Hamilton's principle/Notation - Revision history

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	paired forms. This is a constraint of the platform, not a convention of the		paired forms. This is a constraint of the platform, not a convention of the
	subject.		subject.

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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 ==

{| class="wikitable"
! symbol !! meaning !! introduced at
|-
| <math>L</math> || integrand of a functional; Lagrangian || [[Hamilton's principle/Mathematics/Functionals and the space of paths]]
|-
| <math>\mathcal{L}</math> || Lagrangian density, per unit volume || [[Hamilton's principle/Applications/Classical field theory]]
|-
| <math>J[y]</math> || a general functional || [[Hamilton's principle/Mathematics/Functionals and the space of paths]]
|-
| <math>S</math> || Hamilton's action, <math>\int L\,dt</math> || [[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton]]
|-
| <math>W</math> || abbreviated action, <math>\int \sum p\,dq</math> || [[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton]]
|-
| <math>\eta</math> || a variation direction, vanishing at fixed endpoints || [[Hamilton's principle/Mathematics/The first variation]]
|-
| <math>\varepsilon</math> || the family parameter in <math>y + \varepsilon\eta</math> || [[Hamilton's principle/Mathematics/The first variation]]
|-
| <math>\delta J</math> || first variation || [[Hamilton's principle/Mathematics/The first variation]]
|-
| <math>\delta^2 J</math> || second variation || [[Hamilton's principle/Mathematics/The second variation]]
|-
| <math>P</math>, <math>Q</math> || coefficients of the second variation in Legendre form || [[Hamilton's principle/Mathematics/The second variation]]
|-
| <math>p_i</math> || conjugate momentum, <math>\partial L/\partial \dot q_i</math> || [[Hamilton's principle/Mathematics/The Euler-Lagrange equation]]
|-
| <math>H</math> || Hamiltonian, <math>\sum p\dot q - L</math> || [[Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations]]
|-
| <math>\lambda_\beta(x)</math> || multiplier '''function''', pointwise constraint || [[Hamilton's principle/Mathematics/Lagrange multipliers]]
|-
| <math>\mu_\gamma</math> || multiplier '''constant''', isoperimetric constraint || [[Hamilton's principle/Mathematics/Isoperimetric problems]]
|-
| <math>E(x,y,p,q)</math> || Weierstrass excess function || [[Hamilton's principle/Mathematics/The Weierstrass excess function]]
|-
| <math>\psi_i</math> || Lagrange expression, in Noether's sense || [[Hamilton's principle/Mathematics/Noether's first theorem]]
|-
| <math>\Gamma^m_{ik}</math> || Christoffel symbols || [[Hamilton's principle/Applications/Geodesics]]
|-
| <math>\nu</math> || Maslov index, count of conjugate points || [[Hamilton's principle/Applications/The Feynman path integral]]
|}

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

== Collision 1: <math>V</math> ==

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

{| class="wikitable"
! symbol !! in this volume !! in Hamilton 1834
|-
| <math>V</math> || potential energy || the '''characteristic function''', <math>\int_0^t 2T\,dt</math>
|-
| <math>U</math> || not used || the '''force-function''', with sign opposite to potential energy
|-
| <math>T</math> || kinetic energy || half the living force; same quantity
|}

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

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

== Collision 2: the action itself ==

{| class="wikitable"
! quantity !! this volume !! Hamilton 1834 !! Hamilton 1835 !! common modern usage
|-
| <math>\int L\,dt</math> || <math>S</math> || — || <math>S</math> || <math>S</math>
|-
| <math>\int 2T\,dt = \int\sum p\,dq</math> || <math>W</math> || <math>V</math> || <math>V</math> || <math>S_0</math> or <math>W</math>
|}

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
<math>V</math> 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
<math>S</math> and <math>W</math>, names the principles separately, and uses
''stationary'' rather than ''least'' for both. The distinction is the subject of
[[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton]].

== Collision 3: <math>\delta</math> ==

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

{| class="wikitable"
! usage !! meaning !! where
|-
| <math>\delta J</math> || first variation of a functional || this volume, and Hamilton
|-
| <math>\delta y</math> || an infinitesimal change of path, this volume's <math>\varepsilon\eta</math> || classical literature, and Hamilton
|-
| <math>\delta V/\delta x</math> || '''partial derivative''' || Hamilton throughout, where a modern text writes <math>\partial V/\partial x</math>
|}

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

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

== Noether's notation ==

Her symbols are preserved where her equations are reproduced, on
[[Hamilton's principle/Mathematics/Noether's first theorem]],
[[Hamilton's principle/Mathematics/Noether's second theorem]] and
[[Hamilton's principle/Mathematics/Several independent variables]].

{| class="wikitable"
! Noether !! meaning !! this volume's equivalent
|-
| <math>f</math> || the integrand || <math>\mathcal{L}</math>
|-
| <math>I</math> || the integral || <math>S</math>
|-
| <math>u_i</math> || dependent variables || fields <math>\phi_i</math>
|-
| <math>x_\lambda</math> || independent variables || <math>x^\alpha</math>
|-
| <math>\psi_i</math> || Lagrange expressions || left side of the Euler-Lagrange equation
|-
| <math>\mathrm{Div}\,A</math> || divergence || <math>\partial_\alpha A^\alpha</math>
|-
| <math>G_\rho</math> || finite continuous group, <math>\rho</math> parameters || global symmetry
|-
| <math>G_{\infty\rho}</math> || infinite continuous group, <math>\rho</math> arbitrary functions || gauge symmetry
|-
| <math>\Delta u</math> against <math>\delta u</math> || total change against change at fixed argument || see her equation (9)
|}

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

== Signs and units ==

Metric signature, where it appears, is <math>(+,-,-,-)</math>. Gravitational
acceleration is written <math>g_0</math> to avoid collision with the metric
determinant <math>g</math> and the metric components <math>g_{ik}</math>; the
collision is real and appears on
[[Hamilton's principle/Applications/The catenary]] and
[[Hamilton's principle/Applications/The Einstein-Hilbert action]] within the same
volume.

Equation numbers restart at (1) on each page and are written inside the
mathematics as <math>\qquad\qquad\mathrm{(n)}</math>, because MediaWiki's TeX
checker rejects the usual tagging command. The cross-page index is
[[Hamilton's principle/Equation index]].

== A note on <math>\vert\;\vert</math> ==

Absolute values and norms are written with <math>\vert</math> 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.

← Older revision		Revision as of 04:58, 16 September 2026
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	! symbol !! meaning !! introduced at		! symbol !! meaning !! introduced at
	\|-		\|-
	\| <math>L</math> \|\| integrand of a functional; Lagrangian \|\| [[Hamilton's principle/Mathematics/Functionals and the space of paths]]		\| <math>L</math> \|\| integrand of a functional; Lagrangian \|\| [[Hamilton's principle/Mathematics/Functionals and the space of paths\|Functionals and the space of paths]]
	\|-		\|-
	\| <math>\mathcal{L}</math> \|\| Lagrangian density, per unit volume \|\| [[Hamilton's principle/Applications/Classical field theory]]		\| <math>\mathcal{L}</math> \|\| Lagrangian density, per unit volume \|\| [[Hamilton's principle/Applications/Classical field theory\|Classical field theory]]
	\|-		\|-
	\| <math>J[y]</math> \|\| a general functional \|\| [[Hamilton's principle/Mathematics/Functionals and the space of paths]]		\| <math>J[y]</math> \|\| a general functional \|\| [[Hamilton's principle/Mathematics/Functionals and the space of paths\|Functionals and the space of paths]]
	\|-		\|-
	\| <math>S</math> \|\| Hamilton's action, <math>\int L\,dt</math> \|\| [[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton]]		\| <math>S</math> \|\| Hamilton's action, <math>\int L\,dt</math> \|\| [[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton\|Abbreviated action: Maupertuis versus Hamilton]]
	\|-		\|-
	\| <math>W</math> \|\| abbreviated action, <math>\int \sum p\,dq</math> \|\| [[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton]]		\| <math>W</math> \|\| abbreviated action, <math>\int \sum p\,dq</math> \|\| [[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton\|Abbreviated action: Maupertuis versus Hamilton]]
	\|-		\|-
	\| <math>\eta</math> \|\| a variation direction, vanishing at fixed endpoints \|\| [[Hamilton's principle/Mathematics/The first variation]]		\| <math>\eta</math> \|\| a variation direction, vanishing at fixed endpoints \|\| [[Hamilton's principle/Mathematics/The first variation\|the first variation]]
	\|-		\|-
	\| <math>\varepsilon</math> \|\| the family parameter in <math>y + \varepsilon\eta</math> \|\| [[Hamilton's principle/Mathematics/The first variation]]		\| <math>\varepsilon</math> \|\| the family parameter in <math>y + \varepsilon\eta</math> \|\| [[Hamilton's principle/Mathematics/The first variation\|the first variation]]
	\|-		\|-
	\| <math>\delta J</math> \|\| first variation \|\| [[Hamilton's principle/Mathematics/The first variation]]		\| <math>\delta J</math> \|\| first variation \|\| [[Hamilton's principle/Mathematics/The first variation\|the first variation]]
	\|-		\|-
	\| <math>\delta^2 J</math> \|\| second variation \|\| [[Hamilton's principle/Mathematics/The second variation]]		\| <math>\delta^2 J</math> \|\| second variation \|\| [[Hamilton's principle/Mathematics/The second variation\|the second variation]]
	\|-		\|-
	\| <math>P</math>, <math>Q</math> \|\| coefficients of the second variation in Legendre form \|\| [[Hamilton's principle/Mathematics/The second variation]]		\| <math>P</math>, <math>Q</math> \|\| coefficients of the second variation in Legendre form \|\| [[Hamilton's principle/Mathematics/The second variation\|the second variation]]
	\|-		\|-
	\| <math>p_i</math> \|\| conjugate momentum, <math>\partial L/\partial \dot q_i</math> \|\| [[Hamilton's principle/Mathematics/The Euler-Lagrange equation]]		\| <math>p_i</math> \|\| conjugate momentum, <math>\partial L/\partial \dot q_i</math> \|\| [[Hamilton's principle/Mathematics/The Euler-Lagrange equation\|the Euler-Lagrange equation]]
	\|-		\|-
	\| <math>H</math> \|\| Hamiltonian, <math>\sum p\dot q - L</math> \|\| [[Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations]]		\| <math>H</math> \|\| Hamiltonian, <math>\sum p\dot q - L</math> \|\| [[Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations\|the Hamiltonian and Hamilton's equations]]
	\|-		\|-
	\| <math>\lambda_\beta(x)</math> \|\| multiplier '''function''', pointwise constraint \|\| [[Hamilton's principle/Mathematics/Lagrange multipliers]]		\| <math>\lambda_\beta(x)</math> \|\| multiplier '''function''', pointwise constraint \|\| [[Hamilton's principle/Mathematics/Lagrange multipliers\|Lagrange multipliers]]
	\|-		\|-
	\| <math>\mu_\gamma</math> \|\| multiplier '''constant''', isoperimetric constraint \|\| [[Hamilton's principle/Mathematics/Isoperimetric problems]]		\| <math>\mu_\gamma</math> \|\| multiplier '''constant''', isoperimetric constraint \|\| [[Hamilton's principle/Mathematics/Isoperimetric problems\|Isoperimetric problems]]
	\|-		\|-
	\| <math>E(x,y,p,q)</math> \|\| Weierstrass excess function \|\| [[Hamilton's principle/Mathematics/The Weierstrass excess function]]		\| <math>E(x,y,p,q)</math> \|\| Weierstrass excess function \|\| [[Hamilton's principle/Mathematics/The Weierstrass excess function\|the Weierstrass excess function]]
	\|-		\|-
	\| <math>\psi_i</math> \|\| Lagrange expression, in Noether's sense \|\| [[Hamilton's principle/Mathematics/Noether's first theorem]]		\| <math>\psi_i</math> \|\| Lagrange expression, in Noether's sense \|\| [[Hamilton's principle/Mathematics/Noether's first theorem\|Noether's first theorem]]
	\|-		\|-
	\| <math>\Gamma^m_{ik}</math> \|\| Christoffel symbols \|\| [[Hamilton's principle/Applications/Geodesics]]		\| <math>\Gamma^m_{ik}</math> \|\| Christoffel symbols \|\| [[Hamilton's principle/Applications/Geodesics\|Geodesics]]
	\|-		\|-
	\| <math>\nu</math> \|\| Maslov index, count of conjugate points \|\| [[Hamilton's principle/Applications/The Feynman path integral]]		\| <math>\nu</math> \|\| Maslov index, count of conjugate points \|\| [[Hamilton's principle/Applications/The Feynman path integral\|the Feynman path integral]]
	\|}		\|}

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	A reader who carries this volume's <math>V</math> into Hamilton's equation (A.)		A reader who carries this volume's <math>V</math> into Hamilton's equation (A.)
	will read it as nonsense. The quotations on		will read it as nonsense. The quotations on
	[[Hamilton's principle/Mathematics/Action as a function of its endpoints]] are		[[Hamilton's principle/Mathematics/Action as a function of its endpoints\|Action as a function of its endpoints]] are
	footnoted with this warning at each occurrence.		footnoted with this warning at each occurrence.

Line 94:		Line 94:
	<math>S</math> and <math>W</math>, names the principles separately, and uses		<math>S</math> and <math>W</math>, names the principles separately, and uses
	''stationary'' rather than ''least'' for both. The distinction is the subject of		''stationary'' rather than ''least'' for both. The distinction is the subject of
	[[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton]].		[[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton\|Abbreviated action: Maupertuis versus Hamilton]].

	== Collision 3: <math>\delta</math> ==		== Collision 3: <math>\delta</math> ==
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	The third is the trap. In Hamilton's equations (C.), (D.), (E.), (F.) and (G.),		The third is the trap. In Hamilton's equations (C.), (D.), (E.), (F.) and (G.),
	quoted on		quoted on
	[[Hamilton's principle/Mathematics/Action as a function of its endpoints]] and		[[Hamilton's principle/Mathematics/Action as a function of its endpoints\|Action as a function of its endpoints]] and
	[[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation]], every		[[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation\|the Hamilton-Jacobi equation]], every
	<math>\delta</math> in a quotient is an ordinary partial derivative. Read as		<math>\delta</math> in a quotient is an ordinary partial derivative. Read as
	variations they are meaningless.		variations they are meaningless.
Line 125:		Line 125:

	Her symbols are preserved where her equations are reproduced, on		Her symbols are preserved where her equations are reproduced, on
	[[Hamilton's principle/Mathematics/Noether's first theorem]],		[[Hamilton's principle/Mathematics/Noether's first theorem\|Noether's first theorem]],
	[[Hamilton's principle/Mathematics/Noether's second theorem]] and		[[Hamilton's principle/Mathematics/Noether's second theorem\|Noether's second theorem]] and
	[[Hamilton's principle/Mathematics/Several independent variables]].		[[Hamilton's principle/Mathematics/Several independent variables\|Several independent variables]].

	{\| class="wikitable"		{\| class="wikitable"
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	determinant <math>g</math> and the metric components <math>g_{ik}</math>; the		determinant <math>g</math> and the metric components <math>g_{ik}</math>; the
	collision is real and appears on		collision is real and appears on
	[[Hamilton's principle/Applications/The catenary]] and		[[Hamilton's principle/Applications/The catenary\|the catenary]] and
	[[Hamilton's principle/Applications/The Einstein-Hilbert action]] within the same		[[Hamilton's principle/Applications/The Einstein-Hilbert action\|the Einstein-Hilbert action]] within the same
	volume.		volume.

Line 168:		Line 168:
	mathematics as <math>\qquad\qquad\mathrm{(n)}</math>, because MediaWiki's TeX		mathematics as <math>\qquad\qquad\mathrm{(n)}</math>, because MediaWiki's TeX
	checker rejects the usual tagging command. The cross-page index is		checker rejects the usual tagging command. The cross-page index is
	[[Hamilton's principle/Equation index]].		[[Hamilton's principle/Equation index\|Equation index]].

	== A note on <math>\vert\;\vert</math> ==		== A note on <math>\vert\;\vert</math> ==