The Discrete Hamel’s Formalism and

Energy-Momentum Integrators for the

n-dimensional Spherical Pendulum

Dmitry V. Zenkov, Melvin Leok, and Anthony M. Bloch

This version: January 29, 2013

In honor of Arieh Iserles

Abstract

This paper discusses Hamel’s formalism and its applications to structure-preserving integration of the n-dimensional spherical pendulum. It utilizes redundant coordinates in order to eliminate multiple chartson the configuration space of the pendulum as well as nonphysical artificial singularities induced bylocal coordinates, while keeping the minimal possible degree of redundancy and avoiding integrationof differential-algebraic equations. We show that by a suitable choice of reconstruction equation, thisapproach leads to an energy-momentum integrator for the n-dimensional spherical pendulum. Long-timenumerical simulations are performed that compare the numerical performance of the proposed Hamelintegrator with Stormer–Verlet and RATTLE.

1 Introduction

This paper introduces a new energy-momentum integrator for an n-dimensional spherical pendulum. Theconfiguration space for this pendulum is an (n − 1)-dimensional sphere. Calculations in spherical coordi-nates, in general, are not a good option because of unavoidable artificial singularities introduced by thesecoordinates. In addition, the topology of a sphere makes it impossible to use global singularity-free in-trinsic coordinates. The paper utilizes the discrete Hamel formalism and demonstrates its usefulness instructure-preserving integration of degenerate systems on homogeneous spaces.

Recall that in order to avoid the issues mentioned above in the three-dimensional setting, an integratorthat utilizes the interpretation of a 2-sphere as a homogeneous space was introduced in Lee et al. [2009]. Thisintegrator performs very well, but has a somewhat large degree of redundancy. A recent paper Zenkov et al.[2012] targets the development of an integrator whose performance is similar to that of the integrator in Leeet al. [2009] and whose redundancy is the minimum possible. Here, we extend these results to the generaln-dimensional setting; this work should be viewed as a step towards the construction of Hamel integratorsfor degenerate systems on homogeneous spaces.

While both the present paper and Lee et al. [2009] utilize the interpretation of the pendulum as a rotatingrigid body, it is done in different ways. The algorithm introduced in Lee et al. [2009] is based on the evaluationof the rotation matrix that represents the attitude for this body. The key feature of the dynamics exploitedin the present paper is that, in order to capture the orientation of a pendulum, it is sufficient to evaluate justone column of that rotation matrix. The resulting equations of motion are interpreted as Hamel’s equationswritten in redundant coordinates.

A comprehensive exposition of the discrete Hamel formalism will be a subject of a future publication.Here, we demonstrate the usefulness and flexibility of some of this formalism by constructing an integratorfor a spherical pendulum that is energy- and momentum-preserving. The calculations are carried out in theCartesian coordinates of the n-dimensional Euclidean space. This allows one to avoid singularities and/or

1

multiple coordinate charts that are inevitable for calculations on a sphere. Hamel’s approach allows one,among other things, to represent the dynamics in such a way that the length constraint becomes unnecessary.Thus, one avoids the well-known difficulty of numerically solving differential-algebraic equations.

The paper is organized as follows. Hamel’s formalism and its discretization are briefly discussed inSections 2 and 3. The dynamics of a spherical pendulum is reviewed in Section 4. The discrete modelfor the pendulum based on Hamel’s formalism, its comparison to some other discretization techniques, andsimulations are given in Sections 5, 6, and 7.

2 Lagrangian Mechanics

2.1 The Euler–Lagrange Equations

A Lagrangian mechanical system is specified by a smooth manifold Q called the configuration space and afunction L : TQ→ R called the Lagrangian. In many cases, the Lagrangian is the kinetic minus potentialenergy of the system, with the kinetic energy defined by a Riemannian metric and the potential energy beinga smooth function on the configuration space Q. If necessary, non-conservative forces can be introduced(e.g., gyroscopic forces that are represented by terms in L that are linear in the velocity), but this is notdiscussed in detail in this paper.

In local coordinates q = (q1, . . . , qn) on the configuration space Q, we write L = L(q, q). The dynamicsis given by the Euler–Lagrange equations

d

dt

∂L

∂q=∂L

∂q, (1)

ord

dt

∂L

∂qi=∂L

∂qi, i = 1, . . . , n,

in coordinate notation.These equations were originally derived by Lagrange [1788] by requiring that simple force balance F = ma

be covariant, i.e. expressible in arbitrary generalized coordinates. A variational derivation of the Euler–Lagrange equations, namely Hamilton’s principle (also called the principle of critical action), came laterin the work of Hamilton [1834, 1835]. For more details, see Bloch [2003], Marsden and Ratiu [1999], andTheorem 2.1 below.

2.2 The Hamel Equations

In this subsection, we briefly discuss the Hamel equations. The exposition follows the paper Bloch et al.[2009].

In many cases, the Lagrangian and the equations of motion have a simpler structure when writtenusing velocity components measured against a frame that is unrelated to the system’s local configurationcoordinates. An example of such a system is the rigid body.

Let q = (q1, . . . , qn) be local coordinates on the configuration space Q and ui ∈ TQ, i = 1, . . . , n, besmooth independent local vector fields defined in the same coordinate neighborhood U .1 The componentsof ui relative to the coordinate basis ∂/∂qj will be denoted ψji (q); that is,

ui(q) = ψji (q)∂

∂qj,

where i, j = 1, . . . , n and where summation on the repeated index j is understood.Let ξ = (ξ1, . . . , ξn) ∈ Rn be the components of the velocity vector q ∈ TQ relative to the frame

u(q) := (u1(q), . . . , un(q)), i.e.,q = u(q) · ξ, (2)

1In certain cases, some or all of the ui can be chosen to be global vector fields on Q.

2

where, by definition,u(q) · ξ := ui(q)ξ

i. (3)

Thenl(q, ξ) := L(q, u(q) · ξ) (4)

is the Lagrangian of the system written in the local coordinates (q, ξ) on the tangent bundle TQ. Thecoordinates (q, ξ) are Lagrangian analogues of non-canonical variables in Hamiltonian dynamics. Whenconvenient, we will reverse the order of factors in (3), i.e., we assume that

u(q) · ξ = ξ ·u(q).

Given two elements ξ, ζ ∈ Rn, define the antisymmetric bracket operation [ · , · ]q : Rn × Rn → Rn by

u(q) · [ξ, ζ]q =[u(q) · ξ, u(q) · ζ

]where [ · , · ] is the Jacobi–Lie bracket of vector fields on Q. That is, [ξ, ζ]q consists of the components of[uiξ

i, ujζ](q) relative to the frame u1, . . . , un.Therefore, each tangent space TqU is isomorphic to the Lie algebra Wq := (Rn, [ · , · ]q). Thus, if the

fields u1, . . . , un are independent on U ⊂ Q, the tangent bundle TU is diffeomorphic to a Lie algebra bundleover U .

The dual of [ · , · ]q is, by definition, the operation [ · , · ]∗q : Wq ×W ∗q →W ∗q given by

〈[ξ, α]∗q , ζ〉 ≡ 〈ad∗ξ α, ζ〉 := 〈α, [ξ, ζ]q〉.

Here ad∗ is the dual of the usual ad operator in a Lie algebra; note that in general this operation need notbe associated with a Lie group.

Define the structure functions caij(q) by the equations

[ui(q), uj(q)] = caij(q)ua(q), (5)

i, j,m = 1, . . . , n. These quantities vanish if and only if the vector fields ui(q), i = 1, . . . , n, commute.Viewing ui as vector fields on TQ whose fiber components equal 0 (that is, taking the vertical lift of these

vector fields), it is straightforward to see that the directional derivatives ui[l] for a function l : TQ→ R areevaluated in coordinates by the formula

ui[l] = ψji∂l

∂qj.

By definition, for the frame u = (u1, . . . , un),

u[l] := (u1[l], . . . , un[l]).

The evolution of the variables (q, ξ) is governed by the Hamel equations

d

dt

∂l

∂ξ=

[ξ,∂l

∂ξ

]∗q

+ u[l]. (6)

coupled with equations (2). The coordinate form of (6) reads

d

dt

∂l

∂ξj= caij

∂l

∂ξaξi + uj [l] = caij

∂l

∂ξaξi + ψij

∂l

∂qi.

If ui = ∂/∂qi, then ψij = δij , ξj = qj , caij = 0, and equations (6) become the Euler–Lagrange equations (1).

Equations (6) were introduced in Hamel [1904] (see also Neimark and Fufaev [1972] for details and somehistory).

3

2.3 Hamilton’s Principle for Hamel’s Equations

Let γ : [a, b]→ Q be a smooth curve in the configuration space. A variation of the curve γ(t) is a smoothmap β : [a, b]× [−ε, ε]→ Q that satisfies the condition β(t, 0) = γ(t). This variation defines the vector field

δγ(t) =∂β(t, s)

∂s

∣∣∣∣s=0

along the curve γ(t).

Theorem 2.1 (Bloch et al. [2009]). Let L : TQ→ R be a Lagrangian and l : TQ→ R be its representationin local coordinates (q, ξ). Then, the following statements are equivalent:

(i) The curve q(t), where a ≤ t ≤ b, is a critical point of the action functional∫ b

a

L(q, q) dt (7)

on the space of curves in Q connecting qa to qb on the interval [a, b], where we choose variations of thecurve q(t) that satisfy δq(a) = δq(b) = 0.

(ii) The curve q(t) satisfies the Euler–Lagrange equations (1).

(iii) The curve (q(t), ξ(t)) is a critical point of the functional∫ b

a

l(q, ξ) dt (8)

with respect to variations δξ, induced by the variations δq = u(q) · ζ, and given by

δξ = η + [ξ, ζ]q. (9)

(iv) The curve (q(t), ξ(t)) satisfies the Hamel equations (6) coupled with the equations q = u(q) · ξ.

For the proof of Theorem 2.1 and the early development and history of these equations, as well as othervariational structures associated with Hamel’s equations see Poincare [1901], Hamel [1904], Bloch et al.[2009], and Ball et al. [2012].

3 The Discrete Hamel Equations

Here we present a summary of a recent results on the discrete Hamel formalism. We refer the readers toBall and Zenkov [2011] for details and proofs.

3.1 Discrete Hamilton’s Principle

A discrete analogue of Lagrangian mechanics can be obtained by discretizing Hamilton’s principle; thisapproach underlies the construction of variational integrators. See Marsden and West [2001], and referencestherein, for a more detailed discussion of discrete mechanics. (For a general review of geometric numericalintegration and Lie group integrators, see Hairer et al. [2006] and Iserles et al. [2005], respectively.)

A key notion is that of the discrete Lagrangian, which is a map Ld : Q×Q→ R that approximates theaction integral along an exact solution of the Euler–Lagrange equations joining the configurations qk, qk+1 ∈Q,

Ld(qk, qk+1) ≈ extq∈C([0,h],Q)

∫ h

0

L(q, q) dt, (10)

4

where C([0, h], Q) is the space of curves q : [0, h]→ Q with q(0) = qk, q(h) = qk+1, and ext denotes extremum.In the discrete setting, the action integral of Lagrangian mechanics is replaced by an action sum

Sd(q0, q1, . . . , qN ) =

N−1∑k=0

Ld(qk, qk+1), (11)

where qk ∈ Q, k = 0, 1, . . . , N , is a finite sequence of points in the configuration space. The equations areobtained by the discrete Hamilton’s principle, which extremizes the discrete action given fixed endpointsq0 and qN . Taking the extremum over q1, . . . , qN−1 gives the discrete Euler–Lagrange equations

D1Ld(qk, qk+1) +D2L

d(qk−1, qk) = 0,

for k = 1, . . . , N − 1, where Di denotes the partial derivative with respect to the i-th input. This implicitlydefines the update map Φ : Q×Q→ Q×Q, where Φ(qk−1, qk) = (qk, qk+1) and Q×Q replaces the velocityphase space TQ of Lagrangian mechanics.

In the case that Q is a vector space, it may be convenient to use (qk+1/2, vk,k+1), where qk+1/2 =12 (qk + qk+1) and vk,k+1 = 1

h (qk+1− qk), as a state of a discrete mechanical system. In such a representation,the discrete Euler–Lagrange equations become

12

(D1L

d(qk−1/2, vk−1,k) +D1Ld(qk+1/2, vk,k+1)

)+ 1

h

(D2L

d(qk−1/2, vk−1,k)−D2Ld(qk+1/2, vk,k+1)

)= 0.

These equations are equivalent to the variational principle

δSd =

N−1∑k=0

(D1L

d(qk+1/2, vk,k+1) δqk+1/2 +D2Ld(qk+1/2, vk,k+1) δvk,k+1

)= 0, (12)

where the variations δqk+1/2 and δvk,k+1 are induced by the variations δqk and are given by the formulae

δqk+1/2 = 12

(δqk+1 + δqk

), δvk,k+1 = 1

h

(δqk+1 − δqk

).

3.2 Discrete Hamel’s Equations

Assume the configuration manifold Q is a vector space. In order to construct the discrete Hamel equationsfor a given continuous-time mechanical system, one starts by selecting a frame u(q) = (u1(q), . . . , un(q)) onQ and computing the Lagrangian l(q, ξ) given by (4). One then discretizes this Lagrangian (we only discussthe mid-point rule here) and obtains

ld(qk+1/2, ξk,k+1) = hl(qk+1/2, ξk,k+1). (13)

Here qk+1/2 = 12 (qk+1 + qk) as before, and ξk,k+1 = (ξ1

k,k+1, . . . , ξnk,k+1) are the velocity components relative

to the frame u(q) at qk+1/2. The link between these velocity components and the configurations of thesystem is obtained by discretizing the kinematic equation (2).2 Note that the discretization (13) is carriedout after writing the continuous-time Lagrangian as a function of (q, ξ).

Similar to (11), define the discrete action by the formula

sd =

N−1∑k=0

ld(qk+1/2, ξk,k+1), (14)

which is an approximation of the action integral (8) of the continuous-time system.One of the challenges of discretizing the Hamel equations has been understanding the discrete analogue

of the bracket term in (6). Until recently, it was only known how to handle this for systems on Lie groups

2There is a certain freedom in selecting this discretization.

5

(see e.g. Bobenko and Suris [1999] and Marsden et al. [1999]). Below, we describe the approach of Ball andZenkov [2011] to discretizing the Hamel equations.

Define the discrete conjugate momentum by

µk,k+1 := D2ld(qk+1/2, ξk,k+1). (15)

Theorem 3.1 (Ball and Zenkov [2011]). The sequence(qk+1/2, ξk,k+1

)∈ TQ satisfies the discrete Hamel

equations

1h

(D2l

d(qk−1/2, ξk−1,k)−D2ld(qk+1/2, ξk,k+1)

)+ 1

2

([ξk−1,k, µk−1,k

]∗qk−1/2

+[ξk,k+1, µk,k+1

]∗qk+1/2

)+ 1

2

(u[ld](qk−1/2, ξk−1,k) + u

[ld](qk+1/2, ξk,k+1)

)= 0 (16)

if and only if

δsd = δ

N−1∑k=0

ld(qk+α, ξk,k+1) = 0,

where

δqk+1/2 = u(qk+1/2) · ζk+1/2, (17)

δξk,k+1 = 1h

(ζk+1 − ζk

)+[ξk,k+1, ζk+1/2

]qk+1/2

, (18)

where ζ0 = ζN = 0 and ζk+1/2 = 12 (ζk+1 + ζk), k = 0, . . . , N − 1.

Equations (16) along with the discrete analogue of the kinematic equations (2) define the update map(qk−1/2, ξk−1,k) 7→ (qk+1/2, ξk,k+1).

We refer the readers to Ball and Zenkov [2011] for the proof of Theorem 3.1, the motivation, and theorigins of the definition of variations (17) and (18).

In the discrete model of a spherical pendulum discussed below, the bracket terms in Hamel’s equationsvanish, and we do not need to describe the method for discretizing the bracket terms in this paper (detailson this can be found in Ball and Zenkov [2011]).

4 The Spherical Pendulum

An n-dimensional spherical pendulum is a point mass moving on the sphere of radius r centered atthe origin of Rn. The development here is based on the interpretation of the pendulum as a degeneraten-dimensional heavy rigid body rotating about a fixed point. We thus begin by reviewing the heavy toptheory following Holm et al. [1998].

4.1 The Euler–Poincare Equations and the Heavy Top

The Euler–Poincare Equations. Given a mechanical system on a Lie group G with a left-invariantLagrangian L : G → R, one may write the dynamics in the form of the Euler–Lagrange equations. Thishowever is not the most efficient way, as confirmed by development of the rigid body dynamics in thepioneering work of Euler [1752], Lagrange [1788], and Poincare [1901]. Let g(t) be a curve in G. As followsfrom the aforementioned references, it is much more efficient to use the body angular velocity

Ω = g−1g ∈ g

instead of the Lagrangian velocity g. Here g is the Lie algebra of G, and we will denote both the left actionof the group G on itself and the tangent lifted action on the tangent bundle TG with left multiplication bya group element.

6

Since the Lagrangian is left-invariant, there exists a real-valued function l on the Lie algebra g called thereduced Lagrangian such that

L(g, g) = l(g−1g)

for all (g, g) ∈ TG. Using (g,Ω) as coordinates on the velocity phase space of the system, the dynamics isgiven by the Euler–Poincare equations

d

dt

∂l

∂Ω= ad∗Ω

∂l

∂Ω(19)

coupled with the reconstruction equationg = gΩ.3 (20)

Equations (19) and (20) are a special case of Hamel’s equations with respect to the left-invariant vectorfields obtained by left-translating a frame on the Lie algebra g, and they are of course equivalent to theEuler–Lagrange equations for the Lagrangian L : TG → R. For details, related variational principles, andhistory, see Marsden and Ratiu [1999] and Bloch et al. [2009].

The Heavy Top. A generalized rigid body is a mechanical system whose configuration space is a Liegroup G and whose Lagrangian is a G-invariant Riemann metric on G. For G = SO(3), the Euler–Poincareequations (19) become the dynamic Euler equations that appeared for the first time in Euler [1752].

The heavy top is a rigid body rotating about a fixed point in a gravitational field. Having the n-dimensional pendulum in mind, we consider a generalized heavy top whose kinetic energy is given by aleft-invariant metric on G = SO(n) specified by the inertia tensor I : so(n) → so∗(n), while the potentialenergy is defined by the work due to gravity and is represented by the formula mg〈γ, a〉, just as in the3-dimensional case. Here m is the mass of the body, a ∈ V = Rn is the vector from the fixed point tothe center of mass, and γ is the unit upward vector (i.e., a unit vector in the direction opposite to that ofgravity), all written relative to the body frame. That is, if g ∈ SO(n) is the orientation of the body anden is the unit spatial vector in the direction opposite to the direction of gravity, γ = g−1en. The potentialenergy of course breaks the full SO(n) symmetry of the Lagrangian.

As discussed in Holm et al. [1998], one views the heavy top Lagrangian as a function on TG×V ∗, whichdefines the reduced Lagrangian

l(Ω, γ) = 12 〈IΩ,Ω〉g −mg〈γ, a〉V (21)

on g× V ∗. With the aid of this reduced Lagrangian, the dynamics of the heavy top is given by the Euler–Poincare equations on g× V ∗

d

dt

∂l

∂Ω= ad∗Ω

∂l

∂Ω+∂l

∂γ γ (22)

along with the equationγ = −Ωγ. (23)

Recall that the diamond operation

V × V ∗ 3 (α, γ) 7→ α γ ∈ g∗

is defined by the condition〈α γ,Ω〉g = 〈γ,Ωα〉V .

Omitting details, we just state here that for G = SO(n) and V = Rn the diamond operation, after identifyingV ∗ and V in a standard way, is given by

α γ = γ ∧ α ≡ γ ⊗ α− α⊗ γ.

As before, we refer the reader to Holm et al. [1998] for the derivation, associated variational principles, andmore details on system (22) and (23) and the diamond operation.

3Note that equation (19) decouples from the system (19) and (20).

7

4.2 The Spherical Pendulum as a Degenerate Heavy Top

When the 3-dimensional spherical pendulum is viewed as a degenerate rigid body, the dynamics is representedby the heavy top equations

µ = µ× ξ +mgγ × a, (24)

γ = γ × ξ (25)

with a degenerate inertia tensor diagmr2,mr2, 0, where ξ is the angular velocity of the top, µ is its angularmomentum, both viewed as vectors, γ ∈ R3 is the unit vertical vector (and thus the constraint ‖γ‖ = 1 isimposed), and a is the vector from the origin to the center of mass, which for the spherical pendulum is itsbob, all written relative to the body frame.

Similarly, the n-dimensional pendulum is a heavy top whose degenerate inertia operator is

I Ω = JΩ + ΩJ,

where J is the mass matrix ,

J =

0 · · · 0 0...

. . ....

...

0 · · · 0 0

0 · · · 0 mr2

.Thus, the dynamics of the n-dimensional spherical pendulum is

M = MΩ− ΩM −mga ∧ γ, (26)

γ = −Ωγ, (27)

with

M =∂l

∂Ω= I Ω =

0 · · · 0 M1n

.... . .

......

0 · · · 0 Mn−1n

−M1n · · · −Mn−1n 0

.Since the momentum components Mij , 1 ≤ i, j ≤ n − 1, vanish identically, there are n − 1 independent

equations in (26). Thus, one needs 2n − 1 equations to capture the pendulum dynamics. This reflects thefact that rotations about the direction of the pendulum have no influence on the pendulum’s motion, i.e.,the quantities Ωij , 1 ≤ i, j ≤ n − 1, can be selected as arbitrary functions of time. Various choices ofthese functions correspond to various body frames. The dynamics then is simplified by setting Ωij = 0,1 ≤ i, j ≤ n − 1. We emphasize that this representation, though redundant, eliminates the use of localcoordinates on the sphere, such as spherical coordinates. More details on this appear below. Sphericalcoordinates, while being a nice theoretical tool, introduce artificial singularities (at the north and south polesfor S2). That is, the equations of motion written in spherical coordinates have denominators that vanishat the coordinate singularities, but this has nothing to do with the physics of the problem and is solelycaused by the geometry of the spherical coordinates. Thus, the use of spherical coordinates in numericalcalculations is ill-advised, as even if the numerical solution does not intersect a coordinate singularity, aclose encounter results in numerical ill-conditioning that will significantly compromise the accuracy of thecomputed trajectory.

Another important remark is that the length of the vector γ is a conservation law of equations (26)and (27),

‖γ‖ = const, (28)

and thus adding the constraint ‖γ‖ = 1 does not result in a system of differential-algebraic equations. Thelatter are known to be a nontrivial object for numerical integration.

8

Equations (26) and (27) may be interpreted in a number of ways. Recall that we obtained them asthe equations of motion of a degenerate rigid body. Alternatively, (26) and (27) may be interpreted as thedynamics of the n-dimensional Suslov problem (see Fedorov and Kozlov [1995] for the n-dimensional case,and Neimark and Fufaev [1972] and Bloch [2003] for the classical Suslov problem) for a rigid body with arotationally-invariant inertia tensor and the constraints

Ωij = 0, 1 ≤ i, j ≤ n− 1. (29)

Equations (26) and (27) are in fact Hamel’s equations written in the redundant coordinates (γ1, . . . , γn)on the sphere ‖γ‖ = 1 relative to the frame

uij = γi∂

∂γj− γj ∂

∂γi, 1 ≤ i < j ≤ n− 1. (30)

This is because

1. [uab, ucd] equals 0 if the four indices a, b, c, d are all different; otherwise, it is given (up to a sign) byone of the fields (30), if there is a pair of matching indices, one coming from the field uab and the othercoming from ucd.

2. uij [l] = −uij [mg〈γ, a〉] = −mgrγi if j = n, 1 ≤ i ≤ n− 1, and 0 if i 6= n and j 6= n.

Since one may assume, without loss of generality, that conditions (29) are satisfied, the Hamel equations forthe pendulum reduce to the constrained Hamel equations

mr2ξi = −mgrγi, γi = −ξiγn, γn =∑

ξiγi, 1 ≤ i ≤ n− 1. (31)

Here ξi = Ωin, 1 ≤ i ≤ n− 1, i.e., ξi are the nonzero components of Ω. Setting

ξ = (ξ1, . . . , ξn−1) ≡ (Ω1n, . . . ,Ωi−1n) (32)

and u = u(γ) = (u1n(γ), . . . , un−1n(γ)), the dynamics becomes

d

dt

∂l

∂ξ= u[l], (33)

γ = u · ξ, (34)

where the pendulum’s Lagrangian is given by (21), and, as a function of (ξ, γ), reads

l(ξ, γ) = 12mr

2‖ξ‖2 −mg〈γ, a〉. (35)

The dynamics given by (33) is an n-dimensional analogue of the system (24) and (25). Note that the fieldsu1n(γ), . . . , un−1n(γ) are tangent to, and independent almost everywhere on each sphere ‖γ‖ = const. Themomentum components

mr2(γiξj − γjξi

), 0 ≤ i, j ≤ n− 1,

are preserved, which follows from the invariance of the Lagrangian with respect to rotations about γ. Ourdiscretization will be based on this latter representation of the dynamics.

5 Hamel Discretization for the Spherical Pendulum

The integrator for the spherical pendulum is constructed by discretizing Hamel’s equations (33).Let the positive real constant h be the time step. Applying the mid-point rule to Lagrangian (35), the

discrete Lagrangian is computed to be

ld(ξk,k+1, γk+1/2) = h2mr

2‖ξk,k+1‖2 − hmg〈γk+1/2, a〉. (36)

9

Here, in agreement with the general discrete Hamel’s formalism, ξk,k+1 = (ξ1k,k+1, . . . , ξ

n−1k,k+1) is the discrete

analogue of the angular velocity ξ introduced in (32) and γk+1/2 = 12 (γk+1 + γk). The discrete Hamel

equations for the spherical pendulum then read

1hmr

2(ξk,k+1 − ξk−1,k

)= − 1

2

(Dumg〈γk+1/2, a〉+Dumg〈γk−1/2, a〉

), (37)

1h

(γk+1/2 − γk−1/2

)= 1

2

(uk+1/2 + uk−1/2

)· 1

2

(ξk,k+1 + ξk−1,k

), (38)

or, in components,

1hmr

2(ξik,k+1 − ξik−1,k

)= − 1

2mgr(γik+1/2 + γik−1/2

), (39)

1h

(γik+1/2 − γ

ik−1/2

)= − 1

4

(ξik,k+1 + ξik−1,k

)(γnk+1/2 + γnk−1/2

), (40)

1h

(γnk+1/2 − γ

nk−1/2

)= 1

4

n−1∑i=1

(ξik,k+1 + ξik−1,k

)(γik+1/2 + γik−1/2

), (41)

where 1 ≤ i ≤ n− 1. We reiterate that there is a certain flexibility in setting up the discrete analogue (41)of the continuous-time kinematic equation (34). Our choice may be justified by a number of arguments, oneof them being energy conservation by the discrete dynamics.

Theorem 5.1. The discrete spherical pendulum dynamics (37) and (38) preserves the energy, verticalmomentum, and ‖γ‖.

Proof. Indeed, using the Lie algebra elements Ωk,k+1, equation (38) becomes

γk+1/2 = (I +Ak)−1(I −Ak)γk−1/2,

whereAk = h

4

(Ωk,k+1 + Ωk−1,k

).4

It is straightforward to check that the matrix

(I +Ak)−1(I −Ak)

is orthogonal (it is simply the Cayley transform of Ak), and therefore ‖γk+1/2‖ = ‖γk−1/2‖.Next, either using general symmetry arguments, or by a straightforward calculation, one establishes the

formulaemr2

(γik+1/2ξ

jk,k+1 − γ

jk+1/2ξ

ik,k+1

)= const, 1 ≤ i, j ≤ n− 1,

i.e., the vertical components of spatial momentum are conserved.The energy preservation,

12mr

2‖ξk,k+1‖2 +mgrγnk+1/2 = const, (42)

is confirmed by taking the sum of equations (39) multiplied by(ξik,k+1 +ξik−1,k

)for 1 ≤ i ≤ n−1 and adding

the result to equation (41). This yields the following equality,

12mr

2‖ξk,k+1‖2 +mgrγnk+1/2 − 12mr

2‖ξk−1,k‖2 −mgrγnk−1/2 = 0,

which is equivalent to (42).

To recap, the proposed method preserves the energy, the length constraint, and the momentum, and itis therefore a homogeneous space energy-momentum integrator.

4The matrix I −Ak is invertible if h is sufficiently small.

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6 Comparison with Other Methods

The proposed method takes advantage of the homogeneous space structure of Sn−1, which has a transitive Liegroup action by SO(n). In particular, the vector γ ∈ Sn−1 is updated by the left action of a rotation matrix,given by the Cayley transformation of a skew-symmetric matrix Ak that approximates the angular velocity Ωintegrated over a half-timestep. Interestingly, this falls out naturally from discretizing the Hamel formulationof the spherical pendulum, and it would be interesting to see what general choices of coordinate frames inthe Hamel formulation lead to similar methods for more general flows on homogeneous spaces. We willnow discuss some alternative methods of simulating the spherical pendulum equations in the 3-dimensionalsetting.

6.1 Homogeneous Space Variational Integrators

If one were to instead formulate the spherical pendulum problem directly on S2, it is possible to lift thevariational principle on S2 to SO(3), by relating the curve γ(t) ∈ S2 with a curve g(t) ∈ SO(3), by therelation γ(t) = g(t)γ(0), where g(0) = I. The complication is that the resulting variational principle onSO(3) does not have a unique extremizer, due to the presence of a nontrivial isotropy subgroup associatedwith the action of SO(3) on S2. With a suitable choice of connection (and the associated horizontal lift),this ambiguity can be eliminated, and the resulting problem (and similar problems on homogeneous spaces)can be solved using Lie group variational integrator techniques, as described in Lee et al. [2009].

It should be noted that when applied to the n-dimensional spherical pendulum equations, the homoge-neous space variational integrators of Lee et al. [2009] can be interpreted as an index-reduced version of theRATTLE algorithm of Anderson [1983], where the Lagrange multiplier is eliminated by substitution.

6.2 Nonholonomic Integrators

As mentioned in Section 4, the spherical pendulum equations can be viewed as a Suslov problem, which isan example of a nonholonomic mechanical system. In principle, one could apply a nonholonomic integrator,such as the one described in Fedorov and Zenkov [2005] and McLachlan and Perlmutter [2006]. However,replacing the length constraint with an infinitesimal constraint and a discrete nonholonomic constraint mayresult in loss of structural stability and poor numerical preservation of the constraint properties if the discretenonholonomic constraint is poorly chosen. An alternative approach to simulating nonholonomic mechanicsinvolves a discretization of the forces of constraint, and a careful choice of force discretization has been shownto yield promising results, see Lynch and Zenkov [2009] for details. The proposed integrator agrees in partswith the approach of Lynch and Zenkov [2009], however the latter reference does not take into account thekinematic equation (34) nor the various choices in discretizing the kinematic equation. See Ball and Zenkov[2011] for details.

6.3 Constrained Symplectic Integrators

Given the relatively simple nature of the unit length constraint, it is quite natural to apply the RATTLE algo-rithm (see e.g. Anderson [1983] and McLachlan et al. [2012]) which is a generalization of the Stormer–Verletmethod for constrained Hamiltonian systems that is designed to explicitly preserve holonomic constraints.This method does require the use of a nonlinear solver on a system of nonlinear equations of dimension equalto the number of constraints. The cost of the nonlinear solve can increase significantly as the number ofcopies of the sphere in the configuration space increases.

6.4 Differential-Algebraic Equation Solvers

The proposed discrete Hamel integrator can be easily scaled to an arbitrary number of copies of the sphere,possibly chained together in a n-dimensional spherical pendulum. Such multi-body systems however posesignificant challenges for differential-algebraic equation solvers, since they are examples of what are referred

11

to as high-index DAEs, for which the theory and numerical methods are much less developed. It is possibleto perform index reduction on the system of differential-algebraic equations, but this involves significanteffort, and the numerical results can be mixed.

6.5 Numerical Comparisons

Since the method is a second-order accurate energy-momentum method, it is natural to compare it to theStormer–Verlet method, as well as the RATTLE method (which is a generalization of Stormer–Verlet forconstrained Hamiltonian systems).

For the Stormer–Verlet method, we compute Hamilton’s equations for the spherical pendulum, which isgiven by

x = 1mp, (43)

p = −mge3 +(mgx · e3 − 1

m‖p‖2)x = f(x,p), (44)

and we apply the generalization of the Stormer–Verlet method for general partitioned problems (see (3.4) inHairer et al. [2006]),

pn+1/2 = pn + h2f(xn,pn+1/2),

xn+1 = xn + hmpn+1/2,

pn+1 = pn+1/2 + h2f(xn+1,pn+1/2).

This system of equations is linearly implicit, since the first equation is implicit in pn+ 12, but the rest of the

equations are explicit.The RATTLE method (see (1.26) in Hairer et al. [2006]) can be applied to the particle in a uniform

gravitational field problem,

x = 1mp,

p = −mge3,

subject to the constraint φ(x) = 12 (‖x‖2− 1) = 0. We also introduce Φ(x) = ∂φ

∂x = xT . Then, the RATTLEmethod applied to this problem is given by,

pn+1/2 = pn − h2

(mge3 + Φ(xn)Tλn

),

xn+1 = xn + hmpn+1/2,

0 = φ(xn+1),

pn+1 = pn+1/2 − h2

(mge3 + Φ(xn+1)Tµn

),

0 = 1mΦ(xn+1) ·pn+1.

7 Simulations

In Figures 1 and 2, we present simulations of the dynamics of the spherical pendulum using the theorydeveloped above, which we compare with simulations using the generalized Stormer–Verlet method and theRATTLE method in Figures 3 and 4, respectively.

For simulations, we select the parameters of the system and the initial conditions to be m = 1 kg,r = 9.8 m, h = .2 s, ξ1

0 = .6 rad/s, ξ20 = 0 rad/s, γ1

0 = .3 m, γ20 = .2 m, γ3

0 = −.932738 m. The trajectoryof the bob of the pendulum is shown in Figure 1a. As expected, it reveals the quasiperiodic nature ofpendulum’s dynamics. Theoretically, if one solves the nonlinear equations exactly, and in the absenceof numerical roundoff error, the Hamel integrator should exactly preserve the length constraint, and theenergy. In practice, Figure 1b demonstrates that ‖γ‖ stays to within unit length to about 10−10 after

12

10,000 iterations. Figure 1c demonstrates numerical energy conservation, and the energy error is to about10−10 after 10,000 iterations as well. Indeed, one notices that the energy error tracks the length error of thesimulation, which is presumably due to the relationship between the length of the pendulum and the potentialenergy of the pendulum. The drift in both appear to be due to accumulation of numerical roundoff error,and could possibly be reduced through the use of compensated summation techniques (see, for example,Kahan [1965]).

(a) Pendulum’s trajectory on S2

0 2000 4000 6000 8000 10 000-2. ´ 10-10

1. ´ 10-10

(b) Preservation of the length of γ

0 2000 4000 6000 8000 10 000-2. ´ 10-10

-1. ´ 10-10

0

1. ´ 10-10

2. ´ 10-10

(c) Conservation of energy

Figure 1: Numerical properties of the Hamel integrator.

Figure 2 shows pendulum’s trajectory that crosses the equator. This simulation demonstrates the globalnature of the algorithm, and also seems to do a good job of hinting at the geometric conservation propertiesof the method.

Figure 2: A trajectory with initial conditions above the equator integrated with the Hamel integrator.

We also simulate the spherical pendulum using the generalized Stormer–Verlet method and the RATTLEmethod described in Section 6. The generalized Stormer–Verlet method exhibits surprisingly good unitlength preservation in Figure 3b of 10−11 when applied to index-reduced version of the equations of motion(43)–(44). The energy behavior in Figure 3c is typical of a symplectic integrator, with the characteristic

0.30.2

0.10

0.10.2

0.3

0.30.2

0.10

0.10.2

0.3

0.980.960.94

(a) Pendulum’s trajectory on S2

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

6

4

2

0

2

4

6

8

10

12

x 1012

(b) Preservation of the length of γ

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000.05

0.04

0.03

0.02

0.01

0

(c) Conservation of energy

Figure 3: Numerical properties of the Stormer–Verlet method.

bounded energy oscillations. Even though the RATTLE method is intended to explicitly enforce the unit

13

length constraint, it exhibits a unit length preservation in Figure 4b of 10−7, which is poorer than both theHamel integrator and the generalized Stormer–Verlet method. The energy error for RATTLE in Figure 4cis comparable to that of the generalized Stormer–Verlet method, but both pale in comparison to the energyerror for the Hamel integrator.

0.30.2

0.10

0.10.2

0.3

0.30.2

0.10

0.10.2

0.3

0.980.960.94

(a) Pendulum’s trajectory on S2

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.5

1

1.5

2

2.5x 10

7

(b) Preservation of the length of γ

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

0.05

0.04

0.03

0.02

0.01

0

(c) Conservation of energy

Figure 4: Numerical properties of the RATTLE method.

8 Conclusions and Future Work

In this paper, we constructed an energy-momentum integrator for the n-dimensional spherical pendulum bydiscretizing Hamel’s equations. We showed the integrator preserves key mechanical quantities and illustratedthe method with long-time numerical simulations, and comparisons with the generalized Stormer–Verletmethod and the RATTLE method.

Future work will include further study of discrete systems on Lie group orbits, including the simulationof linked rigid body systems as well as the control of such systems. The excellent numerical properties ofthe proposed Hamel integrator will serve as a basis for constructing numerical optimal control algorithms,which are heavily dependent on the quality of the numerical discretization of the natural dynamics.

An important issue that remains to be studied is the conditions on the constrained variations and thereconstruction equation that will guarantee that the Hamel integrator is symplectic. This has potentialto lead to the construction of a symplectic-energy-momentum integrator, which by a result due to Geand Marsden [1988], would imply that the method recovers the exact trajectory, up to a possible timereparameterization.

As pointed out in Alber and Marsden [1996], the dynamics of the spherical pendulum is integrable. Itwould be interesting to find out if the discrete model introduced here is integrable as well.

We also expect the discrete Hamel’s formalism to be useful in resolving possible loss of structural stabilityby variational and nonholonomic integrators observed in Lynch and Zenkov [2009, 2010] and Peng et al.[2012]. This will be the subject of future publications.

9 Acknowledgments

The research of AMB was partially supported by NSF grants DMS-0806765, DMS-0907949 and DMS-1207993. The research of ML was partially supported by NSF grants DMS-1010687, CMMI-1029445, andDMS-1065972. The research of DVZ was partially supported by NSF grants DMS-0306017, DMS-0604108,DMS-0908995, and DMS-1211454.

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