Wolfram Cloud Document

Here we present geometric equations of motion in the gauge group

SU(2)

for the inverted spherical pendulum and the axially symmetrical top. The derivation is exactly parallel to our prior geometric equations in SO (3) Gauge Theory for Classical Mechanics in , only this time avoiding gimbal lock via better gauge choice. Both derivations are exceptionally concise, expressing the equations and solutions as geodesics on manifolds. It is remarkable that a full description of dynamics arises, yet again, from differential geometry alone. We finish with a side-by-side animation showing numerical equivalence, aside from gimbal lock in the

SO(3)

case, which we can provoke by finding the

SO(3)

poles with the Manipulate sliders. That is how we demonstrate superiority of the

SU(2)

gauge choice over

SO(3)

. We study

SU(2)

here in an intuitive setting of classical mechanics so that we will be better equipped for transition to quantum physics. Instead of studying two abstractions at once in the quantum setting—

SU(2)

and spinors—we will have only spinors to study. This notebook contains a full chapter on the theory of

SU(2)

, requiring only basic University mathematics, to further lower barriers to understanding. Attached are a couple of videos, one showing equivalent trajectories for the symmetrical top between SO(3) and SU(2), and the second showing a singular case for SO(3), where the equations of motion cannot be solved due to coordinate singularity (gimbal lock), but where SU(2) operates normally, avoiding the singularity and respecting the intrinsic topology of 3D space. The notebook shows full development in both gauges, solving dynamics as geodesics in the gauge manifolds.

Preliminaries

Initialize the Relativity Toolkit

Optional: Quit for a Fresh Kernel

◼

Note, if you call Quit in a notebook, evaluation may hang. If you want to “Evaluate Notebook,” mark the following two cells “Evaluatable” in the Cell

Property menu, call Quit and FrontEndTokenExecute once each, then comment out those two cells, or mark them unevaluatable again. Alternatively, you can quit the kernel via the Evaluation menu (last item). At that point, you can “Evaluate Notebook” in a clean environment, or evaluate cells individually.

Quit[];

In[]:=

FrontEndTokenExecute["DeleteGeneratedCells"]

Download the Code

In[]:=

getToolkit[branch_]:=With[{base="https://raw.githubusercontent.com/rebcabin/RelativityToolkit/"<>branch<>"/",bust="?t="<>ToString[Round[AbsoluteTime[]]]},{rules=base<>"RelativityToolkit.wl"<>bust},Get[rules]];getToolkit["v1.13.1"];

Relativity Toolkit version :1.13.1

Relativity Connection Symbol :Γ

Setup

Global Parameters

In[]:=

ClearAll[e1,e2,e3,o,g,plotRg];e1={1,0,0};e2={0,1,0};e3={0,0,1};o={0,0,0};g=9.81;plotRg={-1.25,1.25};ClearAll[x,y,z,u,v,w,t];ClearAll[tLim];tLim=200;

Utility Function

In[]:=

radPerSec[rpm_]:=

2.π

60.

*rpm;

Logic Utility Functions

In[]:=

ClearAll[proof];proof[v_]/;BooleanQ[v]:=(If[v,Print[Style["QED■",StandardGreen,Bold,18]],Print[Style["FAILED PROOF■",StandardRed,Bold,18]]];v);proof[___]:=Print[Style["INVALID INPUT TO PROOF■",StandardYellow,Bold,18]];

Predicates

In[]:=

ClearAll[TracelessQ,HermitianQ,AntiHermitianQ,UnitaryQ];TracelessQ[m_]:=Tr[m]===0;HermitianQ[m_]:=(ComplexExpand[m]===ComplexExpand[m]);AntiHermitianQ[m_]:=(ComplexExpand[m]===ComplexExpand[-m]);UnitaryQ[m_]:=(m.m===m.m===IdentityMatrix[2]);

Graphical and Physical Elements

In[]:=

ClearAll[rig];With[{l=1.2,mass=1.0,d=2.4,r=0.02},{baton=Cylinder[{{0,0,-l},{0,0,+l}},r],batonCenterBodyFrame={0,0,l}},rig["graphics primitives"]=baton;rig["moment of inertia"]=With[{Ib=mass*(MomentOfInertia[baton]/Volume[baton])},{Ibi=Inverse@Ib},{Ib,Ibi}];rig["cb"]:=batonCenterBodyFrame;rig["half length"]:=l;rig["mass"]:=mass;];ClearAll[jack];jack[0,opacity_:0.1,diameter_:0.01]:={Opacity[opacity],{Red,Arrow[Tube[{o,e1}],diameter]},{Darker[Green],Arrow[Tube[{o,e2}],diameter]},{Lighter[Blue],Arrow[Tube[{o,e3}],diameter]},{Lighter[Cyan],Arrow[Tube[{o,-e1}],diameter]},{Magenta,Arrow[Tube[{o,-e2}],diameter]},{Yellow,Arrow[Tube[{o,-e3}],diameter]}};ClearAll[nfm,vfm,font,myFont];myFont[color_,size_:18]:=Style[#,color,Bold,size,FontFamily->"Courier New"]&;nfm[num_,dig_:10,dec_:4]:=ToString[NumberForm[Chop@num,{dig,dec}]];vfm[vec_]:=ToString[NumberForm[#,{7,4},NumberSigns->{"-"," "},NumberPadding->{" ","0"}]&/@Chop@vec];font=myFont[Lighter[Blue],12];

Chapter I:

SO(3)

Recap

In this chapter, we reprise the derivations from SO (3) Gauge Theory for Classical Mechanics in , minus the prose explanations, for two reasons:

To emphasize the brevity of the geometric derivation.

To set up for the final, side-by-side animation of this notebook.

SO(3)
Geometrical Equations of Motion

In[]:=

ClearAll[coords3,vels3,v3,Γ3,I3,I1,cz,m,t,V];coords3={δ,η,ψ};metric3=

I1	0	0
0	(I1 2 Cos[δ] +I3 2 Sin[δ] )	I3Sin[δ]
0	I3Sin[δ]	I3

;g3=MakeIndexer[metric3,coords3];ig3=MakeIndexer[Inverse@metric3,coords3];vels3=(c|->c'[t])/@coords3;v3=MakeIndexer[vels3,coords3];V=mgczCos[δ]Cos[η];Γ3[μ_,ν_,λ_]:=ChristoffelsFromMetricIndexer[g3,ig3,μ,ν,λ,coords3];(ad3=AbsoluteDerivativeFromConnectionIndexer[v3,Γ3,t,coords3]//FullSimplify);(pf3=PotentialScalarFieldFromMetricIndexer[V,ig3,t,coords3]//FullSimplify);(eqns3=MapThread[Equal,{ad3,pf3}]//FullSimplify);

Chapter II:

SU(2)

Theory

SU(2)

normally appears in a Physics curriculum for the first time in quantum mechanics, applied to spinors. However, its origins are in classical mechanics, where it appears as quaternions, specifically to resolve gimbal lock. It took a very long time after the realization that no choice of Euler angles can prevent gimbal lock to synthesize a concise theory of rotations in classical mechanics.

SU(2)

is the culmination of a long chain of discoveries by Euler, Rodrigues, Hamilton, Cayley, Klein, Galois, Lie, and others that go through several areas of mathematics: topology, geometry, differential equations, and algebra.

The surprising outcome is that, despite appearances, 3D space requires two full rotations—

720°

—to restore a system to its original topological state, avoiding twist relative to the environment. A single rotation of

360°

leaves a system in a “twisted” state, requiring another

360°

—in the same direction—to undo. This fact can be demonstrated by loosely strapping an apparatus to a lab bench and then turning the apparatus about a vertical axis. There are numerous other demonstrations such as the Dirac belt trick and the Indonesian candle dance. There is nothing quantum or abstract about this fact.

We cannot see this fact directly in

SO(3)

because the Euler angles of the original state and of the twisted state are the same. However, in

SU(2)

, they are of opposite algebraic signs. The representations of the original and twisted states in

SU(2)

can be smoothly connected by a curve—a smooth trajectory—entirely in

SU(2)

. Such is not possible in

SO(3)

, where to get from one state to the other, we must pass through a coordinate singularity, resulting in gimbal lock.

SO(3)

is literally only half the story of rotation in 3D. Another view is “double cover:” that a single rotation representation in

SO(3)

has exactly two representations in

SU(2)

There is a lot of material in this chapter, but the payoff in the next chapter is a concise and elegant geometric derivation of the equations of motion as geodesics in

SU(2)

, exactly parallel to the derivation vetted previously for

SO(3)

Summary: The Lie Group
SU(2)
and its Lie Algebra,
(2)

This summary presents the basic facts necessary to understand the building blocks of the derivations: the metric tensor, the potential field, the objects put to rotation, and the rotation operators themselves. Most of the facts have proofs in the remainder of this chapter. The proofs require only basic University mathematics. There are many short reminders of even more basic facts, intended to ease understanding.

In[]:=

GridStyle[#,Bold,14]&/@{"Concept","Properties","Basis","Space","Role"},"State","Hermitian, Tr=0","{

}"," (2) ≡

","The Vector being rotated",{"Generator","Anti-Herm, Tr=0","{-

σ

}","(2)","Angular Velocity / Twist Vector"},{"Rotation Q(t)","Unitary, Det=1","exp((2))","SU(2)","Rot. Operator / Path in Group"},"Versor q(t)","Norm = 1","{1, , , }","

≡

","Quaternion Representation",Frame->All,Spacings->{1.5,1.5},Alignment->Left,Background->{None,{Gray}}

◼

Unitarity guarantees that eigenvalues have absolute value 1.

◼

Unit determinant implies a unit norm for the quaternion representation.

◼

A group is a set with an associative binary operation, an identity element, and an inverse for every element.

Adjoint Action

In the equations of motion, we model the baton as one of these vectors.

Manifolds, Curves, Tangents, Geodesics, Trajectories

Proposition 1

Proposition 2

Proposition 3

PROOF: By Jacobi's formula for the derivative of a determinant

Proposition 4

Pauli Matrices as Hermitian Basis for 3-Vectors

Traceless and Hermitian

Pauli matrices are obviously traceless, plus they are Hermitian: each is equal to its conjugate transpose. “Hermitian” is the analogue of “symmetric” amongst real matrices and implies real eigenvalues.

Unitary (Ancillary Fact)

◼

Therefore, the Pauli matrices, as unitary operators, are reflections. Every rotation in 3D is a sequence of two reflections (TODO: elaborate)

The basis of (2) Models Hamilton’s Imaginary Units

Concisely

∴ Pure Imaginary Quaternions are Isomorphic to (2)

The significance of the fact that the vector or imaginary part of a quaternion is in (2) is both geometrical and physical, with two major implications:

Generators of Rotations

Here is a proof that the MatrixExp equals the longhand form:

Example

Geometric Approach

kinetic energy in the metric ⟹

Christoffel symbols and the absolute derivative ⟹

direct potential energy in contravariant form (outside the metric) ⟹

equations of motion (eqns4) ⟹

numerical solutions, manipulable plots and animations.

Coordinates

Angular Velocity in the 4D Basis

Metric Tensor from Kinetic Energy

Add a ‘ghost’ inertia, Iref, for the radial direction; value determined by trial-and-error (TODO: derive by analysis). The metric is both complicated and surprisingly regular. Its validity is demonstrated by the numerical results rather than by analysis. Analysis is omitted, here, for brevity, though analysis is a worthy objective (TODO).

Christoffel Connection with Quaternion Normalization

Absolute Derivative Term

Contravariant Constraint Force Vector

Geometrical Equations of Motion

Convert Geometrical Equations to State-Space Form

For numerical work, we add an equation for the Lagrange multiplier (constraint force) of the quaternion normalization constraint:

Interactive Plots of the Geometrical Solution

Derivation of Angular Velocity in the 4D Basis

This is a derivation of the particular convention we need for physics: the body frame. It justifies the definition of ω above.

which follows from the exponential map

Setup

Define Q and its Derivative in Terms of 4D Coordinates

Hermitian angular Velocity Matrix

ω Components via Pauli Trace Property

Matching the vectors defined in the geometric metric:

Formal Verification

Setup

Potential Energy

Body Angular Velocity

Kinetic Energy Plus Constraint Term

Euler-Lagrange Operator

Equations of Motion from Euler-Lagrange

Take the 2nd time derivative of the constraint:

Solve for the highest derivatives and the constraint force:

Interactive Plots of the Euler-Lagrange Solution

The thing to note here is that the plots are indistinguishable from the geometrical solution for any settings of the sliders.

Conclusion

Future Work and Analytical Validation

While the numerical results are compelling, several areas remain for formal analytical verification and validation to complete the theory. These are marked throughout the text and summarized here as a roadmap for future development: