Appendix A

Fundamental Lemmas of the Calculus

of Variations

Several versions of the fundamental lemma in the calculus of variations are

presented. The classical version of the fundamental lemma when the configu-
ration manifold is Rn is summarized first, and then versions are summarized
when the configuration manifold is an embedded manifold in Rn or a Lie
group embedded in Rn×n .

A.1 Fundamental Lemma of Variational Calculus on Rn

The fundamental lemma of the calculus of variations on Rn [5] is the following

statement. Let x : [t0 , tf ] → Rn be continuous and suppose that F : [t0 , tf ] →
Rn is continuous. If
 tf
F (t) · δx(t) dt = 0,
t0

holds for all continuous variations δx : [t0 , tf ] → Rn satisfying δx(t0 ) =

δx(tf ) = 0, then it follows that

F (t) = 0, t0 ≤ t ≤ tf .

As usual, the notation F · δx indicates the inner or dot product of a covector

F ∈ (Rn )∗ and a vector δx ∈ Rn .
Suppose that the result is not true, that is F (t̄) = 0 for some t0 ≤ t̄ ≤ tf .
Since F is continuous, it is necessarily nonzero and of a fixed sign in some
neighborhood of t̄. Then, an admissible variation can be constructed that is
zero outside of this neighborhood and leads to a violation of the assumption.
This contradiction proves the validity of the statement.

© Springer International Publishing AG 2018 521

T. Lee et al., Global Formulations of Lagrangian and Hamiltonian
Dynamics on Manifolds, Interaction of Mechanics and Mathematics,
DOI 10.1007/978-3-319-56953-6
522 A Fundamental Lemmas of the Calculus of Variations

A.2 Fundamental Lemma of the Calculus of Variations

on an Embedded Manifold

Let M be a differentiable manifold embedded in Rn . The fundamental

lemma of the calculus of variations on M [5] is the following statement.
Let x : [t0 , tf ] → M be continuous and suppose that F : [t0 , tf ] → (Rn )∗ is
continuous. If
 tf
F (t) · ξ(t) dt = 0,
t0

holds for all continuous variations ξ : [t0 , tf ] → Tx(t) M satisfying ξ(t0 ) =

ξ(tf ) = 0, then it follows that

F (t) · ξ = 0, t 0 ≤ t ≤ tf ,

that is F (t) is orthogonal to Tx M for each t0 ≤ t ≤ tf . As usual, the notation

F · ξ indicates the inner or dot product of a covector F ∈ (Rn )∗ and a vector
ξ ∈ Tx(t) M .
The proof is essentially the same as the one above.

A.3 Fundamental Lemma of Variational Calculus

on a Lie Group

The fundamental lemma of the calculus of variations can also be applied to a

Lie group G, since we can associate the tangent space with the corresponding
Lie algebra g. Let g∗ denote the vector space of linear functionals on g. This
leads to the following statement.
The fundamental lemma of the calculus of variations on the Lie group G is
the following statement. Let g : [t0 , tf ] → G be continuous and suppose that
F : [t0 , tf ] → g∗ is continuous. If
 tf
F (t) · η dt = 0,
t0

holds for all continuous variations η : [t0 , tf ] → g satisfying η(t0 ) = η(tf ) = 0,

then it follows that

F (t) · η = 0, t0 ≤ t ≤ t f ,

that is F (t) is orthogonal to g for each t0 ≤ t ≤ tf . As usual, the notation

F · η indicates the pairing of a covector F ∈ g∗ and a vector η ∈ g.
Appendix B
Linearization as an Approximation
to Lagrangian Dynamics on a Manifold

Linearization of a (typically) nonlinear vector field on Rn , in a neighborhood

of an equilibrium of the vector field, is a classical and widely used tech-
nique. It provides a way to approximate the local nonlinear dynamics near
an equilibrium solution, it may provide information about the stability of
the equilibrium according to the stable manifold theorem [33], and it pro-
vides the basis for developing mathematical equations that are widely used
in control applications. This technique is so widely used that we give special
attention to it in this Appendix. A summary of linearization has been given
in Chapter 1. Further background on linearization of a vector field on Rn is
given in [44, 76, 93].
Here, we apply the concept of linearization to a vector field defined on a
manifold in the form developed and studied in this book. We begin with a
vector field defined on an (n − m)-dimensional differentiable manifold M ⊂
Rn , where 1 ≤ m < n; the vector field is denoted by F : M → Rn and it has
the property that for each x ∈ M, F (x) ∈ Tx M .
Consistent with the assumption that M is a manifold embedded in Rn ,
we consider an extension of the vector field F on M to the embedding vector
space. We denote this extension vector field by F e : Rn → Rn . Thus, for
each x ∈ M , F e (x) = F (x). From the perspective of the embedding space,
M can be viewed as an invariant manifold of the vector field F e : Rn → Rn .
If xe ∈ M is an equilibrium of the vector field F , that is F (xe ) = 0, then it
is also an equilibrium of the extended vector field F e .
Linearization of the vector field F : M → Rn in the neighborhood of an
equilibrium solution xe ∈ M leads to a linear vector field on Txe M that is
first-order accurate in approximating F : M → Rn in a neighborhood of
xe ∈ M . It is convenient to describe the linear vector field on the (n − m)-
dimensional tangent space in terms of n − m basis vectors for Txe M .

© Springer International Publishing AG 2018 523

T. Lee et al., Global Formulations of Lagrangian and Hamiltonian
Dynamics on Manifolds, Interaction of Mechanics and Mathematics,
DOI 10.1007/978-3-319-56953-6
524 B Linearization as an Approximation to Lagrangian Dynamics on a Manifold

There are two equivalent approaches that can be followed:

• Linearize the vector field F e on Rn at the equilibrium xe ∈ M ; then restrict
this linear vector field to the tangent space Txe M by introducing a basis
for the tangent space Txe M .
• Introduce n − m local coordinates on M in a neighborhood of the equilib-
rium xe ∈ M ; express the vector field F in terms of these local coordinates
and linearize at the equilibrium.
Either of these approaches leads to a linear vector field defined on an (n−m)-
dimensional subspace of Rn . The two approaches lead to equivalent realiza-
tions of the linearized vector field.
This linearized vector field can be viewed as approximating the original
nonlinear vector field on M , at least in a small neighborhood of the equi-
librium. In this sense, the linearized equations are viewed as describing local
perturbations of the dynamical flow near the equilibrium. This interpretation
provides important motivation for the linearization technique.
The second approach is most common in many engineering applications,
but it requires the introduction of local coordinates on the manifold M and
the description of the resulting vector field on M in terms of local coordinates;
this is often a challenging step. In this book we emphasize the first approach,
but we use whichever approach is most convenient for a particular case.
We now illustrate the linearization process for three classes of Lagrangian
vector fields with configuration manifolds that are studied in this book: S1 ,
S2 , and SO(3). These illustrations provide the necessary background for lin-
earization on configuration manifolds that are products of these. We obtain
Euler–Lagrange equations that describe the dynamics for a particle or rigid
body under the action of a potential. In each case, linearized differential
equations are determined that approximate the dynamics on a manifold in a
neighborhood of an equilibrium solution.

B.1 Linearization on TS1

Consider an ideal particle, of mass m, that moves on a circle of radius r > 0

in a fixed plane in R3 , centered at the origin of an inertial frame, under the
action of a potential. The configuration manifold is S1 , embedded in R2 . As
shown in Chapter 4, the Euler–Lagrange equation, using standard notation,
is given by

2 ∂U (q)
mL2 q̈ + mr2 q̇ q + (I2×2 − qq T ) = 0.
∂q
These differential equations define a Lagrangian vector field on the tangent
bundle of the configuration manifold TS1 . Assume that qe ∈ S1 satisfies
B.1 Linearization on TS1 525
∂U (qe )
∂q = 0 so that (qe , 0) ∈ TS1 is an equilibrium of the Lagrangian vector
field. We obtain linearized equations following the first approach described
above.
We can view the above differential equations as defining an extended La-
grangian vector field on the tangent bundle TR2 , assuming the potential
function is defined everywhere on R2 . We can linearize the extended differen-
tial equations at the equilibrium to obtain the linearized vector field defined
on TR2 as
∂ 2 U (qe )
mr2 ξ¨ + ξ = 0.
∂q 2
This linearization of the extended vector field can be restricted to the tangent
space of TS1 at (qe , 0) ∈ TS1 , which can be described as
 
T(qe ,0) TS1 = (ξ, ξ)
˙ ∈ R4 : q T ξ = 0, ξ˙ ∈ Tq S1 .
e e

The above description of the linearized vector field is in the form of

differential-algebraic equations. It is convenient to describe this linearized
vector field in a more accessible form by introducing a basis for the tangent
space T(qe ,0) TS1 . To this end, select Sqe as a basis vector for Tqe S1 so that
˙ ∈ T(q ,0) TS1 can be written as
any (ξ, ξ) e

ξ = σSqe ,
ξ˙ = σ̇Sqe ,

where σ ∈ R1 can be viewed as a local coordinate for Tqe S1 . Thus, (ξ, ξ)˙ ∈
T(qe ,0) TS1 for all σ ∈ D, where D ⊂ R1 is an open set containing the origin.
Substituting these expressions into the equation for the linearization of the
extended vector field and taking the inner product with Sqe gives
 
∂ 2 U (qe )
mr2 σ̈ + qeT ST Sq e σ = 0.
∂q 2

This scalar second-order differential equation describes the linearized vector

field of the original Lagrangian vector field on TS1 . Thus, this differential
equation, expressed in terms of (σ, σ̇) ∈ TR1 , describes the Lagrangian dy-
namics on the manifold TS1 in a neighborhood of (qe , 0) ∈ TS1 to first order
in the perturbations.
This linearized equation has been described as a second-order differential
equation in σ, consistent with the usual formulation of the Euler–Lagrange
equations. They can also be expressed as a system of first-order differential
equations by including perturbations of the angular velocity or momentum.
526 B Linearization as an Approximation to Lagrangian Dynamics on a Manifold

B.2 Linearization on TS2

Consider an ideal particle, of mass m, that moves on a sphere in R3 of radius

r > 0, centered at the origin of an inertial frame, under the action of a
potential. The configuration manifold is S2 , embedded in R3 . As shown in
Chapter 5, the Euler–Lagrange equation, using standard notation, is

2 ∂U (q)
mr2 q̈ + mr2 q̇ q + (I3×3 − qq T ) = 0.
∂q
These differential equations define a Lagrangian vector field on the tangent
bundle of the configuration manifold TS2 . Assume that qe ∈ S2 satisfies
∂U (qe )
∂q = 0 so that (qe , 0) ∈ TS2 is an equilibrium of the Lagrangian vector
field. We obtain linearized equations following the first approach described
above. The same approach, for different dynamics, is followed in [58].
We can view the above differential equations as defining an extended La-
grangian vector field on the tangent bundle TR3 , assuming the potential
function is defined everywhere on R3 . We can linearize the extended differen-
tial equations at the equilibrium to obtain the linearized vector field defined
on TR3 as described by

∂ 2 U (qe )
mr2 ξ¨ + ξ = 0.
∂q 2
This linearization of the extended vector field can be restricted to the tangent
space of TS2 at (qe , 0) ∈ TS2 , that is
 
T(qe ,0) TS2 = (ξ, ξ)˙ ∈ R6 : q T ξ = 0, ξ˙ ∈ Tq S2 .
e e

The above description of the linearized vector field is in the form of differential-
algebraic equations. It is convenient to describe this linearized vector field in a
more accessible form by introducing a basis for the tangent space T(qe ,0) TS2 .
To this end, select ξ1 , ξ2 as an orthonormal basis for T(qe ,0) TS2 . Thus, any
˙ ∈ T(q ,0) TS2 can be written as
(ξ, ξ) e

ξ = σ 1 ξ1 + σ 2 ξ2 ,
ξ˙ = σ̇1 ξ1 + σ̇2 ξ2 ,

where σ = (σ1 , σ2 ) ∈ R2 can be viewed as local coordinates for Tqe S2 . Thus,

˙ ∈ T(q ,0) TS2 for all σ ∈ D, where D ⊂ R2 is an open set containing the
(ξ, ξ) e
origin. Substituting these expressions into the equation for the linearization of
the extended vector field and taking the inner product with the basis vectors
ξ1 and ξ2 gives
B.3 Linearization on TSO(3) 527
  T ∂ 2 U (qe ) 2
   
2 σ̈1
U (qe )
ξ1 ∂q2 ξ1 ξ1T ∂ ∂q 2 ξ2 σ 1 0
mr + T ∂ 2 U (qe ) 2 = .
σ̈2 U (qe )
ξ2 ∂q2 ξ1 ξ2T ∂ ∂q 2 ξ 2
σ 2 0

This system of second-order linear differential equations describe the lin-

earized vector field of the original Lagrangian vector field on TS2 . Thus, this
differential equation, expressed in terms of (σ, σ̇) ∈ TR2 , describes the La-
grangian dynamics on the manifold TS2 in a neighborhood of (qe , 0) ∈ TS2
to first order in the perturbations.
These linearized equations have been described as a system of second-order
differential equations, consistent with the usual formulation of the Euler–
Lagrange equations. They can also be expressed as a system of first-order
differential equations by including perturbations of the angular velocity vec-
tor or the momentum.

B.3 Linearization on TSO(3)

Consider a rigid body with moment of inertia J that rotates under the action
of a potential. The configuration manifold is SO(3), embedded in R3×3 . As
shown in Chapter 6, the rotational kinematics are

Ṙ = RS(ω),

and the Euler equations are

3
 ∂U (R)
J ω̇ + ω × Jω − ri × = 0.
i=1
∂ri

These differential equations define a vector field on TSO(3). Assume Re ∈

SO(3) satisfies ∂U∂r
(Re )
i
= 0, i = 1, 2, 3, so that (Re , 0) ∈ TSO(3) is an equi-
librium of the vector field defined by the rotational kinematics and Euler’s
equations. We obtain linearized equations following the second approach de-
scribed above. The same approach, for different dynamics, is followed in [20].
We can linearize the above differential equations by first introducing local
coordinates on SO(3). We use the exponential representation

R = Re eS(θ) ,

where θ = (θ1 , θ2 , θ3 ) ∈ D ⊂ R3 , and D is an open set containing the origin.

As indicated in Chapter 1, the exponential map θ ∈ D → R ∈ SO(3) is a
local diffeomorphism.
The kinematics of a rotating rigid body in SO(3) are now expressed in
terms of local coordinates in D ⊂ R3 . The angular velocity vector of the
528 B Linearization as an Approximation to Lagrangian Dynamics on a Manifold

rotating rigid body can be expressed in terms of the time derivatives of the
local coordinates. The following implications

S(ω) = RT Ṙ
= e−S(θ) ReT Re eS(θ) S(θ̇)
= S(θ̇),

demonstrate that ω = θ̇, where we have used the fact that the skew-symmetric
function S : R3 → so(3) is locally invertible.
The Euler equations can be written in terms of local exponential coordi-
nates as
3
 ∂U (Re eS(θ) )
J θ̈ + θ̇ × J θ̇ − Re e−S(θ) ei × = 0.
i=1
∂ri

These equations define the rotational dynamics of the rigid body on a subset
of TSO(3) in terms of local coordinates θ ∈ D ⊂ R3 . These are complicated
equations, but they can be easily linearized in a neighborhood of (Re , 0) ∈
TSO(3), or equivalently in terms of local coordinates in a neighborhood of
the origin in D. Linearization of the differential equations in local coordinates
gives
3
 ∂ 2 U (Re )
J σ̈ − rei × σ = 0.
i=1
∂ri2

Here, rei = ReT ei ∈ S2 , i = 1, 2, 3. This linear vector differential equation

describes the linearized vector field of the Lagrangian vector field on TSO(3);
it is expressed in terms of perturbations σ from the equilibrium (Re , 0) ∈
TSO(3).
These linearized equations have been described as a system of second-order
differential equations, consistent with the usual formulation of the Euler–
Lagrange equations. They can also be expressed as a system of first-order
differential equations by including perturbations of the angular velocity vec-
tor or the momentum.
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Index

action coordinates, vii, 12, 59

left, 381 Coriolis, 121
right, 381 cotangent bundle, 14, 98, 99, 132, 146,
transitive, 381 148, 151, 208, 226, 228, 231,
action integral, viii, 91, 134, 211, 277, 282–284, 325, 326, 368, 377
317, 373 cotangent space, 14, 43, 132, 208
infinitesimal variation, 135, 138, 211, cotangent vector, 14, 208
215, 278, 318, 374, 387 covector, 14, 43, 521
adjoint operator, 372 covector field
angular momentum, 282, 323 on manifold, 43
angular velocity, 132, 275 cross product, vi, 6
atlas, 12 cyclic coordinate, 101

calculus of variations, 521 Darboux’s theorem, 104

fundamental lemma, 136, 139, 212, deformable body, x, 54, 485
350, 355, 375, 521 derivative, 11
canonical transformation, vi directional, 11
Cayley transformation, 8 gradient, 11
centripetal, 121 diffeomorphic, 11, 401
charts, vii, 12 diffeomorphism, 11, 13, 134, 205, 206,
Christoffel, 94, 214 210, 270, 353, 372, 373, 376, 381
closed loop, 85, 86 differential equations, 27, 28
coadjoint operator, 372 differential form, 104
computational dynamics, ix, 42 differential geometry, vi, x, 12, 29, 131,
configuration, vii, 55, 89, 131, 313 207
configuration manifold differential-algebraic equations, 29, 107,
infinite-dimensional, 485 368
conjugate momentum, 95, 99, 143, 147, index one, 30
222, 226, 228, 453, 462 index two, 30, 31, 107, 367, 368
conservative, vii dimension, 2
conserved quantity, 29 discrete-time, ix, 42
control effects, 109 dot product, vi, 2, 521
coordinate-free, vii, 59 dual, 14, 132

© Springer International Publishing AG 2018 535

T. Lee et al., Global Formulations of Lagrangian and Hamiltonian
Dynamics on Manifolds, Interaction of Mechanics and Mathematics,
DOI 10.1007/978-3-319-56953-6
536 Index

dynamics two bodies with a common material

chain pendulum, 486 point, 430
chain pendulum on cart, 492
classical particle, 110 eigenspaces, 3
double planar pendulum, 169 eigenvalues, 3, 34
double spherical pendulum, 255 embedded manifold, vii, 13
elastically connected planar equilibrium, 29, 32, 33, 399, 485, 523
pendulums, 163 asymptotically stable, 33
elastically connected spherical stable, 33
pendulums, 250 unstable, 33
elastically supported rigid body, 331 Euclidean coordinates, vii, 54
fixed-free elastic rod, 507 Euclidean frame, vii, 53, 59
free-free chain, 501 Euclidean motion, 26, 57, 313
full body, 441 Euler angles, 285
Furuta pendulum, 181 Euler equations, 281
horizontal pendulum on a cart, 405 Euler–Lagrange equations, viii, 92, 93,
particle constrained curved surface, 110, 118, 120, 136, 140, 153, 154,
115 158, 159, 164, 165, 170, 171, 176,
particle in Keplerian orbit, 117 177, 182, 184, 189, 190, 213, 218,
particle in uniform gravity, 110 234, 235, 239, 240, 245, 246, 250,
particle on rotating hoop, 157 251, 256, 257, 279, 285, 289, 294,
particle on rotating sphere, 238 300, 321, 322, 327, 332, 337, 375,
particle on torus, 175 376, 383, 399, 402, 405, 410, 416,
planar mechanism, 400 421, 426, 431, 436, 442, 448, 457,
planar pendulum, 152 465, 485, 487, 493, 502, 508
planar pendulum and spherical exponential map, 8, 26, 134, 210, 276,
pendulum, 409 316, 372, 376
quad rotors transporting payload, 464 external disturbances, 109
rigid body planar pendulum, 298 external forces, 108, 387
rotating and translating rigid body,
327 field
rotating and translating rigid body in complex, 2
orbit, 336 real, 2
rotating and translating rigid body flow map, 28
with appendage, 435 frame
rotating rigid body, 285 body-fixed, 55, 274, 313
rotating rigid body in orbit, 293 fixed, 55
rotating rigid body with appendage, inertial, 55
421 non-inertial, 120
rotating spacecraft and CMG, 456 reference, 274
spacecraft with reaction wheels, 446 rotating, 120, 128, 477
spherical pendulum, 233 fundamental lemmas, 521
spherical pendulum and elastic
strings, 243
spherical pendulum on a cart, 415 generalized coordinates, 89
three-dimensional pendulum, 288 geodesic, 128, 129, 204, 268
three-dimensional pendulum on cart, geometric integrator, 42
425 geometric mechanics, 348, 399
three-dimensional revolute joint geometric reduction, vi
robot, 187 global, 8, 12, 28, 39, 151, 284, 327
Index 537

Hamilton’s equations, viii, 98, 99, 110, particle on inclined plane, 59

114, 116, 119, 122, 154, 155, 160, particle on torus, 70
161, 165, 166, 172, 173, 178, 184, particles, 55
185, 191, 192, 227, 235, 236, 241, planar pendulum, 62
247, 252, 253, 258, 259, 284, 286, planar pendulum and spherical
290, 296, 301, 324, 326, 328, 334, pendulum, 68
338, 368, 377, 399, 403, 407, 418, planar rigid body, 75
423, 428, 433, 439, 444, 453, 462, rigid body, 56, 73, 76
470, 485, 489, 496, 504, 511 rotational, 133, 209, 275, 315
Hamilton’s phase space variational rotational and translational, 314
principle, 97, 354 spherical pendulum, 64
Hamilton’s principle, 91, 135, 139, 211, translational, 315
278, 318, 374 kinetic energy, viii, 90, 133, 142, 210,
Hamilton’s variational principle, viii 222, 276, 316
Hamilton-Jacobi theory, vi quadratic, 137, 218, 226, 231, 280,
Hamiltonian flow, 99 284, 321, 325
Hamiltonian function, 95, 99, 107, 110,
114, 116, 119, 122, 143, 147, 154,
155, 160, 161, 165, 166, 172, 173, Lagrange multipliers, vii, 106, 107, 367
178, 179, 184, 185, 191, 192, 223, Lagrange–d’Alembert principle, 108,
227, 241, 252, 253, 258, 259, 281, 109, 387, 402, 403, 452, 468
296, 322, 353, 359, 377, 379, 380, Lagrangian flow, 99
403, 408, 412, 419, 423, 429, 434, Lagrangian function, viii, 90, 98–101,
439, 444, 463, 490, 498, 505, 511 103, 107, 109, 110, 113, 115, 116,
augmented, 368 118, 119, 121, 127, 129, 133, 134,
modified, 148, 151, 228, 232, 282, 284, 137, 138, 140, 142, 143, 148, 150,
323, 325 152, 154, 160, 164, 170, 176, 177,
Hartman–Grobman theorem, 33 189, 190, 205, 206, 209, 211, 213,
hat map, 6 218, 222, 223, 226, 228, 233–235,
holonomic constraints, vii, 105 240, 245, 246, 251, 257, 263,
homogeneous manifold, x, 26, 381 268–270, 276, 277, 281, 289, 310,
homogeneous matrix, 26 316, 317, 327, 337, 346, 347, 349,
homogeneous transformation, vii 373, 381, 382, 402, 406, 410, 416,
hyperregular, 95 422, 427, 437, 450, 458, 461, 466,
487, 488, 495, 503, 508, 510
ideal particles, x augmented, 367
ignorable coordinates, 101 lifted, 382
index, 30 modified, 138, 214, 219, 231, 276, 281,
initial-value problem, 28, 29, 31, 32, 42, 316, 321
92, 98, 100, 109 quadratic, 140, 146
inner product, 2, 43 left trivialization, 376
integral function, 29 Legendre transformation, viii, 95, 99,
invariant function, 29 143, 148, 222, 228, 281, 322, 353,
invariant manifold, 28 376, 377
isotropy, 383 Lie algebra, 25, 371
Lie group, vi, x, 24, 25, 371, 522
kinematics, vii, 372, 382 lifting, 382
constrained rigid link in a plane, 78 linear functional, 2, 14
constrained rigid rod, 80 linear transformation, 2, 274
deformable body, 57 linearization, 32, 33, 441, 523, 524,
double planar pendulum, 65 526–528
double spherical pendulum, 67 Liouville’s theorem, 105
on manifold, 58 local coordinates, 12, 29, 32, 33, 151
particle on hyperbolic paraboloid, 60 Lyapunov function, 34
538 Index

manifold, vii, viii, 1, 11, 16, 18, 21, 23, orthogonal transformation, vii
132 orthogonal vectors, 2
configuration, vii, ix, x, 57, 58, 90, 92,
106, 108, 110, 112, 115, 117, 131, pendulum
133, 137, 138, 142, 151, 152, 157, chain, 486
163, 164, 169, 175, 177, 184, 189, connection of planar and spherical,
204, 207, 209, 213, 214, 222, 409
232–234, 236, 238, 240, 241, 244, double planar, 169, 306
246, 247, 250, 251, 262, 263, 281, double spherical, 255, 266
285, 294, 316, 327, 331, 334, 337, elastic connection of planar links, 163,
347, 348, 350, 355, 367, 371, 372, 199–201
375, 376, 379–382, 384, 385, 387, elastic connection of spherical links,
399, 401, 403–405, 409, 415, 419, 243, 250, 264–266
421, 426, 431, 436, 442, 448, 453, elastic spherical, 472
454, 456, 465, 466, 485, 487, 493, Foucault, 263
501, 507 Furuta, 181
constraint, 106, 107, 367, 368, 481 modified Furuta, 199, 471
embedded, vii, 12, 46, 47, 522 non-planar double, 198
intersection, 15 planar, 152, 306
linear, 360 planar double, 197, 198
product, 15, 22 planar on a cart, 405
mass planar rigid body, 472
distributed, 54 rigid body, 298
lumped, 54 spherical, 233
mass particles, 54 spherical on a cart, 415
matrix, 1 three-dimensional, 288
determinant, 2 three-dimensional on a cart, 425
eigenvalues, 2 perturbations from equilibrium, 32
eigenvectors, 2 phase volume, 105
homogeneous, 9 Poisson’s equations, 53
identity, 1 positive-definite, 4
invariant subspaces, 3 potential energy, viii, 90, 133, 142, 210,
nonsingular, 7 222, 276, 316
null space, 3 principal axes, 280
orthogonal, 7, 26 principle bundle connection, 383
range, 3 projection, 3
rank, 2
singular, 2 quadratic form, 4
skew-symmetric, 4
symmetric, 4 reduction, 101
trace, 3 revolute joint, 298
zero, 1 right translation, 372
matrix identities, 10 right-hand rule, 55
multi-body, x rigid body, x, 54
multi-body system, 53, 399, 485 center of mass, 274, 313, 322, 326
rotational and translational dynamics,
negative-definite, 4 313
nonholonomic constraints, vi, 87, 484 rotational dynamics, 273
norm, 2 Routh reduction, 101
numerical integration algorithms, 42
stability, 33, 523
orthogonal, 24 stable manifold theorem, 34, 523
orthogonal complement, 2 standard basis vectors, 3, 54, 55
orthogonal decomposition, 3 subspace, 2
Index 539

subspaces, 3 variations, viii, 90, 276

symmetry actions, 103 vector, 1
symplectic, 104 spatial, 54
vector field, 27
tangent bundle, viii, 13, 92, 107, 132, linearized, 32–34, 523, 525, 526, 528
138, 142, 208, 209, 214, 221, 276, on manifold, 27, 34, 36, 37, 39, 40
280, 281, 316, 320, 321, 350, 367, differential equations, 28
368, 376 differential-algebraic equations, 29
tangent space, 13, 43, 132, 208, 525, 526 vector space, vi, 1, 2
tangent vector, 13, 28, 132, 208 basis, 2
tensegrity structure, 244, 264, 333, 474, embedding, vii
517 linear independence, 2
time reversal, 105 span, 2
Trammel of Archimedes, 87, 392 vectors
translational momentum, 323 orthonormal, 3
vee map, 6
variation
velocity vector, 55
infinitesimal, 91, 134, 210, 277, 373
variational calculus, vi, viii, 42, 89, 132, virtual work, 387
208, 347
variational integrator, 43 wedge product, 105