abbreviated action, $\int \sum pdq$	Abbreviated action: Maupertuis versus Hamilton
$\eta$	a variation direction, vanishing at fixed endpoints	the first variation
$\epsilon$	the family parameter in $y+\epsilon \eta$	the first variation
$\delta J$	first variation	the first variation
${\delta }^{2}J$	second variation	the second variation
$P$, $Q$	coefficients of the second variation in Legendre form	the second variation
$p_i$	conjugate momentum, $\partial L/\partial {\dot{q}}_i$	the Euler-Lagrange equation
$H$	Hamiltonian, $\sum p\dot{q}-L$	the Hamiltonian and Hamilton's equations
${\lambda }_{\beta }(x)$	multiplier function, pointwise constraint	Lagrange multipliers
${\mu }_{\gamma }$	multiplier constant, isoperimetric constraint	Isoperimetric problems
$E(x,y,p,q)$	Weierstrass excess function	the Weierstrass excess function
${\psi }_i$	Lagrange expression, in Noether's sense	Noether's first theorem
${\Gamma }_{ik}^m$	Christoffel symbols	Geodesics
$\nu$	Maslov index, count of conjugate points	the Feynman path integral

Independent variables are $x$ in the mathematics tracks and $t$ where the subject is mechanics. Greek indices run over spacetime coordinates; Latin indices over coordinates or fields.

Collision 1: $V$

This is the collision that matters most, and it is the one that makes Hamilton's essays unreadable if it is missed.

symbol	in this volume	in Hamilton 1834
$V$	potential energy	the characteristic function, ${\int }_0^{t}2Tdt$
$U$	not used	the force-function, with sign opposite to potential energy
$T$	kinetic energy	half the living force; same quantity

Hamilton's $V$ is not a potential. It is the accumulated action, and his law of living force reads $T=U+H$ rather than $T+V=E$ because his $U$ is minus the potential.

A reader who carries this volume's $V$ into Hamilton's equation (A.) will read it as nonsense. The quotations on Action as a function of its endpoints are footnoted with this warning at each occurrence.

Collision 2: the action itself

quantity	this volume	Hamilton 1834	Hamilton 1835	common modern usage
$\int Ldt$	$S$	—	$S$	$S$
$\int 2Tdt=\int \sum pdq$	$W$	$V$	$V$	$S_0$ or $W$

Hamilton's first essay works entirely with the abbreviated action. The function modern texts call the action enters only in the second essay, as a part of $V$ into which the time enters explicitly.

Both quantities are routinely called "the action" and both principles are routinely called "the principle of least action". This volume writes $S$ and $W$, names the principles separately, and uses stationary rather than least for both. The distinction is the subject of Abbreviated action: Maupertuis versus Hamilton.

Collision 3: $\delta$

$\delta$ carries three distinct meanings in the sources, and Hamilton uses two of them on the same page.

usage	meaning	where
$\delta J$	first variation of a functional	this volume, and Hamilton
$\delta y$	an infinitesimal change of path, this volume's $\epsilon \eta$	classical literature, and Hamilton
$\delta V/\delta x$	partial derivative	Hamilton throughout, where a modern text writes $\partial V/\partial x$

The third is the trap. In Hamilton's equations (C.), (D.), (E.), (F.) and (G.), quoted on Action as a function of its endpoints and the Hamilton-Jacobi equation, every $\delta$ in a quotient is an ordinary partial derivative. Read as variations they are meaningless.

This volume writes $\partial$ for partial derivatives everywhere in its own text and preserves Hamilton's $\delta$ inside quotations, with a footnote at each occurrence.

Noether's notation

Her symbols are preserved where her equations are reproduced, on Noether's first theorem, Noether's second theorem and Several independent variables.

Noether	meaning	this volume's equivalent
$f$	the integrand	$ℒ$
$I$	the integral	$S$
$u_i$	dependent variables	fields ${\varphi }_i$
$x_{\lambda }$	independent variables	$x^{\alpha }$
${\psi }_i$	Lagrange expressions	left side of the Euler-Lagrange equation
$\mathrm{D}\mathrm{i}\mathrm{v}A$	divergence	${\partial }_{\alpha }{A}^{\alpha }$
$G_{\rho }$	finite continuous group, $\rho$ parameters	global symmetry
$G_{\mathrm{\infty }\rho }$	infinite continuous group, $\rho$ arbitrary functions	gauge symmetry
$\Delta u$ against $\delta u$	total change against change at fixed argument	see her equation (9)

The last row is not a notational nicety. The two differ by $\sum (\partial u/\partial x_{\lambda })\Delta x_{\lambda }$, and compressing them is where sign errors in derivations of Noether's theorem come from.

Signs and units

Metric signature, where it appears, is $(+,-,-,-)$. Gravitational acceleration is written $g_0$ to avoid collision with the metric determinant $g$ and the metric components $g_{ik}$; the collision is real and appears on the catenary and the Einstein-Hilbert action within the same volume.

Equation numbers restart at (1) on each page and are written inside the mathematics as $\mathrm{(}\mathrm{n}\mathrm{)}$, because MediaWiki's TeX checker rejects the usual tagging command. The cross-page index is Equation index.

A note on $||$

Absolute values and norms are written with $|$ rather than the usual paired delimiters, throughout, for the same reason: the checker rejects the paired forms. This is a constraint of the platform, not a convention of the subject.