Hamilton's principle/Mathematics/The Legendre transform - Revision history

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	functions of the velocities and (10) cannot be inverted. Such Lagrangians are		functions of the velocities and (10) cannot be inverted. Such Lagrangians are
	'''degenerate''', and they are not pathological — they are the rule in gauge		'''degenerate''', and they are not pathological — they are the rule in gauge
	theories, where ~~the electromagnetic Lagrangian on~~		theories, where [[Hamilton's principle/Applications/The electromagnetic Lagrangian\|the electromagnetic Lagrangian]] is exactly
	[[Hamilton's principle/Applications/The electromagnetic Lagrangian\|the electromagnetic Lagrangian]] is exactly
	this case.		this case.

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			{{DISPLAYTITLE:The Legendre transform}}
	{{Plebeian\|Hamilton's principle/Explainers/Reading a curve two ways}}		{{Plebeian\|Hamilton's principle/Explainers/Reading a curve two ways}}

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	<references />		<references />

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{{Plebeian|Hamilton's principle/Explainers/Reading a curve two ways}}

The '''Legendre transform''' exchanges a variable for the slope of a function
with respect to that variable. Applied to a Lagrangian in its velocity slot it
produces the Hamiltonian, and it converts <math>n</math> second-order equations
into <math>2n</math> first-order ones. This page treats the transform on its own,
before it is put to work on
[[Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations]].

== The construction ==

[[File:OPL-HP-F-10.png|thumb|right|400px|The Legendre transform as a tangent-line description.]]

Let <math>f(v)</math> be a real function of one variable, twice differentiable
and strictly convex, so <math>f''(v) > 0</math>. Define the slope

:<math>p = f'(v) \qquad\qquad\mathrm{(1)}</math>

Strict convexity makes <math>f'</math> strictly increasing and therefore
invertible, so (1) can be solved for <math>v = v(p)</math>. The '''Legendre
transform''' of <math>f</math> is

:<math>g(p) = p\,v(p) - f\bigl(v(p)\bigr)
\qquad\qquad\mathrm{(2)}</math>

'''Convexity is the hypothesis that makes the transform well defined.''' Without
it, (1) has no unique solution and (2) is not a function. This is why the
Legendre condition of
[[Hamilton's principle/Mathematics/The Legendre condition]] reappears here: it is
the same second-derivative requirement, met again for a different reason.

== The geometric reading ==

For fixed <math>p</math>, consider the vertical gap between the line
<math>pv</math> and the curve <math>f(v)</math>:

:<math>h(v) = pv - f(v) \qquad\qquad\mathrm{(3)}</math>

Then <math>h'(v) = p - f'(v)</math>, which vanishes exactly at the
<math>v</math> solving (1), and <math>h'' = -f'' < 0</math>, so that stationary
point is a maximum. Therefore

:<math>g(p) = \max_v \bigl[ pv - f(v) \bigr]
\qquad\qquad\mathrm{(4)}</math>

<math>-g(p)</math> is the intercept of the tangent line of slope <math>p</math>.
So the transform re-describes the curve: instead of listing its points, it lists
its tangent lines by slope and intercept. Both descriptions carry the same
information, which is why the transform is invertible.

== It is an involution ==

Differentiate (2) with respect to <math>p</math>:

:<math>\frac{dg}{dp} = v + p\,\frac{dv}{dp}
- f'(v)\,\frac{dv}{dp}
= v + \bigl[ p - f'(v) \bigr]\frac{dv}{dp}
= v
\qquad\qquad\mathrm{(5)}</math>

the bracket vanishing by (1). So <math>dg/dp = v</math>, mirroring
<math>df/dv = p</math>. The relation between <math>f</math> and <math>g</math> is
symmetric, and transforming twice returns <math>f</math>.

This cancellation — the same one that made <math>y''</math> drop out of the
Beltrami derivation on
[[Hamilton's principle/Mathematics/The Beltrami identity]] — is why the transform
is clean. The chain-rule term is annihilated by the defining relation.

Differentiating (5) again gives

:<math>\frac{d^2 g}{dp^2} = \frac{dv}{dp}
= \frac{1}{f''(v)}
\qquad\qquad\mathrm{(6)}</math>

so <math>g</math> is convex too, and sharply curved where <math>f</math> is flat.

== Several variables ==

For <math>f(v_1, \ldots, v_n)</math> set
<math>p_i = \partial f / \partial v_i</math>. Solvability for the
<math>v_i</math> requires the Hessian
<math>\partial^2 f/\partial v_i \partial v_k</math> to be invertible, which is
the non-degeneracy condition of
[[Hamilton's principle/Mathematics/Several dependent variables]]. Then

:<math>g(p) = \sum_{i=1}^n p_i v_i - f(v)
\qquad\qquad\mathrm{(7)}</math>

and <math>\partial g/\partial p_i = v_i</math> by the same cancellation.

== Passive variables ==

If <math>f = f(u, v)</math> and only <math>v</math> is transformed, then
<math>u</math> is a '''passive''' variable and

:<math>\frac{\partial g}{\partial u} = -\frac{\partial f}{\partial u}
\qquad\qquad\mathrm{(8)}</math>

'''The sign flip in (8) is the source of every sign in Hamilton's equations.'''
Proof: differentiating <math>g = pv - f</math> at fixed <math>p</math>,

:<math>\frac{\partial g}{\partial u}
= p\,\frac{\partial v}{\partial u}
- \frac{\partial f}{\partial u}
- \frac{\partial f}{\partial v}\frac{\partial v}{\partial u}
= -\frac{\partial f}{\partial u}
\qquad\qquad\mathrm{(9)}</math>

again by <math>p = \partial f/\partial v</math>.

== The transform in mechanics ==

Apply it to <math>L(x, y, y')</math> with <math>y'</math> active and
<math>x, y</math> passive. The slope variable is the conjugate momentum of
[[Hamilton's principle/Mathematics/The Euler-Lagrange equation]],

:<math>p = \frac{\partial L}{\partial y'}
\qquad\qquad\mathrm{(10)}</math>

and the transform is

:<math>H(x, y, p) = p\,y' - L(x, y, y')
\qquad\qquad\mathrm{(11)}</math>

with <math>y'</math> expressed through <math>p</math>. Two facts drop out at
once.

<math>H</math> is the conserved quantity of the Beltrami identity, so
[[Hamilton's principle/Mathematics/The Beltrami identity]] is the statement that
the Legendre transform of <math>L</math> is constant when <math>L</math> has no
explicit <math>x</math>.

And by (8),

:<math>\frac{\partial H}{\partial y} = -\frac{\partial L}{\partial y}
\qquad\qquad\mathrm{(12)}</math>

which is the minus sign in <math>\dot p = -\partial H/\partial y</math> on the
next page. It comes from the transform, not from mechanics.

== Where the transform fails ==

If the Hessian in the velocities is singular, the momenta are not independent
functions of the velocities and (10) cannot be inverted. Such Lagrangians are
'''degenerate''', and they are not pathological — they are the rule in gauge
theories, where the electromagnetic Lagrangian on
[[Hamilton's principle/Applications/The electromagnetic Lagrangian]] is exactly
this case.

Degeneracy is the same phenomenon that
[[Hamilton's principle/Mathematics/Noether's second theorem]] characterises:
invariance under a group depending on arbitrary functions produces identities
among the Lagrange expressions, and those identities are why the Hessian is
singular. The constrained Hamiltonian formalism that handles such systems is
named as further reading on [[Hamilton's principle/Sources]] and is not treated
in this volume.

== Elsewhere in physics ==

The same transform relates thermodynamic potentials. Internal energy and
Helmholtz free energy are Legendre transforms in the entropy-temperature pair,
exactly as <math>L</math> and <math>H</math> are in the velocity-momentum pair,
and the sign convention differs by a historical accident rather than by any
mathematics.

== Next ==

[[Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations]]
applies the transform to the Euler-Lagrange system.

== References ==

<references />

← Older revision		Revision as of 04:57, 16 September 2026
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	into <math>2n</math> first-order ones. This page treats the transform on its own,		into <math>2n</math> first-order ones. This page treats the transform on its own,
	before it is put to work on		before it is put to work on
	[[Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations]].		[[Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations\|the Hamiltonian and Hamilton's equations]].

	== The construction ==		== The construction ==
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	it, (1) has no unique solution and (2) is not a function. This is why the		it, (1) has no unique solution and (2) is not a function. This is why the
	Legendre condition of		Legendre condition of
	[[Hamilton's principle/Mathematics/The Legendre condition]] reappears here: it is		[[Hamilton's principle/Mathematics/The Legendre condition\|the Legendre condition]] reappears here: it is
	the same second-derivative requirement, met again for a different reason.		the same second-derivative requirement, met again for a different reason.

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	This cancellation — the same one that made <math>y''</math> drop out of the		This cancellation — the same one that made <math>y''</math> drop out of the
	Beltrami derivation on		Beltrami derivation on
	[[Hamilton's principle/Mathematics/The Beltrami identity]] — is why the transform		[[Hamilton's principle/Mathematics/The Beltrami identity\|the Beltrami identity]] — is why the transform
	is clean. The chain-rule term is annihilated by the defining relation.		is clean. The chain-rule term is annihilated by the defining relation.

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	<math>\partial^2 f/\partial v_i \partial v_k</math> to be invertible, which is		<math>\partial^2 f/\partial v_i \partial v_k</math> to be invertible, which is
	the non-degeneracy condition of		the non-degeneracy condition of
	[[Hamilton's principle/Mathematics/Several dependent variables]]. Then		[[Hamilton's principle/Mathematics/Several dependent variables\|Several dependent variables]]. Then

	:<math>g(p) = \sum_{i=1}^n p_i v_i - f(v)		:<math>g(p) = \sum_{i=1}^n p_i v_i - f(v)
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	Apply it to <math>L(x, y, y')</math> with <math>y'</math> active and		Apply it to <math>L(x, y, y')</math> with <math>y'</math> active and
	<math>x, y</math> passive. The slope variable is the conjugate momentum of		<math>x, y</math> passive. The slope variable is the conjugate momentum of
	[[Hamilton's principle/Mathematics/The Euler-Lagrange equation]],		[[Hamilton's principle/Mathematics/The Euler-Lagrange equation\|the Euler-Lagrange equation]],

	:<math>p = \frac{\partial L}{\partial y'}		:<math>p = \frac{\partial L}{\partial y'}
Line 129:		Line 129:

	<math>H</math> is the conserved quantity of the Beltrami identity, so		<math>H</math> is the conserved quantity of the Beltrami identity, so
	[[Hamilton's principle/Mathematics/The Beltrami identity]] is the statement that		[[Hamilton's principle/Mathematics/The Beltrami identity\|the Beltrami identity]] is the statement that
	the Legendre transform of <math>L</math> is constant when <math>L</math> has no		the Legendre transform of <math>L</math> is constant when <math>L</math> has no
	explicit <math>x</math>.		explicit <math>x</math>.
Line 147:		Line 147:
	'''degenerate''', and they are not pathological — they are the rule in gauge		'''degenerate''', and they are not pathological — they are the rule in gauge
	theories, where the electromagnetic Lagrangian on		theories, where the electromagnetic Lagrangian on
	[[Hamilton's principle/Applications/The electromagnetic Lagrangian]] is exactly		[[Hamilton's principle/Applications/The electromagnetic Lagrangian\|the electromagnetic Lagrangian]] is exactly
	this case.		this case.

	Degeneracy is the same phenomenon that		Degeneracy is the same phenomenon that
	[[Hamilton's principle/Mathematics/Noether's second theorem]] characterises:		[[Hamilton's principle/Mathematics/Noether's second theorem\|Noether's second theorem]] characterises:
	invariance under a group depending on arbitrary functions produces identities		invariance under a group depending on arbitrary functions produces identities
	among the Lagrange expressions, and those identities are why the Hessian is		among the Lagrange expressions, and those identities are why the Hessian is
	singular. The constrained Hamiltonian formalism that handles such systems is		singular. The constrained Hamiltonian formalism that handles such systems is
	named as further reading on [[Hamilton's principle/Sources]] and is not treated		named as further reading on [[Hamilton's principle/Sources\|Sources]] and is not treated
	in this volume.		in this volume.

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	== Next ==		== Next ==

	[[Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations]]		[[Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations\|the Hamiltonian and Hamilton's equations]]
	applies the transform to the Euler-Lagrange system.		applies the transform to the Euler-Lagrange system.