Hamilton's principle/Mathematics/The Beltrami identity - Revision history

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	{{Plebeian\|Hamilton's principle/Explainers/Free ends}}		{{Plebeian\|Hamilton's principle/Explainers/Free ends}}

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{{Plebeian|Hamilton's principle/Explainers/Free ends}}

When the integrand does not contain the independent variable, the
Euler-Lagrange equation admits a first integral. It reduces a second-order
equation to a first-order one and is the single most useful labour-saving result
in the applications track. It is called the '''Beltrami identity''', and in
mechanics it is conservation of energy.

== Statement ==

[[File:OPL-HP-F-43.png|thumb|right|400px|A quantity conserved along the whole path.]]

'''Theorem.''' Let <math>L = L(y, y')</math> contain no explicit dependence on
<math>x</math>, so <math>\partial L/\partial x = 0</math>. Then along any
extremal,

:<math>L - y'\,\frac{\partial L}{\partial y'} = C
\qquad\qquad\mathrm{(1)}</math>

for some constant <math>C</math>.

== Derivation ==

The proof does not integrate the Euler-Lagrange equation. It differentiates the
left side of (1) and shows the result vanishes.

Take the total derivative of <math>L</math> along a path, treating
<math>L</math> as a function of its three slots and using the chain rule:

:<math>\frac{dL}{dx}
= \frac{\partial L}{\partial x}
+ \frac{\partial L}{\partial y}\, y'
+ \frac{\partial L}{\partial y'}\, y''
\qquad\qquad\mathrm{(2)}</math>

Now differentiate the product term:

:<math>\frac{d}{dx}\!\left( y'\, \frac{\partial L}{\partial y'} \right)
= y''\, \frac{\partial L}{\partial y'}
+ y'\, \frac{d}{dx}\!\left( \frac{\partial L}{\partial y'} \right)
\qquad\qquad\mathrm{(3)}</math>

Subtract (3) from (2). The <math>y''</math> terms cancel exactly:

:<math>\frac{d}{dx}\!\left( L - y'\,\frac{\partial L}{\partial y'} \right)
= \frac{\partial L}{\partial x}
+ y'\left[ \frac{\partial L}{\partial y}
- \frac{d}{dx}\!\left( \frac{\partial L}{\partial y'} \right) \right]
\qquad\qquad\mathrm{(4)}</math>

Equation (4) is an identity. It holds for any <math>L</math> and any path,
extremal or not, and it is worth recording in that generality.

Two hypotheses now kill the two terms on the right. Along an extremal the bracket
vanishes, by equation (2) of
[[Hamilton's principle/Mathematics/The Euler-Lagrange equation]]. And
<math>\partial L/\partial x = 0</math> by assumption. So the left side is zero
and the bracketed quantity is constant. <math>\blacksquare</math>

The cancellation of <math>y''</math> between (2) and (3) is the mechanism, and
it is the reason the combination <math>L - y'L_{y'}</math> and not some other
combination is the conserved one.

== The two hypotheses are separable ==

Equation (4) shows the structure clearly:

{| class="wikitable"
! condition dropped !! what survives
|-
| neither dropped || <math>L - y'L_{y'}</math> is constant
|-
| <math>L</math> depends on <math>x</math> || <math>\frac{d}{dx}(L - y'L_{y'}) = \partial L/\partial x</math>, still useful
|-
| path is not an extremal || no conclusion
|}

The middle row is the honest general statement, and it is how the quantity is
handled in time-dependent mechanics.

== Relation to the Hamiltonian ==

The conserved quantity in (1) is, up to sign, the Hamiltonian. Writing
<math>p = L_{y'}</math> for the conjugate momentum,

:<math>H = y'p - L = -C \qquad\qquad\mathrm{(5)}</math>

This is not a coincidence and it is not merely notation. The map from
<math>(y, y', L)</math> to <math>(y, p, H)</math> is the Legendre transform,
developed on
[[Hamilton's principle/Mathematics/The Legendre transform]], and Beltrami's
identity is the statement that <math>H</math> is conserved when <math>L</math>
carries no explicit <math>x</math>. When <math>x</math> is time, that is
conservation of energy, and it is one entry in the symmetry-to-conservation table
on [[Hamilton's principle/Mathematics/Noether's first theorem]].

The pairing is worth setting out, since the machinery track has now produced both
halves:

{| class="wikitable"
! <math>L</math> independent of !! first integral !! mechanical reading
|-
| <math>y</math> || <math>L_{y'} = \text{const}</math> || momentum conserved
|-
| <math>x</math> || <math>L - y'L_{y'} = \text{const}</math> || energy conserved
|}

== Worked example: the catenary ==

A uniform chain of fixed length hangs between two points. It minimises potential
energy, which up to constants is

:<math>J[y] = \int_a^b y\,\sqrt{1 + y'^2}\; dx
\qquad\qquad\mathrm{(6)}</math>

<math>L = y\sqrt{1 + y'^2}</math> is free of <math>x</math>, so (1) applies:

:<math>y\sqrt{1 + y'^2} - y'\cdot\frac{y\,y'}{\sqrt{1 + y'^2}} = C
\qquad\qquad\mathrm{(7)}</math>

Put the left side over a common denominator. The numerator is
<math>y(1 + y'^2) - y\,y'^2 = y</math>, so (7) collapses to

:<math>\frac{y}{\sqrt{1 + y'^2}} = C
\qquad\qquad\mathrm{(8)}</math>

A second-order problem has become a first-order separable one in two lines.
Rearranging, <math>y' = \sqrt{y^2/C^2 - 1}</math>, which separates and integrates
to

:<math>y = C \cosh\!\left( \frac{x - x_0}{C} \right)
\qquad\qquad\mathrm{(9)}</math>

The hanging chain is a catenary. The constants <math>C</math> and
<math>x_0</math> are fixed by the two endpoint conditions.

Two caveats. First, the length constraint has been suppressed here; treating it
properly requires a multiplier, and the full problem is worked on
[[Hamilton's principle/Mathematics/Isoperimetric problems]] and
[[Hamilton's principle/Applications/The catenary]]. Second, (9) is an extremal.
That it is the minimum is a separate claim, addressed on
[[Hamilton's principle/Mathematics/The second variation]].

== Naming ==

The identity is named for Eugenio Beltrami, though the result is older and the
attribution is conventional rather than historical. This volume uses the name
because it is standard, and records here that it is a label of convenience.

== Next ==

[[Hamilton's principle/Mathematics/Several dependent variables]] extends the
machinery from one unknown function to many.

== References ==

<references />

← Older revision		Revision as of 04:57, 16 September 2026
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	Two hypotheses now kill the two terms on the right. Along an extremal the bracket		Two hypotheses now kill the two terms on the right. Along an extremal the bracket
	vanishes, by equation (2) of		vanishes, by equation (2) of
	[[Hamilton's principle/Mathematics/The Euler-Lagrange equation]]. And		[[Hamilton's principle/Mathematics/The Euler-Lagrange equation\|the Euler-Lagrange equation]]. And
	<math>\partial L/\partial x = 0</math> by assumption. So the left side is zero		<math>\partial L/\partial x = 0</math> by assumption. So the left side is zero
	and the bracketed quantity is constant. <math>\blacksquare</math>		and the bracketed quantity is constant. <math>\blacksquare</math>
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	<math>(y, y', L)</math> to <math>(y, p, H)</math> is the Legendre transform,		<math>(y, y', L)</math> to <math>(y, p, H)</math> is the Legendre transform,
	developed on		developed on
	[[Hamilton's principle/Mathematics/The Legendre transform]], and Beltrami's		[[Hamilton's principle/Mathematics/The Legendre transform\|the Legendre transform]], and Beltrami's
	identity is the statement that <math>H</math> is conserved when <math>L</math>		identity is the statement that <math>H</math> is conserved when <math>L</math>
	carries no explicit <math>x</math>. When <math>x</math> is time, that is		carries no explicit <math>x</math>. When <math>x</math> is time, that is
	conservation of energy, and it is one entry in the symmetry-to-conservation table		conservation of energy, and it is one entry in the symmetry-to-conservation table
	on [[Hamilton's principle/Mathematics/Noether's first theorem]].		on [[Hamilton's principle/Mathematics/Noether's first theorem\|Noether's first theorem]].

	The pairing is worth setting out, since the machinery track has now produced both		The pairing is worth setting out, since the machinery track has now produced both
Line 138:		Line 138:
	Two caveats. First, the length constraint has been suppressed here; treating it		Two caveats. First, the length constraint has been suppressed here; treating it
	properly requires a multiplier, and the full problem is worked on		properly requires a multiplier, and the full problem is worked on
	[[Hamilton's principle/Mathematics/Isoperimetric problems]] and		[[Hamilton's principle/Mathematics/Isoperimetric problems\|Isoperimetric problems]] and
	[[Hamilton's principle/Applications/The catenary]]. Second, (9) is an extremal.		[[Hamilton's principle/Applications/The catenary\|the catenary]]. Second, (9) is an extremal.
	That it is the minimum is a separate claim, addressed on		That it is the minimum is a separate claim, addressed on
	[[Hamilton's principle/Mathematics/The second variation]].		[[Hamilton's principle/Mathematics/The second variation\|the second variation]].

	== Naming ==		== Naming ==
Line 151:		Line 151:
	== Next ==		== Next ==

	[[Hamilton's principle/Mathematics/Several dependent variables]] extends the		[[Hamilton's principle/Mathematics/Several dependent variables\|Several dependent variables]] extends the
	machinery from one unknown function to many.		machinery from one unknown function to many.