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			{{DISPLAYTITLE:The rigid body}}
	The '''rigid body''' is the first system in this volume whose configuration space		The '''rigid body''' is the first system in this volume whose configuration space
	is not a piece of Euclidean space. Its orientations form a curved,		is not a piece of Euclidean space. Its orientations form a curved,
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	<references />		<references />

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The '''rigid body''' is the first system in this volume whose configuration space
is not a piece of Euclidean space. Its orientations form a curved,
three-dimensional group manifold, and that fact — rather than any difficulty in
the forces — is what makes rigid-body motion instructive. It also makes the
variational treatment worth doing carefully, because the naive coordinate
approach hides the structure and introduces singularities that are not physical.

== The configuration space ==

[[File:OPL-HP-F-27.png|thumb|right|400px|Stability of rotation about the three principal axes.]]

A rigid body is a system of particles whose mutual distances are fixed. Those
constraints are holonomic in the sense of
[[Hamilton's principle/Mathematics/Holonomic constraints]] — relations among
coordinates, not velocities — so elimination is legitimate.

Eliminating them leaves six degrees of freedom: three for the position of the
centre of mass and three for the orientation. The kinetic energy separates,

:<math>T = \tfrac{1}{2}M\vert \dot{R} \vert^2 + T_{\text{rot}}
\qquad\qquad\mathrm{(1)}</math>

exactly as Hamilton separates the motion of the centre of gravity from the
relative motion in the two-body problem of
[[Hamilton's principle/Applications/Central force and the Kepler problem]]. The
translational part is a free particle. Everything of interest is in
<math>T_{\text{rot}}</math>.

The orientation is an element of the rotation group. That space is compact and
curved, and it is not covered by any single coordinate patch — a fact that will
reappear below as a defect of the Euler angles rather than of the body.

== Rotational kinetic energy ==

[[File:OPL-HP-F-47.png|thumb|right|400px|The effective potential of a central-force problem.]]

In a frame fixed in the body, with axes along the principal axes of inertia, the
angular velocity <math>\omega = (\omega_1, \omega_2, \omega_3)</math> gives

:<math>T_{\text{rot}} = \tfrac{1}{2}\bigl(
I_1\omega_1^2 + I_2\omega_2^2 + I_3\omega_3^2 \bigr)
\qquad\qquad\mathrm{(2)}</math>

with <math>I_1, I_2, I_3</math> the principal moments of inertia. This is a
metric on the orientation space, and it is a '''left-invariant''' one: the
kinetic energy of a motion does not change if the whole body is first rotated.
That invariance is a symmetry of the action, so
[[Hamilton's principle/Mathematics/Noether's first theorem]] applies and the
conserved quantity is the angular momentum in the '''space''' frame.

== Euler angles, and their defect ==

The usual coordinates are the Euler angles
<math>(\phi, \vartheta, \psi)</math>, in terms of which

:<math>\omega_1 = \dot\phi\sin\vartheta\sin\psi + \dot\vartheta\cos\psi,
\quad
\omega_2 = \dot\phi\sin\vartheta\cos\psi - \dot\vartheta\sin\psi,
\quad
\omega_3 = \dot\phi\cos\vartheta + \dot\psi
\qquad\qquad\mathrm{(3)}</math>

Substituting (3) into (2) gives a Lagrangian in three coordinates and their
velocities, and the Euler-Lagrange equations of
[[Hamilton's principle/Mathematics/Several dependent variables]] apply directly.

'''But the coordinates are singular at <math>\vartheta = 0</math>.''' There
<math>\phi</math> and <math>\psi</math> generate the same rotation, the map from
angles to orientations degenerates, and the equations blow up. Nothing happens to
the body at that configuration; the singularity is in the chart. It is the same
gimbal lock that afflicts mechanical gyroscope mountings, and it is a consequence
of trying to cover a curved compact space with three angles.

This is worth stating plainly because it is a general lesson about the
variational method: '''the Euler-Lagrange equations are only as good as the
coordinates, and a singular chart produces singular equations for a
non-singular motion.'''

== The torque-free case ==

Take the body free, so <math>V = 0</math> and <math>L = T_{\text{rot}}</math>.
Two conservation laws follow without solving anything.

<math>L</math> carries no explicit time, so by
[[Hamilton's principle/Mathematics/The Beltrami identity]] the energy is
conserved:

:<math>2T = I_1\omega_1^2 + I_2\omega_2^2 + I_3\omega_3^2 = \text{const}
\qquad\qquad\mathrm{(4)}</math>

<math>L</math> is invariant under rotations of the body in space, so by Noether
the space-frame angular momentum is conserved, and so is its magnitude:

:<math>\vert \mathbf{L} \vert^2
= I_1^2\omega_1^2 + I_2^2\omega_2^2 + I_3^2\omega_3^2 = \text{const}
\qquad\qquad\mathrm{(5)}</math>

Equations (4) and (5) are two surfaces in the space of body-frame angular
velocity: an ellipsoid and a second ellipsoid with squared semi-axes. '''The
motion is confined to their intersection''', a closed curve, and the entire
qualitative behaviour can be read off without integrating anything.

The full equations of motion, obtained from (2) and (3), are Euler's:

:<math>I_1\dot\omega_1 = (I_2 - I_3)\,\omega_2\omega_3
\qquad\qquad\mathrm{(6)}</math>

with the other two by cyclic permutation. Note that (6) is written in the body
frame, where the moments of inertia are constant; in the space frame the inertia
tensor varies and the equations are worse.

== The intermediate axis theorem ==

The intersection curves of (4) and (5) supply a result that is easy to state and
easy to demonstrate.

Take <math>I_1 < I_2 < I_3</math> and consider rotation about a principal axis,
perturbed slightly. Linearising (6) about rotation near axis 1 gives a
perturbation equation with growth rate squared

:<math>\lambda^2 = \frac{(I_2 - I_1)(I_3 - I_1)}{I_2 I_3}\,\omega_1^2
\qquad\qquad\mathrm{(7)}</math>

up to sign, and the analogous expressions for the other axes. For the smallest
moment, both factors in the numerator are positive; for the largest, both are
negative; in each case the product of the corresponding rates is negative,
<math>\lambda</math> is imaginary, and the perturbation oscillates. '''Stable.'''

For the '''intermediate''' axis the two factors have opposite signs,
<math>\lambda</math> is real, and the perturbation grows exponentially.
'''Unstable.'''

This is the tennis-racket theorem, and it is visible in the geometry as well: the
intersection curves near the intermediate axis are hyperbolic rather than closed
loops around the axis. A book or a phone spun about its intermediate axis tumbles;
about either of the others it does not.

The result required no integration. It came from two conserved quantities and the
shape of their intersection, which is the characteristic payoff of the
conservation-law machinery.

== The heavy symmetric top ==

Add gravity, with the body supported at a point on its symmetry axis. Then
<math>I_1 = I_2</math>, and

:<math>L = \tfrac{1}{2}I_1\bigl( \dot\vartheta^2
+ \dot\phi^2\sin^2\vartheta \bigr)
+ \tfrac{1}{2}I_3\bigl( \dot\phi\cos\vartheta + \dot\psi \bigr)^2
- Mg_0 \lambda \cos\vartheta
\qquad\qquad\mathrm{(8)}</math>

with <math>\lambda</math> the distance from support to centre of mass.

Both <math>\phi</math> and <math>\psi</math> are cyclic, so two momenta are
conserved immediately, and the energy is a third constant. Three conserved
quantities for three degrees of freedom: the system is integrable, and the
problem reduces to a single first-order equation in <math>\vartheta</math>,

:<math>\tfrac{1}{2}I_1\dot\vartheta^2 + V_{\text{eff}}(\vartheta) = E'
\qquad\qquad\mathrm{(9)}</math>

which is a one-dimensional problem in an effective potential, exactly as the
Kepler problem reduced on
[[Hamilton's principle/Applications/Central force and the Kepler problem]].

<math>V_{\text{eff}}</math> rises to infinity at <math>\vartheta = 0</math> and
<math>\pi</math> when the two conserved momenta differ, so <math>\vartheta</math>
oscillates between turning points: the axis '''nutates''' while it precesses. The
qualitative motion follows from the shape of a one-dimensional potential, without
solving (9), which requires elliptic functions.

<blockquote>
Provenance: every equation on this page is derived here or is standard.
Euler's equations (6) are named for their author; no primary source for them has
been verified for this volume, and none is quoted. See
[[Hamilton's principle/Sources]].
</blockquote>

== Next ==

[[Hamilton's principle/Applications/Continuum systems: the vibrating string]]
takes the step from finitely many degrees of freedom to infinitely many.

== References ==

<references />

← Older revision		Revision as of 04:58, 16 September 2026
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	A rigid body is a system of particles whose mutual distances are fixed. Those		A rigid body is a system of particles whose mutual distances are fixed. Those
	constraints are holonomic in the sense of		constraints are holonomic in the sense of
	[[Hamilton's principle/Mathematics/Holonomic constraints]] — relations among		[[Hamilton's principle/Mathematics/Holonomic constraints\|Holonomic constraints]] — relations among
	coordinates, not velocities — so elimination is legitimate.		coordinates, not velocities — so elimination is legitimate.

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	exactly as Hamilton separates the motion of the centre of gravity from the		exactly as Hamilton separates the motion of the centre of gravity from the
	relative motion in the two-body problem of		relative motion in the two-body problem of
	[[Hamilton's principle/Applications/Central force and the Kepler problem]]. The		[[Hamilton's principle/Applications/Central force and the Kepler problem\|Central force and the Kepler problem]]. The
	translational part is a free particle. Everything of interest is in		translational part is a free particle. Everything of interest is in
	<math>T_{\text{rot}}</math>.		<math>T_{\text{rot}}</math>.
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	kinetic energy of a motion does not change if the whole body is first rotated.		kinetic energy of a motion does not change if the whole body is first rotated.
	That invariance is a symmetry of the action, so		That invariance is a symmetry of the action, so
	[[Hamilton's principle/Mathematics/Noether's first theorem]] applies and the		[[Hamilton's principle/Mathematics/Noether's first theorem\|Noether's first theorem]] applies and the
	conserved quantity is the angular momentum in the '''space''' frame.		conserved quantity is the angular momentum in the '''space''' frame.

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	Substituting (3) into (2) gives a Lagrangian in three coordinates and their		Substituting (3) into (2) gives a Lagrangian in three coordinates and their
	velocities, and the Euler-Lagrange equations of		velocities, and the Euler-Lagrange equations of
	[[Hamilton's principle/Mathematics/Several dependent variables]] apply directly.		[[Hamilton's principle/Mathematics/Several dependent variables\|Several dependent variables]] apply directly.

	'''But the coordinates are singular at <math>\vartheta = 0</math>.''' There		'''But the coordinates are singular at <math>\vartheta = 0</math>.''' There
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	<math>L</math> carries no explicit time, so by		<math>L</math> carries no explicit time, so by
	[[Hamilton's principle/Mathematics/The Beltrami identity]] the energy is		[[Hamilton's principle/Mathematics/The Beltrami identity\|the Beltrami identity]] the energy is
	conserved:		conserved:

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	which is a one-dimensional problem in an effective potential, exactly as the		which is a one-dimensional problem in an effective potential, exactly as the
	Kepler problem reduced on		Kepler problem reduced on
	[[Hamilton's principle/Applications/Central force and the Kepler problem]].		[[Hamilton's principle/Applications/Central force and the Kepler problem\|Central force and the Kepler problem]].

	<math>V_{\text{eff}}</math> rises to infinity at <math>\vartheta = 0</math> and		<math>V_{\text{eff}}</math> rises to infinity at <math>\vartheta = 0</math> and
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	Euler's equations (6) are named for their author; no primary source for them has		Euler's equations (6) are named for their author; no primary source for them has
	been verified for this volume, and none is quoted. See		been verified for this volume, and none is quoted. See
	[[Hamilton's principle/Sources]].		[[Hamilton's principle/Sources\|Sources]].
	</blockquote>		</blockquote>

	== Next ==		== Next ==

	[[Hamilton's principle/Applications/Continuum systems: the vibrating string]]		[[Hamilton's principle/Applications/Continuum systems: the vibrating string\|Continuum systems: the vibrating string]]
	takes the step from finitely many degrees of freedom to infinitely many.		takes the step from finitely many degrees of freedom to infinitely many.