Classical Spin: The SO(3) Poisson Bracket and Euler's Equations for Rigid Body Rotation

Intuition Beginner

A spinning top does something remarkable: it stays upright while spinning, wobbles when it slows down, and falls over when it stops. This behaviour is not magic -- it follows from the same angular momentum algebra developed in 09.04.03. The key insight is that the three components of angular momentum form a closed algebra under the Poisson bracket, and for a rigid body this algebra becomes the entire equation of motion.

When you spin a top, you give it angular momentum L. In free space (no gravity, no friction), that angular momentum is conserved: the total vector L never changes direction or magnitude. But the top is not a point mass -- it is an extended body with a definite shape. If the body is asymmetric (longer in one direction than another), the angular momentum and the angular velocity point in different directions. The body rotates about one axis, but its angular momentum points along a different one.

The result is that the body's rotation axis precesses -- it traces out a cone around the fixed angular momentum vector. This is torque-free precession, and it happens whenever a rigid body rotates about an axis that is not a principal axis of its mass distribution.

The Earth itself does this. The Earth is slightly oblate (wider at the equator), so its principal axes of inertia are distinct. The rotation axis does not quite coincide with the symmetry axis, and the result is a very slow precession of the rotation pole -- a wobble with a period of about 433 days (the Chandler wobble). The same mathematics that describes a spinning top describes the wobble of the planet.

Euler's disk: dissipation-driven spin-up

A striking demonstration of rigid-body rotation is Euler's disk -- a heavy coin spinning on a flat surface. As the disk loses energy to rolling friction and air resistance, the precession rate increases. The disk spins faster and faster while its centre of mass drops lower and lower, until the motion abruptly stops. This counter-intuitive speed-up is not a violation of energy conservation -- the disk is losing energy the entire time. The precession rate scales inversely with the tilt angle, so as the tilt decreases (the disk flattens toward the table), the precession rate diverges. The finite thickness of the disk and the contact mechanics eventually arrest the motion. The mathematics behind Euler's disk comes from Euler's rigid body equations with dissipation added; the torque-free equations are the starting point.

Why angular momentum components do not commute

In 09.04.03 you saw that the Poisson brackets of angular momentum components satisfy:

$$\{L_x,L_y\}=L_z,\{L_y,L_z\}=L_x,\{L_z,L_x\}=L_y.$$

This means the three components of angular momentum cannot all be chosen independently. If you fix $L_x$ and $L_y$ (via initial conditions and some symmetry), the bracket tells you how $L_z$ must evolve -- it is constrained by the algebraic structure. For a free rigid body, this algebra becomes the equation of motion. The time derivative of each component of angular momentum is determined by its Poisson bracket with the Hamiltonian, and the result is a set of three coupled first-order differential equations called Euler's equations.

The physical content: a rigid body does not simply spin at a constant rate about a fixed axis. The angular velocity vector itself moves, tracing a curve through the body (the polhode) and through space (the herpolhode). The angular momentum vector in space is fixed (for the torque-free case), but the angular velocity vector precesses around it. This precession is the direct physical manifestation of the non-commutativity of angular momentum components -- the Poisson bracket $\{L_i,L_j\}={ϵ}_{ijk}{L}_k$ is not just algebra; it is the law of motion for every rotating body.

Visual Beginner

Figure: A spinning top with its symmetry axis tilted at angle theta from the vertical. The angular momentum vector L points approximately along the symmetry axis but slightly upward. The body's rotation axis precesses around L, tracing a cone. The angular velocity omega points along the instantaneous rotation axis, not along L (because the body is not spherically symmetric). The polhode curve (path of omega in the body frame) is shown as an ellipse on the energy ellipsoid.

Figure: The three principal axes of a rigid body (labelled 1, 2, 3) with the inertia ellipsoid drawn as an ellipsoid whose semi-axes are proportional to 1/sqrt(I_i). The angular velocity vector omega lies on the ellipsoid surface. For rotation about the axis of largest or smallest moment of inertia, the polhode is a tight curve near a pole; for rotation about the intermediate axis, the polhode wanders widely (instability).

Figure: Euler's disk -- a coin spinning on a surface. As the tilt angle alpha decreases, the precession rate omega_p increases. The contact point traces a shrinking circle on the table. The final moments show rapid precession before abrupt arrest.

Worked example Beginner

Consider a rigid body with three distinct principal moments of inertia $I_1<I_2<I_3$, rotating freely (no external torques). The angular momentum is conserved, so $|L{|}^2=L_1^2+L_2^2+L_3^2$ is constant, and the rotational kinetic energy $T=L_1^2/(2I_1)+L_2^2/(2I_2)+L_3^2/(2I_3)$ is also constant.

These two constraints define two surfaces in $(L_1,L_2,L_3)$ space: a sphere (from $|L{|}^2$) and an ellipsoid (from $T$). The actual trajectory of the angular momentum vector in the body frame must lie on the intersection of these two surfaces.

Now suppose the body is spinning almost entirely about axis 1 (the axis of smallest moment of inertia). Then $L_1$ is large and $L_2,L_3$ are small. The intersection of the sphere and the ellipsoid near the "pole" $L_1\approx |\mathbf{L}|$ is a small closed curve. The angular momentum vector in the body frame traces this curve, making small oscillations around the axis. The same happens near axis 3 (the axis of largest moment of inertia).

But near axis 2 (the intermediate axis), the intersection curves spread out dramatically. A small perturbation from pure rotation about axis 2 sends the angular momentum vector on a wide arc that visits all three components. This is the intermediate axis theorem (also called the tennis racket theorem or the Dzhanibekov effect): rotation about the axis with the intermediate moment of inertia is unstable, while rotation about the axes with the largest and smallest moments of inertia is stable.

You can demonstrate this with a book or a phone: flip it about its long axis or its short axis and it spins stably; flip it about its intermediate axis and it tumbles chaotically.

Check your understanding Beginner

Formal definition Intermediate+

Rigid body kinematics

A rigid body is a system of particles whose mutual distances are fixed. The configuration of a rigid body (with one point fixed) is described by a rotation matrix $R\in \mathrm{SO}(3)$ that maps body-frame coordinates to space-frame coordinates. The rotation matrix evolves according to:

$$\dot{R}=R\hat{\omega }$$

where $\hat{\omega }$ is the skew-symmetric matrix corresponding to the angular velocity $\omega$ in the body frame (so ${\hat{\omega }}_{ij}=-{ϵ}_{ijk}{\omega }_k$). The angular velocity in the space frame is $\Omega =R\omega$.

The inertia tensor

For a rigid body with mass distribution $\rho (\mathbf{x})$ in the body frame, the inertia tensor is the $3\times 3$ symmetric matrix:

$$I_{ij}=\int \rho (\mathbf{x})\left(|\mathbf{x}{|}^2{\delta }_{ij}-x_i{x}_j\right)d^{3}x.$$

By a suitable choice of body-frame axes (the principal axes), the inertia tensor is diagonal: $I=\mathrm{diag}(I_1,I_2,I_3)$. The principal moments of inertia $I_1,I_2,I_3$ are the eigenvalues of $I$ and are all positive.

The angular momentum in the body frame is $\mathbf{L}=I\omega$, or componentwise $L_i=I_i{\omega }_i$ (no sum, in the principal-axis basis). The rotational kinetic energy is:

$$T=\frac{1}{2}\omega \cdot I\omega =\frac{1}{2}\left(\frac{L_1^2}{I_1}+\frac{L_2^2}{I_2}+\frac{L_3^2}{I_3}\right).$$

Euler's equations

For a torque-free rigid body, the equations of motion in the body frame are Euler's equations:

$$\boxed{I_i\frac{d{\omega }_i}{dt}+{ϵ}_{ijk}{\omega }_j{I}_k{\omega }_k=0}$$

or equivalently in vector form:

$$I\frac{d\omega }{dt}+\omega \times (I\omega )=0.$$

Writing these out component by component:

$$I_1{\dot{\omega }}_1=(I_2-I_3){\omega }_2{\omega }_3$$ $$I_2{\dot{\omega }}_2=(I_3-I_1){\omega }_3{\omega }_1$$ $$I_3{\dot{\omega }}_3=(I_1-I_2){\omega }_1{\omega }_2$$

These are three coupled nonlinear first-order ODEs. They conserve two quantities: the squared angular momentum $|\mathbf{L}{|}^2=L_1^2+L_2^2+L_3^2$ and the kinetic energy $T$. The existence of two conserved quantities on a three-dimensional phase space means the system is integrable (in the Liouville sense).

The so(3) Lie-Poisson bracket

In terms of the angular momentum components $L_i=I_i{\omega }_i$, Euler's equations become:

$${\dot{L}}_1=\left(\frac{1}{I_3}-\frac{1}{I_2}\right)L_2{L}_3,{\dot{L}}_2=\left(\frac{1}{I_1}-\frac{1}{I_3}\right)L_3{L}_1,{\dot{L}}_3=\left(\frac{1}{I_2}-\frac{1}{I_1}\right)L_1{L}_2.$$

These equations can be written compactly using the so(3) Lie-Poisson bracket. Define the bracket of two functions $f,g$ on $\mathfrak{so}(3{)}^{\ast }\cong {\mathbb{R}}^3$ by:

$$\boxed{\{f,g\}=-L\cdot (\nabla f\times \nabla g)}$$

or in components:

$$\{L_i,L_j\}=-{ϵ}_{ijk}{L}_k.$$

(The minus sign here is a convention choice; different authors use different sign conventions. The physics is the same either way.)

With this bracket, the Hamiltonian for the free rigid body is:

$$H(\mathbf{L})=\frac{1}{2}\left(\frac{L_1^2}{I_1}+\frac{L_2^2}{I_2}+\frac{L_3^2}{I_3}\right),$$

and the equation of motion ${\dot{L}}_i=\{L_i,H\}$ reproduces Euler's equations exactly. The time derivative of any function $f$ on angular momentum space is $\dot{f}=\{f,H\}$, which is the Poisson-bracket form of Hamilton's equations on the dual of a Lie algebra.

This bracket is degenerate -- it does not come from a symplectic form on all of ${\mathbb{R}}^3$. The degeneracy is measured by the Casimir function $C(\mathbf{L})=\frac{1}{2}(L_1^2+L_2^2+L_3^2)$, which satisfies $\{C,f\}=0$ for every function $f$. The level sets of $C$ are spheres in angular momentum space, and on each sphere the bracket restricts to a genuine (non-degenerate) symplectic form.

Stability of rotation about principal axes

Euler's equations admit three families of equilibrium solutions: pure rotation about each principal axis. The stability of these equilibria is:

• Axis 1 ($\omega =({\omega }_1,0,0)$): Stable if $I_1$ is the smallest or largest moment of inertia. Linearising around the equilibrium gives characteristic exponents that are purely imaginary (bounded oscillations).
• Axis 3 ($\omega =(0,0,{\omega }_3)$): Stable under the same conditions (by symmetry of the argument).
• Axis 2 ($\omega =(0,{\omega }_2,0)$): Unstable when $I_1\ne I_2\ne I_3$. Linearising gives one positive real characteristic exponent (exponential growth).

The stability can be understood geometrically: the energy ellipsoid $T=\mathrm{const}$ and the angular momentum sphere $|\mathbf{L}{|}^2=\mathrm{const}$ intersect near the "poles" (axes 1 and 3) in small closed curves, but near the "equator" (axis 2) the intersection curves are open (the separatrix).

More precisely, linearise about $\omega =({\omega }_1^0,0,0)$ by setting ${\omega }_1={\omega }_1^0+\delta {\omega }_1$, ${\omega }_2=\delta {\omega }_2$, ${\omega }_3=\delta {\omega }_3$ and keeping only first-order terms in $\delta {\omega }_i$. Euler's equations give:

$$\delta {\dot{\omega }}_2=\frac{(I_3-I_1)}{I_2}{\omega }_1^0\delta {\omega }_3,\delta {\dot{\omega }}_3=\frac{(I_1-I_2)}{I_3}{\omega }_1^0\delta {\omega }_2.$$

Combining these: ${\ddot{\delta \omega }}_2=\frac{(I_3-I_1)(I_1-I_2)}{I_2{I}_3}({\omega }_1^0{)}^2\delta {\omega }_2$. The coefficient is negative when $I_1$ is the smallest or largest moment, giving imaginary characteristic exponents and oscillatory solutions (stable). When $I_1$ is the intermediate moment, the coefficient is positive, giving real characteristic exponents and exponential growth (unstable).

Key theorem with proof Intermediate+

Theorem (Euler's equations from the Lie-Poisson bracket). The Euler equations for a torque-free rigid body are equivalent to the Hamiltonian equations $\dot{f}=\{f,H\}$ on $\mathfrak{so}(3{)}^{\ast }$ equipped with the Lie-Poisson bracket $\{L_i,L_j\}=-{ϵ}_{ijk}{L}_k$ and the Hamiltonian $H=\frac{1}{2}{\sum }_k{L}_k^2/I_k$.

Proof. The equation of motion for $L_1$ is:

$${\dot{L}}_1=\{L_1,H\}=-{ϵ}_{1jk}{L}_k\frac{\partial H}{\partial L_j}=-{ϵ}_{123}{L}_3\frac{L_2}{I_2}-{ϵ}_{132}{L}_2\frac{L_3}{I_3}.$$

Since ${ϵ}_{123}=1$ and ${ϵ}_{132}=-1$:

$${\dot{L}}_1=-L_3\cdot \frac{L_2}{I_2}+L_2\cdot \frac{L_3}{I_3}=L_2{L}_3\left(\frac{1}{I_3}-\frac{1}{I_2}\right).$$

Substituting $L_i=I_i{\omega }_i$:

$$I_1{\dot{\omega }}_1=(I_2{\omega }_2)(I_3{\omega }_3)\left(\frac{1}{I_3}-\frac{1}{I_2}\right)=I_2{I}_3{\omega }_2{\omega }_3\cdot \frac{I_2-I_3}{I_2{I}_3}=(I_2-I_3){\omega }_2{\omega }_3.$$

This is the first of Euler's equations. The other two follow by cyclic permutation. $\square$

Theorem (Casimir conservation). The function $C(\mathbf{L})=\frac{1}{2}(L_1^2+L_2^2+L_3^2)$ is a Casimir for the so(3) Lie-Poisson bracket: $\{C,f\}=0$ for every smooth function $f$.

Proof. Compute $\{C,L_i\}$ for a basis:

$$\{C,L_1\}=-{ϵ}_{1jk}{L}_k\frac{\partial C}{\partial L_j}=-{ϵ}_{1jk}{L}_k{L}_j=-{ϵ}_{123}{L}_3{L}_2-{ϵ}_{132}{L}_2{L}_3=-L_3{L}_2+L_2{L}_3=0.$$

Since the functions $L_i$ generate all smooth functions on ${\mathbb{R}}^3$ via products and linear combinations, and the bracket is a derivation in each argument, $\{C,f\}=0$ for all $f$. $\square$

The Casimir conservation means that angular momentum trajectories are confined to spheres $|\mathbf{L}{|}^2=\mathrm{const}$. Combined with energy conservation, each trajectory lies on the intersection of a sphere and an energy ellipsoid -- a closed curve (for stable equilibria) or an open separatrix (for the unstable case).

Worked example (torque-free precession) Intermediate+

Consider an axisymmetric rigid body with $I_1=I_2\ne I_3$ (e.g., a symmetric top or a gyroscope). Euler's equations simplify to:

$$I_1{\dot{\omega }}_1=(I_1-I_3){\omega }_2{\omega }_3,I_1{\dot{\omega }}_2=(I_3-I_1){\omega }_1{\omega }_3,I_3{\dot{\omega }}_3=0.$$

From the third equation, ${\omega }_3={\omega }_3^0=\mathrm{const}$. Define the body precession rate:

$${\Omega }_b=\frac{(I_3-I_1)}{I_1}{\omega }_3^0.$$

Then the first two equations become:

$${\dot{\omega }}_1=-{\Omega }_b{\omega }_2,{\dot{\omega }}_2={\Omega }_b{\omega }_1.$$

This is the equation of uniform circular motion with angular frequency ${\Omega }_b$. The solution is:

$${\omega }_1(t)=A\cos ({\Omega }_{b}t+\varphi ),{\omega }_2(t)=A\sin ({\Omega }_{b}t+\varphi ).$$

The angular velocity vector in the body frame precesses around the symmetry axis (axis 3) at constant rate ${\Omega }_b$. Meanwhile, ${\omega }_3$ is constant. The angular velocity vector traces a cone around axis 3 -- this is body-frame precession.

The angular momentum in the body frame is $L_1=I_1{\omega }_1$, $L_2=I_1{\omega }_2$, $L_3=I_3{\omega }_3^0$. The components $L_1$ and $L_2$ precess at the same rate ${\Omega }_b$, while $L_3$ is constant. The total angular momentum $\mathbf{L}$ is fixed in the space frame, but its components in the body frame rotate -- this is the content of torque-free precession.

For an oblate body like the Earth ($I_3>I_1$), ${\Omega }_b>0$ and the precession is prograde. For a prolate body like a long rod ($I_3<I_1$), ${\Omega }_b<0$ and the precession is retrograde.

Exercises Intermediate+

Lean formalization Intermediate+

Formalising the rigid body and Euler's equations in Lean requires several components that are not yet in Mathlib:

1. The Lie-Poisson bracket on ${\mathfrak{g}}^{\ast }$. For a Lie algebra $\mathfrak{g}$, define the Poisson bracket $\{f,g\}(\mu )=-⟨\mu ,[d_{\mu }f,d_{\mu }g]⟩$ on $C^{\infty }({\mathfrak{g}}^{\ast })$. Verify bilinearity, antisymmetry, the Leibniz rule, and the Jacobi identity (which follows from the Jacobi identity of $\mathfrak{g}$).

2. Specialisation to $\mathfrak{so}(3{)}^{\ast }\cong {\mathbb{R}}^3$. Identify the bracket as $\{f,g\}=-\mathbf{L}\cdot (\nabla f\times \nabla g)$ and prove the fundamental relations $\{L_i,L_j\}=-{ϵ}_{ijk}{L}_k$.

3. The inertia tensor as a positive-definite symmetric bilinear form on $\mathfrak{so}(3)$. Diagonalise it to obtain principal axes and principal moments of inertia.

4. The rigid body Hamiltonian. Define $H(\mathbf{L})=\frac{1}{2}{\sum }_k{L}_k^2/I_k$ and prove that the Hamiltonian vector field $X_H$ (with respect to the Lie-Poisson bracket) generates Euler's equations.

5. Casimir conservation. Prove that $C=|\mathbf{L}{|}^2/2$ is a Casimir function: $\{C,f\}=0$ for all $f$.

6. Stability analysis. Formalise the linear stability analysis of equilibria and prove the intermediate axis theorem.

7. Symplectic reduction. Prove that the dynamics on each Casimir level set (sphere) is Hamiltonian with respect to the symplectic form induced by the Lie-Poisson structure.

Items (1)--(3) are prerequisites that would establish the Lie-Poisson bracket as a general infrastructure. Items (4)--(5) would follow for $\mathfrak{so}(3)$ specifically. Item (6) requires ODE theory in Mathlib that is under active development. Item (7) requires the Marsden-Weinstein reduction theorem, which is a substantial formalisation effort.

Advanced results Master

The Lie-Poisson bracket on the dual of a Lie algebra

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra with Lie bracket $[\cdot ,\cdot {]}_{\mathfrak{g}}$. The dual space ${\mathfrak{g}}^{\ast }$ carries a natural Poisson structure called the Lie-Poisson bracket (also called the Kirillov-Kostant-Souriau bracket). For $f,g\in C^{\infty }({\mathfrak{g}}^{\ast })$ and $\mu \in {\mathfrak{g}}^{\ast }$:

$$\{f,g\}(\mu )=-⟨\mu ,[d_{\mu }f,d_{\mu }g{]}_{\mathfrak{g}}⟩$$

where $d_{\mu }f\in {\mathfrak{g}}^{\ast \ast }\cong \mathfrak{g}$ is the functional derivative of $f$ at $\mu$. (The isomorphism ${\mathfrak{g}}^{\ast \ast }\cong \mathfrak{g}$ holds for finite-dimensional $\mathfrak{g}$.)

The Jacobi identity for this bracket follows directly from the Jacobi identity of $\mathfrak{g}$:

$$\{f,\{g,h\}\}(\mu )=⟨\mu ,[df,[dg,dh]]⟩.$$

The cyclic sum vanishes by the Jacobi identity of $[\cdot ,\cdot {]}_{\mathfrak{g}}$.

There are two sign conventions: the minus (or right) Lie-Poisson bracket (written above) and the plus (or left) Lie-Poisson bracket with the opposite sign. The free rigid body uses the minus bracket on $\mathfrak{so}(3{)}^{\ast }$. The choice of sign corresponds to whether one identifies $\mathfrak{so}(3{)}^{\ast }$ with left-invariant or right-invariant vector fields on $\mathrm{SO}(3)$.

Coadjoint orbits as symplectic manifolds

The coadjoint action of a Lie group $G$ on its dual Lie algebra ${\mathfrak{g}}^{\ast }$ is defined by ${\mathrm{Ad}}_g^{\ast }(\mu )=\mu \circ {\mathrm{Ad}}_{g^{-1}}$ for $g\in G$, $\mu \in {\mathfrak{g}}^{\ast }$. The coadjoint orbit through $\mu$ is ${\mathcal{O}}_{\mu }=\{{\mathrm{Ad}}_g^{\ast }\mu :g\in G\}$.

Each coadjoint orbit carries a natural symplectic form, the Kirillov-Kostant-Souriau form:

$${\omega }_{\mu }({\mathrm{ad}}_{\xi }^{\ast }\mu ,{\mathrm{ad}}_{\eta }^{\ast }\mu )=⟨\mu ,[\xi ,\eta ]⟩$$

where ${\mathrm{ad}}_{\xi }^{\ast }\mu =-\mu \circ {\mathrm{ad}}_{\xi }$ is the infinitesimal coadjoint action and $\xi ,\eta \in \mathfrak{g}$ are tangent vectors to the orbit at $\mu$.

For $G=\mathrm{SO}(3)$, the coadjoint orbits are the spheres $|\mathbf{L}{|}^2=\mathrm{const}$ (for $\mathbf{L}\ne 0$) and the origin $\{0\}$ (a degenerate orbit). The symplectic form on the sphere of radius $R$ is:

$$\omega =\frac{1}{R}dA$$

where $dA$ is the area form on the sphere. The total area is ${\int }_{\mathcal{O}}\omega =4\pi R$, which scales with the angular momentum magnitude. (For the quantum theory, the quantisation condition $\int \omega =2\pi n\hslash$ gives the familiar quantisation $|\mathbf{L}|=n\hslash$.)

Symplectic reduction for the rigid body

The rigid body provides the paradigmatic example of symplectic reduction. Start from the phase space $T^{\ast }\mathrm{SO}(3)$ (position = rotation matrix $R$, momentum = body angular momentum $p$). This is a 6-dimensional symplectic manifold.

The left action of $\mathrm{SO}(3)$ on itself by multiplication lifts to a Hamiltonian action on $T^{\ast }\mathrm{SO}(3)$ with momentum map $J(R,p)=Rp$ (the space-frame angular momentum). For the free rigid body, the Hamiltonian $H(R,p)=\frac{1}{2}p\cdot I^{-1}p$ is invariant under the left action, so $J$ is conserved.

Left reduction at $\mu =0$: quotient $T^{\ast }\mathrm{SO}(3)$ by the left $\mathrm{SO}(3)$ action to obtain $T^{\ast }\mathrm{SO}(3)/\mathrm{SO}(3)\cong \mathfrak{so}(3{)}^{\ast }$. The reduced Poisson bracket on $\mathfrak{so}(3{)}^{\ast }$ is the Lie-Poisson bracket. The reduced Hamiltonian is $H(\mathbf{L})=\frac{1}{2}(L_1^2/I_1+L_2^2/I_2+L_3^2/I_3)$. The reduced equations of motion are Euler's equations.

This reduction procedure explains why the Lie-Poisson bracket appears: it is the Poisson bracket inherited from the canonical bracket on $T^{\ast }\mathrm{SO}(3)$ after quotienting by the symmetry group. The Casimir $|\mathbf{L}{|}^2$ arises as the value of the momentum map $J$ (the space-frame angular momentum is fixed at $\mu$, and $|\mu {|}^2=|\mathbf{L}{|}^2$ in the body frame after reduction).

The unreduced dynamics on $T^{\ast }\mathrm{SO}(3)$ can be reconstructed from the reduced dynamics on $\mathfrak{so}(3{)}^{\ast }$ by solving the kinematic equation $\dot{R}=R\hat{\omega }$, where $\omega =I^{-1}\mathbf{L}$ is determined by the reduced flow.

The heavy top and semidirect product reduction

When gravity acts on a rigid body with a fixed point, the symmetry is reduced from $\mathrm{SO}(3)$ to $\mathrm{SO}(2)$ (rotations about the vertical). The phase space can be described by the semidirect product $\mathfrak{so}(3{)}^{\ast }\times {\mathbb{R}}^3$, where the ${\mathbb{R}}^3$ factor records the direction of gravity in the body frame.

The Lie-Poisson bracket on $\mathfrak{so}(3{)}^{\ast }\times {\mathbb{R}}^3$ (the semidirect product of $\mathfrak{so}(3)$ with ${\mathbb{R}}^3$ under the vector representation) is:

$$\{L_i,L_j\}=-{ϵ}_{ijk}{L}_k,\{L_i,{\Gamma }_j\}=-{ϵ}_{ijk}{\Gamma }_k,\{{\Gamma }_i,{\Gamma }_j\}=0$$

where $\mathbf{\Gamma }=R^{-1}\hat{\mathbf{g}}$ is the unit vector pointing in the direction of gravity, expressed in the body frame. The Hamiltonian for the heavy symmetric top (Lagrange case, $I_1=I_2$) is:

$$H(\mathbf{L},\mathbf{\Gamma })=\frac{L_1^2+L_2^2}{2I_1}+\frac{L_3^2}{2I_3}+mg\ell {\Gamma }_3.$$

The Casimirs are $|\mathbf{\Gamma }{|}^2=1$ (the gravity direction is a unit vector) and $\mathbf{L}\cdot \mathbf{\Gamma }$ (the component of angular momentum along the gravity direction). With two Casimirs on a six-dimensional space, the reduced dynamics lives on two-dimensional level sets, making the system integrable (three conserved quantities: $H$, $|\mathbf{\Gamma }{|}^2$, $\mathbf{L}\cdot \mathbf{\Gamma }$, on a six-dimensional phase space; equivalently, three conserved quantities on the four-dimensional reduced space).

The energy-Casimir method for stability

The linear stability analysis of Euler's equations can be strengthened to nonlinear stability using the energy-Casimir method. The idea is to construct a Lyapunov function from the Hamiltonian and the Casimir:

$$V(\mathbf{L})=H(\mathbf{L})+\Phi (C(\mathbf{L}))$$

where $\Phi :\mathbb{R}\to \mathbb{R}$ is a smooth function chosen so that $V$ has a strict local minimum at the equilibrium. Since both $H$ and $C$ are conserved, $V$ is conserved, and a strict local minimum of $V$ guarantees nonlinear Lyapunov stability.

For rotation about axis 3 of an axisymmetric body ($I_1=I_2<I_3$), choose $\Phi (C)=\lambda C$ with $\lambda =(1/I_1-1/I_3)/2$. Then $V=H+\lambda C$ has a strict minimum at $L_1=L_2=0$, $L_3=\mathrm{const}$, proving that rotation about the symmetry axis is nonlinearly stable. This extends the linear result to all perturbation sizes, as long as the trajectory remains on the same Casimir level set.

Integrability and the Jacobi elliptic functions

For a general (asymmetric) rigid body with $I_1\ne I_2\ne I_3$, the solution of Euler's equations can be written in terms of Jacobi elliptic functions. The equations for ${\omega }_1(t)$ reduce to:

$${\left(\frac{d{\omega }_1}{dt}\right)}^2=\frac{(I_2-I_3)(I_1-I_3)}{I_1^2}\left(\frac{2T}{I_3-I_1}\right)\left(1-\frac{{\omega }_1^2}{{\omega }_1^{\max 2}}\right)\left(1-k^2\frac{{\omega }_1^2}{{\omega }_1^{\max 2}}\right)$$

where $k^2=(I_2-I_1)(I_3-I_2)/((I_3-I_2)(I_2-I_1))$ is the elliptic modulus (assuming the trajectory does not cross the separatrix). The solution is ${\omega }_1(t)={\omega }_1^{\max }\mathrm{cn}(\lambda t,k)$ where $\mathrm{cn}$ is the Jacobi cosine and $\lambda$ is determined by the moments of inertia and the energy. This exact solvability is a consequence of the integrability of the system: two conserved quantities ($H$ and $C$) on a three-dimensional phase space leave only one degree of freedom, which is always solvable by quadrature.

Synthesis. The rigid body is the simplest mechanical system whose phase space is not a cotangent bundle but rather the dual of a Lie algebra equipped with the Lie-Poisson bracket. The reduction from $T^{\ast }\mathrm{SO}(3)$ to $\mathfrak{so}(3{)}^{\ast }$ is the prototype of symplectic reduction, and the resulting equations (Euler's equations) are the prototype of Lie-Poisson dynamics. The Casimir $|\mathbf{L}{|}^2$ confines dynamics to coadjoint orbits (spheres), and the intersection of the energy level set with each orbit gives the trajectories. This structure -- Lie-Poisson bracket, Casimir functions, coadjoint orbits, reduction from a cotangent bundle -- repeats throughout geometric mechanics: in fluid dynamics (the Euler equations for ideal fluids as Lie-Poisson equations on the dual of the diffeomorphism group), in plasma physics (the Vlasov-Poisson equations), and in magnetohydrodynamics. The rigid body is the simplest example of a pattern that pervades continuum mechanics.

Full proof set Master

Proposition 1 (Jacobi identity for the so(3) Lie-Poisson bracket). The bracket $\{f,g\}(\mathbf{L})=-\mathbf{L}\cdot (\nabla f\times \nabla g)$ on $C^{\infty }({\mathbb{R}}^3)$ satisfies the Jacobi identity.

Proof. For linear functions $f=a_i{L}_i$, $g=b_j{L}_j$, $h=c_k{L}_k$:

$$\{f,g\}=-a_i{b}_j{ϵ}_{ijk}{L}_k.$$

Then $\{f,\{g,h\}\}=-a_i(-b_j{c}_k{ϵ}_{jkl}{ϵ}_{imn}{\delta }_{j{m}^{\prime }}{\delta }_{k{n}^{\prime }}\cdot \text{(derivative of }{L}_l\text{ w.r.t. }{L}_{m^{\prime }}))$. Since the functions are linear, only first derivatives appear, and the calculation reduces to:

$$\{f,\{g,h\}\}+cyc.=a_i{b}_j{c}_k({ϵ}_{jlm}{ϵ}_{ilm}+{ϵ}_{klm}{ϵ}_{jlm}+{ϵ}_{ilm}{ϵ}_{klm})L_m=0$$

by the Jacobi identity for the Levi-Civita symbol ${ϵ}_{ijk}$ (which is equivalent to the Jacobi identity for $\mathfrak{so}(3)$). For general smooth functions, expand in Taylor series; the terms involving second and higher derivatives cancel because the bracket depends only on first derivatives, and the linear result extends to polynomials by the derivation property. $\square$

Proposition 2 (Coadjoint orbits of SO(3) are spheres). The coadjoint orbits of SO(3) acting on $\mathfrak{so}(3{)}^{\ast }\cong {\mathbb{R}}^3$ are the spheres $|\mathbf{L}{|}^2=r^2$ (for $r>0$) and the origin.

Proof. The coadjoint action of $R\in \mathrm{SO}(3)$ on $\mathbf{L}\in {\mathbb{R}}^3$ is ${\mathrm{Ad}}_R^{\ast }\mathbf{L}=R\mathbf{L}$ (the standard vector representation, since $\mathfrak{so}(3{)}^{\ast }\cong {\mathbb{R}}^3$ via the cross product). Since $R$ is orthogonal, $|R\mathbf{L}|=|\mathbf{L}|$. Conversely, for any two vectors $\mathbf{L},{\mathbf{L}}^{\prime }$ with $|\mathbf{L}|=|{\mathbf{L}}^{\prime }|$, there exists $R\in \mathrm{SO}(3)$ with ${\mathbf{L}}^{\prime }=R\mathbf{L}$ (rotate the plane spanned by $\mathbf{L}$ and ${\mathbf{L}}^{\prime }$). So the orbit through $\mathbf{L}\ne 0$ is the sphere of radius $|\mathbf{L}|$, and the orbit through $0$ is the origin. $\square$

Proposition 3 (Symplectic form on coadjoint orbits of SO(3)). The Kirillov-Kostant-Souriau symplectic form on the coadjoint orbit ${\mathcal{O}}_r=\{\mathbf{L}\in {\mathbb{R}}^3:|\mathbf{L}|=r\}$ is $\omega =(1/r)dA$ where $dA$ is the area form on the sphere of radius $r$.

Proof. At a point $\mathbf{L}\in {\mathcal{O}}_r$, the tangent space is spanned by vectors $\mathbf{v}=\xi \times \mathbf{L}$ for $\xi \in {\mathbb{R}}^3\cong \mathfrak{so}(3)$. The KKS form evaluated on two tangent vectors $\xi \times \mathbf{L}$ and $\eta \times \mathbf{L}$ is:

$$\omega (\xi \times \mathbf{L},\eta \times \mathbf{L})=⟨\mathbf{L},[\xi ,\eta ]⟩=\mathbf{L}\cdot (\xi \times \eta ).$$

The standard area form on the sphere of radius $r$ evaluated on the same tangent vectors is $dA(\xi \times \mathbf{L},\eta \times \mathbf{L})=(\xi \times \mathbf{L})\times (\eta \times \mathbf{L})\cdot \hat{\mathbf{L}}=\mathbf{L}\cdot (\xi \times \eta )\cdot r$ (by the BAC-CAB rule and $|\mathbf{L}|=r$). Hence $\omega =(1/r)dA$. $\square$

Proposition 4 (Lie-Poisson bracket restricts to the orbit symplectic form). Let $f,g\in C^{\infty }({\mathbb{R}}^3)$ and let $\bar{f},\bar{g}$ be their restrictions to the coadjoint orbit ${\mathcal{O}}_r$. Then $\{f,g\}{|}_{{\mathcal{O}}_r}=\{\bar{f},\bar{g}{\}}_{\omega }$ where the right side is the symplectic Poisson bracket on $({\mathcal{O}}_r,\omega )$.

Proof. The Hamiltonian vector field of $f$ with respect to the Lie-Poisson bracket is $X_f=\nabla f\times \mathbf{L}$ (restricted to the orbit). The symplectic Poisson bracket on $({\mathcal{O}}_r,\omega )$ satisfies $\{\bar{f},\bar{g}{\}}_{\omega }=\omega (X_{\bar{f}},X_{\bar{g}})$. Since $X_{\bar{f}}=(\nabla f\times \mathbf{L}){|}_{{\mathcal{O}}_r}$ (the restriction commutes with taking the Hamiltonian vector field), we have:

$$\{\bar{f},\bar{g}{\}}_{\omega }=\omega (\nabla f\times \mathbf{L},\nabla g\times \mathbf{L})=\frac{1}{r}\mathbf{L}\cdot ((\nabla f\times \mathbf{L})\times (\nabla g\times \mathbf{L})).$$

By the BAC-CAB rule and $|\mathbf{L}|=r$:

$$=\frac{1}{r}\left[(\nabla g\cdot \nabla f)|\mathbf{L}{|}^2-(\nabla f\cdot \mathbf{L})(\nabla g\cdot \mathbf{L})\right]\cdot \frac{1}{r}=-\mathbf{L}\cdot (\nabla f\times \nabla g)=\{f,g\}.$$

$\square$

Proposition 5 (Nonlinear stability via energy-Casimir method). Rotation of an axisymmetric rigid body ($I_1=I_2<I_3$) about the symmetry axis is nonlinearly stable in the sense of Lyapunov.

Proof. The equilibrium is ${\mathbf{L}}_0=(0,0,L_3^0)$ with $L_3^0\ne 0$. Define $V(\mathbf{L})=H(\mathbf{L})+\lambda C(\mathbf{L})$ with $C=|\mathbf{L}{|}^2/2$ and $\lambda =(1/I_1-1/I_3)/2>0$. Then:

$$V=\frac{L_1^2+L_2^2}{2I_1}+\frac{L_3^2}{2I_3}+\frac{\lambda }{2}(L_1^2+L_2^2+L_3^2)=\frac{L_1^2+L_2^2}{2}\left(\frac{1}{I_1}+\lambda \right)+\frac{L_3^2}{2}\left(\frac{1}{I_3}+\lambda \right).$$

With the chosen $\lambda$: $1/I_1+\lambda =(1/I_1+1/I_1-1/I_3)/2=(2/I_1-1/I_3)/2$ and $1/I_3+\lambda =(1/I_3+1/I_1-1/I_3)/2=1/(2I_1)$. So:

$$V=\frac{2/I_1-1/I_3}{4}(L_1^2+L_2^2)+\frac{1}{4I_1}{L}_3^2.$$

On the Casimir level set $L_1^2+L_2^2+L_3^2=(L_3^0{)}^2$, write $L_3=L_3^0+\delta L_3$. Then:

$$V=\frac{1}{4I_1}\left[(L_3^0{)}^2+2L_3^0\delta L_3+(\delta L_3{)}^2\right]+\frac{2/I_1-1/I_3}{4}(L_1^2+L_2^2).$$

Since $2/I_1-1/I_3>0$ (because $I_1<I_3$), $V$ has a strict minimum at $L_1=L_2=0$, $L_3=L_3^0$ on each Casimir level set. As $V$ is conserved and has a strict minimum at the equilibrium, the equilibrium is Lyapunov stable. $\square$

Connections Master

• 09.04.01 The Hamiltonian formalism on $T^{\ast }\mathrm{SO}(3)$ is the starting point; reduction to $\mathfrak{so}(3{)}^{\ast }$ produces the Lie-Poisson bracket and Euler's equations. The Legendre transform from the Lagrangian on $T\mathrm{SO}(3)$ gives the phase space $T^{\ast }\mathrm{SO}(3)$.

• 09.04.03 The angular momentum Poisson brackets $\{L_i,L_j\}=-{ϵ}_{ijk}{L}_k$ developed in 09.04.03 are exactly the Lie-Poisson bracket on $\mathfrak{so}(3{)}^{\ast }$. This unit promotes those algebraic relations to a complete dynamics.

• 09.04.05 The symplectic form on each coadjoint orbit (sphere in angular momentum space) is a concrete example of a symplectic manifold that is not a cotangent bundle. The symplectic form unit develops the general theory; this unit provides the key physical example.

• 09.06.01 The free rigid body is Liouville-integrable: two conserved quantities ($H$ and $C$) on the three-dimensional phase space $\mathfrak{so}(3{)}^{\ast }$ leave one degree of freedom, solvable by quadrature. The Liouville-Arnold theorem applies and the motion is conjugate to linear flow on a circle.

• 09.03.01 Noether's theorem identifies the conservation of space-frame angular momentum with the rotational symmetry of the free rigid body. The momentum map for the $\mathrm{SO}(3)$ action on $T^{\ast }\mathrm{SO}(3)$ is the angular momentum, and its conservation is the Noether theorem in the Hamiltonian setting.

• 12.03.01 The Lie-Poisson bracket $\{L_i,L_j\}=-{ϵ}_{ijk}{L}_k$ becomes the quantum spin commutation relations $[{\hat{S}}_i,{\hat{S}}_j]=i\hslash {ϵ}_{ijk}{\hat{S}}_k$ under canonical quantisation. The Casimir $|\mathbf{L}{|}^2$ becomes ${\hat{S}}^2$ with eigenvalues $s(s+1){\hslash }^2$. The coadjoint orbit quantisation gives the spin-s representations of $\mathrm{SU}(2)$.

Historical & philosophical context Master

Leonhard Euler derived the equations of motion for a rigid body with a fixed point in his Decouverte d'un nouveau principe de mecanique (1758), published in the Memoirs of the Berlin Academy. Euler's approach was purely vectorial: he wrote the equations in terms of angular velocity components in the body frame and derived the cross-product terms from the kinematics of rotating reference frames. He did not have the concept of a Poisson bracket or a Lie algebra; his derivation was a tour de force of Newtonian mechanics applied to an extended body.

Louis Poinsot, in his Theorie nouvelle de la rotation des corps (1834), gave the geometric interpretation that bears his name: the Poinsot construction. Poinsot showed that the angular velocity vector traces a curve (the polhode) on the inertia ellipsoid, and that this ellipsoid rolls without slipping on a fixed plane perpendicular to the angular momentum vector (the invariable plane). The herpolhode is the curve traced on the invariable plane. Poinsot's construction gives a complete qualitative picture of rigid body motion without solving any equations -- it is a purely geometric description of the dynamics.

The algebraic structure behind Euler's equations was not recognised until the development of Lie algebra theory in the late 19th century. Sophus Lie's general theory of continuous groups (1880s) contained the Lie-Poisson bracket on ${\mathfrak{g}}^{\ast }$ as a special case of the symplectic structure on coadjoint orbits, but the connection to rigid body mechanics was not made explicitly until much later.

The modern understanding emerged in the 1960s-1970s through several independent streams. Kirillov (1962) and Kostant (1970) identified the symplectic structure on coadjoint orbits. Arnold (1966) showed that the Euler equations for ideal fluids are Lie-Poisson equations on the dual of the diffeomorphism group, with the rigid body as the finite-dimensional prototype. Marsden and Weinstein (1974) developed the general reduction theory that includes the passage from $T^{\ast }\mathrm{SO}(3)$ to $\mathfrak{so}(3{)}^{\ast }$ as a special case.

The intermediate axis theorem was known empirically for centuries (any spinning object demonstrates it) but was first analysed rigorously by Klein and Sommerfeld in their Ueber die Theorie des Kreisels (1897-1910), the definitive four-volume treatise on the theory of the spinning top. The theorem gained renewed attention in 1985 when cosmonaut Vladimir Dzhanibekov observed the effect with a wing-nut in zero gravity aboard the Salyut-7 space station; the "Dzhanibekov effect" became a popular demonstration of the instability of rotation about the intermediate axis.

The philosophical significance of the rigid body in the history of mechanics is that it was the first system whose phase space is not a cotangent bundle. The angular momentum space $\mathfrak{so}(3{)}^{\ast }\cong {\mathbb{R}}^3$ with the Lie-Poisson bracket is a Poisson manifold that is not symplectic (the bracket is degenerate, with $|\mathbf{L}{|}^2$ as Casimir). The symplectic leaves are the coadjoint orbits (spheres), and the dynamics on each leaf is Hamiltonian. This structure -- a degenerate Poisson bracket foliating into symplectic leaves -- is the generic situation in reduced mechanical systems, and the rigid body is the simplest and most physically transparent example.

Bibliography Master

• Euler, L., "Decouverte d'un nouveau principe de mecanique," Memoires de l'Academie des Sciences de Berlin 6 (1758), 185-217. The original derivation of Euler's rigid body equations.

• Poinsot, L., Theorie nouvelle de la rotation des corps (Bachelier, Paris, 1834). The geometric construction of rigid body motion.

• Klein, F. & Sommerfeld, A., Ueber die Theorie des Kreisels (4 vols., Teubner, Leipzig, 1897-1910). The definitive classical treatise on the spinning top.

• Kirillov, A. A., "Unitary representations of nilpotent Lie groups," Russian Math. Surveys 17 (1962), 53-104. The orbit method and coadjoint orbit symplectic structure.

• Arnold, V. I., "Sur la geometrie differentielle des groupes de Lie de dimension infinie et ses applications a l'hydrodynamique des fluides parfaits," Ann. Inst. Fourier 16 (1966), 319-361. Euler's fluid equations as Lie-Poisson equations, with the rigid body as prototype.

• Marsden, J. E. & Weinstein, A., "Reduction of symplectic manifolds with symmetry," Reports on Math. Phys. 5 (1974), 121-130. The general reduction theorem.

• Arnold, V. I., Mathematical Methods of Classical Mechanics, 2nd ed. (Springer GTM 60, 1989), §29, §45.

• Marsden, J. E. & Ratiu, T. S., Introduction to Mechanics and Symmetry, 2nd ed. (Springer TAM 17, 1999), Ch. 15.

• Holm, D. D., Geometric Mechanics Part I (Imperial College Press, 2008). Lie-Poisson structures with emphasis on examples.

• Goldstein, H., Poole, C. P. & Safko, J., Classical Mechanics, 3rd ed. (Pearson, 2002), Ch. 5.5-5.7.

• Landau, L. D. & Lifshitz, E. M., Mechanics, 3rd ed. (Course of Theoretical Physics Vol. 1, Pergamon, 1976), §32-36.