Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations - Revision history

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	<math>\partial^2 L / \partial \dot q_i \partial \dot q_k</math> is invertible, so		<math>\partial^2 L / \partial \dot q_i \partial \dot q_k</math> is invertible, so
	(1) inverts to give <math>\dot q_i</math> in terms of		(1) inverts to give <math>\dot q_i</math> in terms of
	<math>(t, q, p)</math>. The '''Hamiltonian''' is ~~the Legendre transform of~~		<math>(t, q, p)</math>. The '''Hamiltonian''' is [[Hamilton's principle/Mathematics/The Legendre transform\|the Legendre transform]]:
	[[Hamilton's principle/Mathematics/The Legendre transform\|the Legendre transform]]:

	:<math>H(t, q, p) = \sum_{i=1}^n p_i \dot q_i - L		:<math>H(t, q, p) = \sum_{i=1}^n p_i \dot q_i - L
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	the sum cancelling pairwise by (7). So <math>H</math> is conserved exactly when		the sum cancelling pairwise by (7). So <math>H</math> is conserved exactly when
	it carries no explicit time. This is ~~the Beltrami identity of~~		it carries no explicit time. This is [[Hamilton's principle/Mathematics/The Beltrami identity\|the Beltrami identity]] in canonical form, and
	[[Hamilton's principle/Mathematics/The Beltrami identity\|the Beltrami identity]] in canonical form, and
	the cancellation in (8) is the sign structure earning its keep.		the cancellation in (8) is the sign structure earning its keep.

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			{{DISPLAYTITLE:The Hamiltonian and Hamilton's equations}}
	{{Plebeian\|Hamilton's principle/Explainers/The Hamiltonian}}		{{Plebeian\|Hamilton's principle/Explainers/The Hamiltonian}}

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	<references />		<references />

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{{Plebeian|Hamilton's principle/Explainers/The Hamiltonian}}

Applying the Legendre transform to a Lagrangian replaces <math>n</math>
second-order equations by <math>2n</math> first-order ones. The new equations are
symmetric in form, treat position and momentum on the same footing, and describe
motion as a flow in a <math>2n</math>-dimensional space. This page derives them
and states what the change of description buys.

== Setup ==

[[File:OPL-HP-F-11.png|thumb|right|400px|Phase portrait of a pendulum.]]

Take <math>L(t, q_i, \dot q_i)</math> with <math>i = 1, \ldots, n</math>, writing
<math>t</math> for the independent variable and <math>q_i</math> for the
coordinates, as the mechanical literature does. Define the momenta

:<math>p_i = \frac{\partial L}{\partial \dot q_i}
\qquad\qquad\mathrm{(1)}</math>

and assume the Hessian
<math>\partial^2 L / \partial \dot q_i \partial \dot q_k</math> is invertible, so
(1) inverts to give <math>\dot q_i</math> in terms of
<math>(t, q, p)</math>. The '''Hamiltonian''' is the Legendre transform of
[[Hamilton's principle/Mathematics/The Legendre transform]]:

:<math>H(t, q, p) = \sum_{i=1}^n p_i \dot q_i - L
\qquad\qquad\mathrm{(2)}</math>

with every <math>\dot q_i</math> eliminated in favour of <math>p</math>. That
elimination is not optional. <math>H</math> is a function of
<math>(t,q,p)</math>, and leaving a <math>\dot q</math> in it makes the
derivatives below meaningless.

== Derivation ==

[[File:OPL-HP-F-24.png|thumb|right|400px|Lagrangian and Hamiltonian descriptions compared.]]

Take the differential of (2), treating all variables as free:

:<math>dH = \sum_i \Bigl( \dot q_i\, dp_i + p_i\, d\dot q_i \Bigr)
- \sum_i \Bigl( \frac{\partial L}{\partial q_i}\, dq_i
+ \frac{\partial L}{\partial \dot q_i}\, d\dot q_i \Bigr)
- \frac{\partial L}{\partial t}\, dt
\qquad\qquad\mathrm{(3)}</math>

The <math>d\dot q_i</math> terms cancel by (1). This is the cancellation of
[[Hamilton's principle/Mathematics/The Legendre transform]] and it is the whole
derivation:

:<math>dH = \sum_i \dot q_i\, dp_i
- \sum_i \frac{\partial L}{\partial q_i}\, dq_i
- \frac{\partial L}{\partial t}\, dt
\qquad\qquad\mathrm{(4)}</math>

Since <math>H</math> is a function of <math>(t,q,p)</math>, its differential is
also

:<math>dH = \sum_i \frac{\partial H}{\partial q_i}\, dq_i
+ \sum_i \frac{\partial H}{\partial p_i}\, dp_i
+ \frac{\partial H}{\partial t}\, dt
\qquad\qquad\mathrm{(5)}</math>

Matching coefficients of the independent differentials in (4) and (5):

:<math>\frac{\partial H}{\partial p_i} = \dot q_i,
\qquad
\frac{\partial H}{\partial q_i} = -\frac{\partial L}{\partial q_i},
\qquad
\frac{\partial H}{\partial t} = -\frac{\partial L}{\partial t}
\qquad\qquad\mathrm{(6)}</math>

Nothing dynamical has been used yet; (6) is pure Legendre transform. Now impose
the Euler-Lagrange equation of
[[Hamilton's principle/Mathematics/The Euler-Lagrange equation]],
<math>\partial L/\partial q_i = \dot p_i</math>. Substituting into the middle
relation of (6) gives '''Hamilton's canonical equations''':

:<math>\dot q_i = \frac{\partial H}{\partial p_i},
\qquad
\dot p_i = -\frac{\partial H}{\partial q_i}
\qquad\qquad\mathrm{(7)}</math>

<math>\blacksquare</math>

The asymmetry of the minus sign is inherited from the passive-variable rule (8)
of [[Hamilton's principle/Mathematics/The Legendre transform]]. It is a fact
about the transform, not about physics.

== What changed ==

{| class="wikitable"
! !! Lagrangian !! Hamiltonian
|-
| unknowns || <math>n</math> || <math>2n</math>
|-
| order || second || first
|-
| arguments || <math>(t, q, \dot q)</math> || <math>(t, q, p)</math>
|-
| space || configuration space || phase space
|-
| a state is || a point and a velocity || a point
|}

The count of constants is unchanged: <math>n</math> second-order equations and
<math>2n</math> first-order equations both carry <math>2n</math> constants. No
information is gained or lost. What changes is that in phase space a single point
determines the entire future, so trajectories never cross, and the motion is a
flow.

That geometric fact is what the reformulation buys, and it is the basis of every
later development — canonical transformations, Liouville's theorem, and the
Hamilton-Jacobi method of
[[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation]].

== Conservation ==

Along a solution,

:<math>\frac{dH}{dt}
= \sum_i \left( \frac{\partial H}{\partial q_i}\dot q_i
+ \frac{\partial H}{\partial p_i}\dot p_i \right)
+ \frac{\partial H}{\partial t}
= \frac{\partial H}{\partial t}
\qquad\qquad\mathrm{(8)}</math>

the sum cancelling pairwise by (7). So <math>H</math> is conserved exactly when
it carries no explicit time. This is the Beltrami identity of
[[Hamilton's principle/Mathematics/The Beltrami identity]] in canonical form, and
the cancellation in (8) is the sign structure earning its keep.

If in addition <math>H</math> does not contain <math>q_k</math>, then (7) gives
<math>\dot p_k = 0</math> immediately, and the coordinate is cyclic. In the
Hamiltonian description a conservation law is read off by inspection.

== When is <math>H</math> the energy? ==

Not always, and the conditions are worth stating because the identification is
usually made silently.

If the kinetic energy is a homogeneous quadratic in the velocities and the
potential depends on position only, then <math>L = T - V</math> and Euler's
theorem on homogeneous functions gives
<math>\sum_i p_i\dot q_i = 2T</math>, so

:<math>H = 2T - (T - V) = T + V
\qquad\qquad\mathrm{(9)}</math>

and <math>H</math> is the energy. If the coordinates are time-dependent — a
rotating frame, a moving constraint — then <math>T</math> is not homogeneous
quadratic and <math>H \neq T + V</math>, though <math>H</math> may still be
conserved. Conserved and equal to the energy are two different claims.

== Hamilton's own route ==

Hamilton did not arrive at (7) through the Legendre transform. He introduced the
characteristic function first and obtained the canonical form as a consequence,
remarking that his transformed equations coincide in all respects, a slight
difference of notation excepted, with the elegant canonical forms in the
''Mécanique Analytique'' of Lagrange.<ref>Hamilton, ''On a General Method in
Dynamics'' (1834), §8, following equation (Z.). Verbatim phrase. ESTABLISHED
source.</ref>

The modern presentation inverts his order for good pedagogical reasons, and the
cost is that the characteristic function looks like an afterthought when it was
the point. [[Hamilton's principle/Mathematics/Action as a function of its endpoints]]
restores his ordering.

== Worked example: the harmonic oscillator ==

:<math>L = \tfrac{1}{2}m\dot q^2 - \tfrac{1}{2}m\omega^2 q^2
\qquad\qquad\mathrm{(10)}</math>

Then <math>p = m\dot q</math>, so <math>\dot q = p/m</math>, and

:<math>H = p\cdot\frac{p}{m}
- \left( \frac{p^2}{2m} - \tfrac{1}{2}m\omega^2 q^2 \right)
= \frac{p^2}{2m} + \tfrac{1}{2}m\omega^2 q^2
\qquad\qquad\mathrm{(11)}</math>

Hamilton's equations are

:<math>\dot q = \frac{p}{m},
\qquad
\dot p = -m\omega^2 q
\qquad\qquad\mathrm{(12)}</math>

Eliminating <math>p</math> recovers <math>\ddot q + \omega^2 q = 0</math>. But
(11) read directly gives the phase portrait without solving anything: level sets
of <math>H</math> are ellipses, so every trajectory is a closed curve traversed
at constant energy. Reading the motion off the level sets, rather than
integrating, is the characteristic Hamiltonian move.

== Next ==

[[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation]] seeks a
canonical transformation that makes <math>H</math> vanish, converting the flow
into a single partial differential equation.

== References ==

<references />

← Older revision		Revision as of 04:57, 16 September 2026
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	(1) inverts to give <math>\dot q_i</math> in terms of		(1) inverts to give <math>\dot q_i</math> in terms of
	<math>(t, q, p)</math>. The '''Hamiltonian''' is the Legendre transform of		<math>(t, q, p)</math>. The '''Hamiltonian''' is the Legendre transform of
	[[Hamilton's principle/Mathematics/The Legendre transform]]:		[[Hamilton's principle/Mathematics/The Legendre transform\|the Legendre transform]]:

	:<math>H(t, q, p) = \sum_{i=1}^n p_i \dot q_i - L		:<math>H(t, q, p) = \sum_{i=1}^n p_i \dot q_i - L
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	The <math>d\dot q_i</math> terms cancel by (1). This is the cancellation of		The <math>d\dot q_i</math> terms cancel by (1). This is the cancellation of
	[[Hamilton's principle/Mathematics/The Legendre transform]] and it is the whole		[[Hamilton's principle/Mathematics/The Legendre transform\|the Legendre transform]] and it is the whole
	derivation:		derivation:

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	Nothing dynamical has been used yet; (6) is pure Legendre transform. Now impose		Nothing dynamical has been used yet; (6) is pure Legendre transform. Now impose
	the Euler-Lagrange equation of		the Euler-Lagrange equation of
	[[Hamilton's principle/Mathematics/The Euler-Lagrange equation]],		[[Hamilton's principle/Mathematics/The Euler-Lagrange equation\|the Euler-Lagrange equation]],
	<math>\partial L/\partial q_i = \dot p_i</math>. Substituting into the middle		<math>\partial L/\partial q_i = \dot p_i</math>. Substituting into the middle
	relation of (6) gives '''Hamilton's canonical equations''':		relation of (6) gives '''Hamilton's canonical equations''':
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	The asymmetry of the minus sign is inherited from the passive-variable rule (8)		The asymmetry of the minus sign is inherited from the passive-variable rule (8)
	of [[Hamilton's principle/Mathematics/The Legendre transform]]. It is a fact		of [[Hamilton's principle/Mathematics/The Legendre transform\|the Legendre transform]]. It is a fact
	about the transform, not about physics.		about the transform, not about physics.

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	later development — canonical transformations, Liouville's theorem, and the		later development — canonical transformations, Liouville's theorem, and the
	Hamilton-Jacobi method of		Hamilton-Jacobi method of
	[[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation]].		[[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation\|the Hamilton-Jacobi equation]].

	== Conservation ==		== Conservation ==
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	the sum cancelling pairwise by (7). So <math>H</math> is conserved exactly when		the sum cancelling pairwise by (7). So <math>H</math> is conserved exactly when
	it carries no explicit time. This is the Beltrami identity of		it carries no explicit time. This is the Beltrami identity of
	[[Hamilton's principle/Mathematics/The Beltrami identity]] in canonical form, and		[[Hamilton's principle/Mathematics/The Beltrami identity\|the Beltrami identity]] in canonical form, and
	the cancellation in (8) is the sign structure earning its keep.		the cancellation in (8) is the sign structure earning its keep.

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	The modern presentation inverts his order for good pedagogical reasons, and the		The modern presentation inverts his order for good pedagogical reasons, and the
	cost is that the characteristic function looks like an afterthought when it was		cost is that the characteristic function looks like an afterthought when it was
	the point. [[Hamilton's principle/Mathematics/Action as a function of its endpoints]]		the point. [[Hamilton's principle/Mathematics/Action as a function of its endpoints\|Action as a function of its endpoints]]
	restores his ordering.		restores his ordering.

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	== Next ==		== Next ==

	[[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation]] seeks a		[[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation\|the Hamilton-Jacobi equation]] seeks a
	canonical transformation that makes <math>H</math> vanish, converting the flow		canonical transformation that makes <math>H</math> vanish, converting the flow
	into a single partial differential equation.		into a single partial differential equation.