#
Geometric Phases of Nonlinear Planar $N$-Pendula
via Cartan’s Moving Frames

Francesco Fedele  School of Civil and Environmental Engineering, Georgia
Institute of Technology, Atlanta, GA 30332, USA Arash Yavari Corresponding
author, e-mail<EMAIL_ADDRESS>School of Civil and Environmental
Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA The
George W. Woodruff School of Mechanical Engineering, Georgia Institute of
Technology, Atlanta, GA 30332, USA

###### Abstract

We study the geometric phases of nonlinear planar $N$-pendula with continuous
rotational symmetry. In the Hamiltonian framework, the geometric structure of
the phase space is a principal fiber bundle, i.e., a base, or shape manifold
$\mathcal{B}$, and fibers $\mathcal{F}$ along the symmetry direction attached
to it. The symplectic structure of the Hamiltonian dynamics determines the
connection and curvature forms of the shape manifold. Using Cartan’s
structural equations with zero torsion we find an intrinsic (pseudo)
Riemannian metric for the shape manifold. For a double pendulum the metric is
pseudo-Riemannian if the total angular momentum $\mathsf{A}<0$, and the shape
manifold is an expanding spacetime with the Robertson-Walker metric and
positive curvature. For $\mathsf{A}>0$, the shape manifold is the hyperbolic
plane $\mathbb{H}^{2}$ with negative curvature. We then generalize our results
to free $N$-pendula. We show that the associated shape manifold $\mathcal{B}$
is reducible to the product manifold of $(N-1)$ hyperbolic planes
$\mathbb{H}^{2}$ ($\mathsf{A}>0$), or Robertson-Walker $2$D spacetimes
($\mathsf{A}<0$). We then consider $N$-pendula subject to time-dependent self-
equilibrated moments. The extended autonomous Hamiltonian system is
considered. The associated shape manifold is a product space of $N$ hyperbolic
planes $\mathbb{H}^{2}$ ($\mathsf{A}>0$), or Robertson-Walker $2$D spacetimes
($\mathsf{A}<0$). In either case, the geometric phase follows by integrating a
$2$-form given by the sum of the sectional curvature forms of $\mathcal{B}$.
The Riemannian structure of the shape manifold provides an intrinsic measure
of the closeness of one shape to another in terms of curvature, or induced
geometric phase.

Keywords:

Geometric phase, Berry’s phase, geometric drift, Cartan’s moving frames.

###### Contents

  1. 1 Introduction
  2. 2 Differential geometry via Cartan’s moving frames
    1. 2.1 Non-metricity
    2. 2.2 Torsion
    3. 2.3 Curvature
    4. 2.4 Pseudo-Riemannian manifolds
    5. 2.5 Riemannian product spaces
    6. 2.6 Cartan’s curvature $2$-forms of an $N$-dimensional pseudo-Riemannian manifold with a diagonal metric
    7. 2.7 Curvature $2$-forms of a $2$-dimensional pseudo-Riemannian manifold with a diagonal metric
  3. 3 Dynamics of (free) nonlinear planar double pendula
    1. 3.1 The Hamiltonian structure
    2. 3.2 Hamiltonian reduction and geometric phases
    3. 3.3 Curvature and intrinsic metric of the shape manifold
    4. 3.4 Geodesics of the metric
      1. 3.4.1 Negative angular momentum: Robertson-Walker spacetime
      2. 3.4.2 Positive angular momentum: The hyperbolic plane $\mathbb{H}^{2}$
      3. 3.4.3 The set of all geodesics
  4. 4 Dynamics of (free) nonlinear planar $N$-pendula
    1. 4.1 Curvature and intrinsic metric of the shape manifold
    2. 4.2 Negative angular momentum: Multi-universe
    3. 4.3 Positive angular momentum: The hyperbolic product space $\mathbb{H}^{2(N-1)}$
  5. 5 Dynamics of nonlinear planar $N$-pendula under self-equilibrated external moments
    1. 5.1 Extended autonomous Hamiltonian system
    2. 5.2 Curvature and intrinsic metric of the shape manifold
  6. 6 Conclusions

##  1 Introduction

A classical example in which geometric phases arise is the parallel transport
of a vector tangent to a sphere. The change in the vector direction is equal
to the solid angle of the closed path spanned by the vector and it can be
described by Hannay’s angles (Hannay, 1985). The rate at which the angle, or
geometric phase, changes in time is the geometric phase velocity. In physics,
the rotation of Foucault’s pendulum can also be explained by means of
geometric phases. Pancharatnam (1956) discovered their effects in polarized
light, and later Berry (1984a) rediscovered it for quantum-mechanical systems
(see also (Berry, 1990; Simon, 1983a; Aharonov and Anandan, 1987; Garrison and
Chiao, 1988)). Berry (1984b, 1990) showed that a quantum mechanical system
that undergoes an adiabatic evolution acquires a phase factor that is purely
geometric.

Another example drawn from classical mechanics is the spinning body in a
dissipationless medium, which has a rotational symmetry with respect to the
axis of rotation. The associated angular, or geometric phase velocity $\Omega$
follows from the conservation of angular momentum $I\Omega^{2}$, where $I$ is
the mass moment of inertia. If the body changes shape, $I$ varies over time
and so does the angular speed $\Omega$. In the frame rotating at that speed,
one only observes the body shape-changing dynamics and the rotational symmetry
is reduced. In a fixed frame one cannot distinguish between the body
deformation and spinning motion. In general, geometric phases are observed in
classical mechanical systems with internal variables that rule their shape
deformations, and variables that rule their rigid translation of the system as
a whole. A cyclic motion of the shape variables can induce a rigid translation
if the total momentum is conserved.

In classical and quantum mechanics the key geometrical structure is the
symplectic form of a Hamiltonian. The Riemannian structure and a metric are
traditionally associated to the theories of General Relativity and
gravitation. In quantum mechanics, the scalar product on the Hilbert space
induces naturally a distance between quantum states, but the interest is not
in the local properties of the manifold of states. The physically relevant
quantities are transition probability amplitudes between quantum states, which
do not depend on their relative distance. However, Provost and Vallee (1980)
argued that for macroscopic systems exhibiting collective behaviour, the
possibility of going from one state to another is not described by a direct
transition amplitude (scalar product in Hilbert space) but rather through a
succession of infinitesimal steps on the manifold of collective states. The
relevant distance between distinct states is then the distance measured along
geodesics on the manifold.

In quantum mechanics, the Riemannian metric is the Fubini-Study metric of
complex projective spaces (Provost and Vallee, 1980; Anandan, 1991). The
importance of the associated geodesic curves stems from the fact that Berry’s
phase between two quantum states can be expressed by integrating the
associated connection form along the geodesic between the two states (Samuel
and Bhandari, 1988; Wilczek and Shapere, 1989). As a matter of fact, the
quantum metric provides the infinitesimal distance between two nearby states
differing by a Berry phase. Such a distance measures the quantum fluctuations
between the two states (Provost and Vallee, 1980).

In fluid mechanics, the motion of a swimmer at low Reynolds numbers can be
explained in terms of geometric phases (Shapere and Wilczek, 1987, 1989).
Swimmers can cyclically change their shape (internal variables) to move
forward (translation variables). Since inertia is neglected the swimmer’s
velocity is uniquely determined by the geometry of the sequence of its body’s
shapes, which lead to a net translation, i.e., the geometric phase. A fixed
observer sees the swimmer drifting as its body shape cyclically changes over
time, but it is hard to distinguish between the two motions. On the contrary,
an observer moving with the swimmer sees only its body deformations and
translation symmetry is reduced in the (symmetry-reduced) moving frame. In
wave mechanics, the slowdown of large oceanic wave groups can be explained in
terms of geometric phases (Fedele, 2014; Banner et al., 2014; Fedele et al.,
2020). Channel flow turbulence governed by the Navier-Stokes equations admits
a continuous translation symmetry. Vortical structures, i.e., packets of
vorticity, advect downstream at a speed that depends on their intrinsic
inertia (dynamical phase) and on the way their shape varies over time
(geometric phase). Fedele et al. (2015) showed that the geometric phase
component of the vortex speed can be interpreted as a self-propulsion velocity
induced by the shape-changing vortex deformations similar to the motion of a
swimmer at low Reynolds numbers (Shapere and Wilczek, 1989).

In the literature, geometric phases have been understood in terms of holonomy
of connections on vector bundles (Simon, 1983b). In this paper we study
geometric phases of nonlinear $N$-pendula in the Hamiltonian framework
(Marsden et al., 1990) exploiting Cartan’s moving frames to characterize the
Riemannian structure of the reduced dynamics. We first present a complete
analysis of the geometric phases of a coupled planar double pendulum, which
conserves total angular momentum. This problem was discussed by Marsden et al.
(1990) to introduce the approach of Hamiltonian reduction for mechanical
systems with a continuous Lie symmetry. Such a symmetry implies that the
associated phase space has the structure of a principal fiber bundle, i.e., a
shape manifold and transversal fibers attached to it. The symplectic form of
the Hamiltonian dynamics yields the connection form on the shape manifold,
which thus determines the horizontal transport through the fiber bundle. A
cyclic flow on the shape manifold induces a drift along the fibers. This
includes dynamic and geometric phases. The dynamic phase increases with the
time spent by the flow to wander around the phase space and answers the
question: “How long did your trip take?” (Berry, 1984a). On the contrary, the
geometric phase is independent of time and it depends only upon the curvature
of the shape manifold, and answers the question: “Where have you been?”
(Berry, 1984a). The geometric phase is defined by the connection form. Marsden
et al. (1990) define the associated geometric phases and relate them to the
curvature form of the shape manifold. Here, we present a new analysis
exploiting Cartan’s first structural equations with zero torsion and derive
the intrinsic Riemannian structure of the shape manifold, which to the best of
our knowledge, has not been investigated to this date. The use of Cartan’s
moving frames in studying the geometric phases of nonlinear $N$-pendula is
motivated by the success of the applications of Cartan’s machinery in the
analysis of distributed defects in nonlinear solids by the second author and
co-workers (Yavari and Goriely, 2012a, b, 2013, 2014).

This paper is organized as follows. We first review the theory of Cartan’s
moving frames and associated connection and curvature forms. The theory is
then applied to pseudo-Riemannian manifolds. As a special case we derive the
Cartan curvature forms of an $N$-dimensional manifold with a diagonal metric.
We then introduce the problem of a planar two-pendulum in the Hamiltonian
setting. The geometric phases of the system are then studied and an intrinsic
metric of the shape manifold is derived. We then extend our study to the
geometric phases of free nonlinear $N$-pendula and $N$-pendula subject to
self-equilibrating external moments. Finally, we discuss the physical
relevance of the intrinsic metric for applications, and in particular, to
fluid turbulence.

##  2 Differential geometry via Cartan’s moving frames

Given an $N$-manifold $\mathcal{B}$ with a metric $\mathbf{G}$ and an affine
connection $\nabla$, $(\mathcal{B},\nabla,\mathbf{G})$ is called a metric-
affine manifold (Gordeeva et al., 2010). Here we mainly follow Hehl and
Obukhov (2003) and Sternberg (2013). Let us consider an orthonormal frame
field $\\{\mathbf{e}_{1}(X),\ldots,\mathbf{e}_{N}(X)\\}$ that at every point
$X\in\mathcal{B}$ forms a basis for the tangent space $T_{X}\mathcal{B}$. A
moving frame is, in general, a non-coordinate basis for the tangent space. The
moving frame field $\\{\mathbf{e}_{\alpha}\\}$ defines the moving co-frame
field $\\{\vartheta^{1},\ldots,\vartheta^{N}\\}$ such that
$\vartheta^{\alpha}(\mathbf{e}_{\beta})=\delta^{\alpha}_{\beta}$, where
$\delta^{\alpha}_{\beta}$ is the Kronecker delta. As the moving frame is
assumed to be orthonormal, i.e., $\mathopen{\hbox{\set@color${\langle}$}\mkern
2.0mu\kern-3.49998pt\leavevmode\hbox{\set@color${\langle}$}}\mathbf{e}_{\alpha},\mathbf{e}_{\beta}\mathclose{\hbox{\set@color${\rangle}$}\mkern
2.0mu\kern-3.49998pt\leavevmode\hbox{\set@color${\rangle}$}}_{\mathbf{G}}=\delta_{\alpha\beta}$,
where $\mathopen{\hbox{\set@color${\langle}$}\mkern
2.0mu\kern-3.49998pt\leavevmode\hbox{\set@color${\langle}$}}.,.\mathclose{\hbox{\set@color${\rangle}$}\mkern
2.0mu\kern-3.49998pt\leavevmode\hbox{\set@color${\rangle}$}}_{\mathbf{G}}$ is
the inner product induced by the metric $\mathbf{G}$, with respect to the
moving frame the metric has the representation

$\mathbf{G}=\delta_{\alpha\beta}\,\vartheta^{\alpha}\otimes\vartheta^{\beta}\,,$
(2.1)

where summation over repeated indices is assumed.

An affine (linear) connection is an operation
$\nabla:\mathcal{X}(\mathcal{B})\times\mathcal{X}(\mathcal{B})\rightarrow\mathcal{X}(\mathcal{B})$,
where $\mathcal{X}(\mathcal{B})$ is the set of vector fields on $\mathcal{B}$,
with certain properties, namely, a)
$\nabla_{f_{1}\mathbf{X}_{1}+f_{2}\mathbf{X}_{2}}\mathbf{Y}=f_{1}\nabla_{\mathbf{X}_{1}}\mathbf{Y}+f_{2}\nabla_{\mathbf{X}_{2}}\mathbf{Y}$,
b)
$\nabla_{f_{1}\mathbf{X}_{1}+f_{2}\mathbf{X}_{2}}\mathbf{Y}=f_{1}\nabla_{\mathbf{X}_{1}}\mathbf{Y}+f_{2}\nabla_{\mathbf{X}_{2}}\mathbf{Y}$,
and c)
$\nabla_{\mathbf{X}}(f\mathbf{Y})=f\nabla_{\mathbf{X}}\mathbf{Y}+(\mathbf{X}f)\mathbf{Y}$,
where $\mathbf{X}$, $\mathbf{Y}$, $\mathbf{X}_{1}$, $\mathbf{X}_{2}$,
$\mathbf{Y}_{1}$, and $\mathbf{Y}_{2}$ are arbitrary vector fields,
$f,f_{1},f_{2}$ are arbitrary functions, and $a_{1},a_{2}$ are arbitrary
scalars. The vector $\nabla_{\mathbf{X}}\mathbf{Y}$ is the covariant
derivative of $\mathbf{Y}$ along $\mathbf{X}$. Given the connection $\nabla$,
the connection $1$-forms are defined as

$\nabla\mathbf{e}_{\alpha}=\mathbf{e}_{\gamma}\otimes\omega^{\gamma}{}_{\alpha}\,.$
(2.2)

The connection coefficients are defined as
$\nabla_{\mathbf{e}_{\beta}}\mathbf{e}_{\alpha}=\left\langle\omega^{\gamma}{}_{\alpha},\mathbf{e}_{\beta}\right\rangle\mathbf{e}_{\gamma}=\omega^{\gamma}{}_{\beta\alpha}\,\mathbf{e}_{\gamma}$.111$\langle.,.\rangle$
is the natural pairing of $1$-forms and vectors. Thus, the connection
$1$-forms have the representation
$\omega^{\gamma}{}_{\alpha}=\omega^{\gamma}{}_{\beta\alpha}\,\vartheta^{\beta}$.
It is straightforward to show that
$\nabla\vartheta^{\alpha}=-\omega^{\alpha}{}_{\gamma}\,\vartheta^{\gamma}$,
and
$\nabla_{\mathbf{e}_{\beta}}\vartheta^{\alpha}=-\omega^{\alpha}{}_{\beta\gamma}\,\vartheta^{\gamma}$.

A coordinate chart $\\{X^{A}\\}$ for $\mathcal{B}$ defines a coordinate basis
$\left\\{\partial_{A}=\frac{\partial}{\partial X^{A}}\right\\}$ for
$T_{X}\mathcal{B}$. The moving frame field $\\{\mathbf{e}_{\alpha}\\}$ is
related to the coordinate basis by a $GL(N,\mathbb{R})$-rotation:
$\mathbf{e}_{\alpha}=\mathsf{F}_{\alpha}{}^{A}\,\partial_{A}$. In order to
preserve orientation, it is assumed that $\det\mathsf{F}_{\alpha}{}^{A}>0$.
The relation between the moving and coordinate co-frames is
$\vartheta^{\alpha}=\mathsf{F}^{\alpha}{}_{A}\,dX^{A}$, where
$[\mathsf{F}^{\alpha}{}_{A}]$ is the inverse of $[\mathsf{F}_{\alpha}{}^{A}]$.
For the coordinate frame $[\partial_{A},\partial_{B}]=0$, where
$[\mathbf{X},\mathbf{Y}]=\mathbf{X}\mathbf{Y}-\mathbf{Y}\mathbf{X}$ is the Lie
bracket (commutator) of the vector fields $\mathbf{X}$ and $\mathbf{Y}$. For
an arbitrary scalar field $f$,
$[\mathbf{X},\mathbf{Y}][f]=\mathbf{X}[f]\mathbf{Y}-\mathbf{Y}[f]\mathbf{X}$.
For the moving frame field one has

$[\mathbf{e}_{\alpha},\mathbf{e}_{\beta}]=-c^{\gamma}{}_{\alpha\beta}\,\mathbf{e}_{\gamma}\,,$
(2.3)

where $c^{\gamma}{}_{\alpha\beta}$ are components of the _object of
anhonolomy_ $c^{\gamma}=d\vartheta^{\gamma}$. Noting that

$c^{\gamma}=d\left(\mathsf{F}^{\gamma}{}_{B}\,dX^{B}\right)=\sum_{\alpha<\beta}c^{\gamma}{}_{\alpha\beta}\,\vartheta^{\alpha}\wedge\vartheta^{\beta}\,,$
(2.4)

one can show that

$c^{\gamma}{}_{\alpha\beta}=\mathsf{F}_{\alpha}{}^{A}\,\mathsf{F}_{\beta}{}^{B}\left(\partial_{A}\mathsf{F}^{\gamma}{}_{B}-\partial_{B}\mathsf{F}^{\gamma}{}_{A}\right)\,.$
(2.5)

In the local chart $\\{X^{A}\\}$,
$\nabla_{\partial_{A}}\partial_{B}=\Gamma^{C}{}_{AB}\,\partial_{C}$, where
$\Gamma^{C}{}_{AB}$ are the Christoffel symbols of the connection.

###  2.1 Non-metricity

For a metric-affine manifold $(\mathcal{B},\nabla,\mathbf{G})$, non-metricity
$\boldsymbol{\mathcal{Q}}:\mathcal{X}(\mathcal{B})\times\mathcal{X}(\mathcal{B})\times\mathcal{X}(\mathcal{B})\rightarrow\mathcal{X}(\mathcal{B})$
is defined as

$\boldsymbol{\mathcal{Q}}(\mathbf{X},\mathbf{Y},\mathbf{Z})=\mathopen{\hbox{\set@color${\langle}$}\mkern
2.0mu\kern-3.49998pt\leavevmode\hbox{\set@color${\langle}$}}\nabla_{\mathbf{X}}\mathbf{Y},\mathbf{Z}\mathclose{\hbox{\set@color${\rangle}$}\mkern
2.0mu\kern-3.49998pt\leavevmode\hbox{\set@color${\rangle}$}}_{\mathbf{G}}+\mathopen{\hbox{\set@color${\langle}$}\mkern
2.0mu\kern-3.49998pt\leavevmode\hbox{\set@color${\langle}$}}\mathbf{Y},\nabla_{\mathbf{X}}\mathbf{Z}\mathclose{\hbox{\set@color${\rangle}$}\mkern
2.0mu\kern-3.49998pt\leavevmode\hbox{\set@color${\rangle}$}}_{\mathbf{G}}-\mathbf{X}\big{[}\mathopen{\hbox{\set@color${\langle}$}\mkern
2.0mu\kern-3.49998pt\leavevmode\hbox{\set@color${\langle}$}}\mathbf{Y},\mathbf{Z}\mathclose{\hbox{\set@color${\rangle}$}\mkern
2.0mu\kern-3.49998pt\leavevmode\hbox{\set@color${\rangle}$}}_{\mathbf{G}}\big{]}\,.$
(2.6)

In the moving frame $\\{\mathbf{e}_{\alpha}\\}$,
$\mathcal{Q}_{\gamma\alpha\beta}=\boldsymbol{\mathcal{Q}}(\mathbf{e}_{\gamma},\mathbf{e}_{\alpha},\mathbf{e}_{\beta})$.
Non-metricity $1$-forms are defined as
$\mathcal{Q}_{\alpha\beta}=\mathcal{Q}_{\gamma\alpha\beta}\,\vartheta^{\gamma}$.
One can show that
$\mathcal{Q}_{\gamma\alpha\beta}=\omega^{\xi}{}_{\gamma\alpha}\,G_{\xi\beta}+\omega^{\xi}{}_{\gamma\beta}\,G_{\xi\alpha}-\langle
dG_{\alpha\beta},\mathbf{e}_{\gamma}\rangle=\omega_{\beta\gamma\alpha}+\omega_{\alpha\gamma\beta}-\langle
dG_{\alpha\beta},\mathbf{e}_{\gamma}\rangle$, where $d$ is the exterior
derivative. Hence

$\mathcal{Q}_{\alpha\beta}=\omega_{\alpha\beta}+\omega_{\beta\alpha}-dG_{\alpha\beta}\,.$
(2.7)

This is _Cartan’s zeroth structural equation_. For an orthonormal frame
$G_{\alpha\beta}=\delta_{\alpha\beta}$ and hence

$\mathcal{Q}_{\alpha\beta}=\omega_{\alpha\beta}+\omega_{\beta\alpha}.$ (2.8)

The connection $\nabla$ is compatible with the metric $\mathbf{G}$ if non-
metricity vanishes, i.e.,

$\nabla_{\mathbf{X}}\mathopen{\hbox{\set@color${\langle}$}\mkern
2.0mu\kern-3.49998pt\leavevmode\hbox{\set@color${\langle}$}}\mathbf{Y},\mathbf{Z}\mathclose{\hbox{\set@color${\rangle}$}\mkern
2.0mu\kern-3.49998pt\leavevmode\hbox{\set@color${\rangle}$}}_{\mathbf{G}}=\mathopen{\hbox{\set@color${\langle}$}\mkern
2.0mu\kern-3.49998pt\leavevmode\hbox{\set@color${\langle}$}}\nabla_{\mathbf{X}}\mathbf{Y},\mathbf{Z}\mathclose{\hbox{\set@color${\rangle}$}\mkern
2.0mu\kern-3.49998pt\leavevmode\hbox{\set@color${\rangle}$}}_{\mathbf{G}}+\mathopen{\hbox{\set@color${\langle}$}\mkern
2.0mu\kern-3.49998pt\leavevmode\hbox{\set@color${\langle}$}}\mathbf{Y},\nabla_{\mathbf{X}}\mathbf{Z}\mathclose{\hbox{\set@color${\rangle}$}\mkern
2.0mu\kern-3.49998pt\leavevmode\hbox{\set@color${\rangle}$}}_{\mathbf{G}}\,.$
(2.9)

This is equivalent to $\nabla\mathbf{G}=\mathbf{0}$, which in a coordinate
chart reads
$G_{AB|C}=G_{AB,C}-\Gamma^{D}{}_{CA}G_{DB}-\Gamma^{D}{}_{CB}G_{AD}=0$. With
respect to the moving frame, $\omega_{\alpha\beta}+\omega_{\beta\alpha}=0$,
i.e., the connection $1$-forms of a metric-compatible connection are anti-
symmetric.

###  2.2 Torsion

Torsion
$\boldsymbol{T}:\mathcal{X}(\mathcal{B})\times\mathcal{X}(\mathcal{B})\rightarrow\mathcal{X}(\mathcal{B})$
of the connection $\nabla$ is defined as

$\boldsymbol{T}(\mathbf{X},\mathbf{Y})=\nabla_{\mathbf{X}}\mathbf{Y}-\nabla_{\mathbf{Y}}\mathbf{X}-[\mathbf{X},\mathbf{Y}]\,.$
(2.10)

In a local chart $\\{X^{A}\\}$, torsion has components
$T^{A}{}_{BC}=\Gamma^{A}{}_{BC}-\Gamma^{A}{}_{CB}$. With respect to the moving
frame torsion has the components
$T^{\alpha}{}_{\beta\gamma}=\omega^{\alpha}{}_{\beta\gamma}-\omega^{\alpha}{}_{\gamma\beta}+c^{\alpha}{}_{\beta\gamma}$.
The torsion $2$-forms have the following relations with the connection
$1$-forms

$\mathcal{T}^{\alpha}=d\vartheta^{\alpha}+\omega^{\alpha}{}_{\beta}\wedge\vartheta^{\beta}\,.$
(2.11)

These are called _Cartan’s first structural equations_. The connection
$\nabla$ is symmetric if it is torsion-free, i.e.,
$\nabla_{\mathbf{X}}\mathbf{Y}-\nabla_{\mathbf{Y}}\mathbf{X}=[\mathbf{X},\mathbf{Y}]$.
With respect to the moving frame,
$d\vartheta^{\alpha}+\omega^{\alpha}{}_{\beta}\wedge\vartheta^{\beta}=0$.

###  2.3 Curvature

The curvature
$\boldsymbol{\mathcal{R}}:\mathcal{X}(\mathcal{B})\times\mathcal{X}(\mathcal{B})\times\mathcal{X}(\mathcal{B})\rightarrow\mathcal{X}(\mathcal{B})$
of the affine connection $\nabla$ is defined as

$\boldsymbol{\mathcal{R}}(\mathbf{X},\mathbf{Y})\mathbf{Z}=\nabla_{\mathbf{X}}\nabla_{\mathbf{Y}}\mathbf{Z}-\nabla_{\mathbf{Y}}\nabla_{\mathbf{X}}\mathbf{Z}-\nabla_{[\mathbf{X},\mathbf{Y}]}\mathbf{Z}\,.$
(2.12)

In a coordinate chart,
$\mathcal{R}^{A}{}_{BCD}=\Gamma^{A}{}_{CD,B}-\Gamma^{A}{}_{BD,C}+\Gamma^{A}{}_{BM}\,\Gamma^{M}{}_{CD}-\Gamma^{A}{}_{CM}\,\Gamma^{M}{}_{BD}$.
With respect to the moving frame, the curvature tensor has the components
$\mathcal{R}^{\alpha}{}_{\beta\lambda\mu}=\partial_{\beta}\omega^{\alpha}{}_{\lambda\mu}-\partial_{\lambda}\omega^{\alpha}{}_{\beta\mu}+\omega^{\alpha}{}_{\beta\xi}\,\omega^{\xi}{}_{\lambda\mu}-\omega^{\alpha}{}_{\lambda\xi}\,\omega^{\xi}{}_{\beta\mu}+\omega^{\alpha}{}_{\xi\mu}\,c^{\xi}{}_{\beta\lambda}$.
The curvature $2$-forms are defined as

$\mathcal{R}^{\alpha}{}_{\beta}=d\omega^{\alpha}{}_{\beta}+\omega^{\alpha}{}_{\gamma}\wedge\omega^{\gamma}{}_{\beta}\,.$
(2.13)

These are called _Cartan’s second structural equations_.

Requiring that $\nabla$ be both metric compatible and torsion free determines
it uniquely. This is the Levi-Civita connection. With respect to a coordinate
chart $\\{X^{A}\\}$ it has the connection coefficients (Christoffel symbols)
$\Gamma^{C}{}_{AB}=\frac{1}{2}G^{CD}(G_{BD,A}+G_{AD,B}-G_{AB,D})$. The Levi-
Civita connection $1$-forms can be explicitly calculated (O’Neill, 2014).
Using Cartan’s first structural equations
$d\vartheta^{\alpha}=-\omega^{\alpha}{}_{\beta}\wedge\vartheta^{\beta}$. Thus

$d\vartheta^{\alpha}(\mathbf{e}_{\beta},\mathbf{e}_{\gamma})=-(\omega^{\alpha}{}_{\beta}\wedge\vartheta^{\beta})(\mathbf{e}_{\beta},\mathbf{e}_{\gamma})=-\omega^{\alpha}{}_{\beta}(\mathbf{e}_{\beta})\,\vartheta^{\beta}(\mathbf{e}_{\gamma})+\omega^{\alpha}{}_{\beta}(\mathbf{e}_{\gamma})\,\vartheta^{\beta}(\mathbf{e}_{\beta})=-\omega^{\alpha}{}_{\beta\gamma}+\omega^{\alpha}{}_{\gamma\beta}\,.$
(2.14)

Similarly,

$d\vartheta^{\beta}(\mathbf{e}_{\gamma},\mathbf{e}_{\alpha})=-\omega^{\beta}{}_{\gamma\alpha}+\omega^{\beta}{}_{\alpha\gamma}\,,\quad
d\vartheta^{\gamma}(\mathbf{e}_{\alpha},\mathbf{e}_{\beta})=-\omega^{\gamma}{}_{\beta\alpha}+\omega^{\gamma}{}_{\alpha\beta}\,.$
(2.15)

Thus

$d\vartheta^{\alpha}(\mathbf{e}_{\beta},\mathbf{e}_{\gamma})+d\vartheta^{\beta}(\mathbf{e}_{\gamma},\mathbf{e}_{\alpha})-d\vartheta^{\gamma}(\mathbf{e}_{\alpha},\mathbf{e}_{\beta})=2\,\omega^{\alpha}{}_{\gamma\beta}\,,$
(2.16)

where use was made of the fact that for a metric compatible connection
$\omega^{\alpha}{}_{\gamma\beta}+\omega^{\beta}{}_{\gamma\alpha}=0$. Thus

$\omega^{\alpha}{}_{\gamma\beta}=\frac{1}{2}\left[d\vartheta^{\alpha}(\mathbf{e}_{\beta},\mathbf{e}_{\gamma})+d\vartheta^{\beta}(\mathbf{e}_{\gamma},\mathbf{e}_{\alpha})-d\vartheta^{\gamma}(\mathbf{e}_{\alpha},\mathbf{e}_{\beta})\right]\,.$
(2.17)

The components of the Riemann curvature and the Ricci tensor are related to
the curvature $2$-forms as

$\operatorname{Riem}^{\alpha}{}_{\beta\xi\eta}=\mathcal{R}^{\alpha}{}_{\beta}(\mathbf{e}_{\xi},\mathbf{e}_{\eta})\,,\qquad\operatorname{Ric}_{\alpha\beta}=\mathcal{R}^{\gamma}{}_{\alpha}(\mathbf{\mathbf{e}_{\gamma}},\mathbf{e}_{\beta})\,.$
(2.18)

The Ricci scalar is defined as
$\operatorname{R}=\operatorname{Ric}_{\alpha\beta}\delta^{\alpha\beta}$. Note
that with respect to the moving frame $g_{\alpha\beta}=\delta_{\alpha\beta}$,
and hence $g^{\alpha\beta}=\delta^{\alpha\beta}$.

In the coordinate chart $\\{X^{A}\\}$ metric has the components
$G_{AB}=\mathsf{F}_{A}{}^{\alpha}\,\mathsf{F}_{B}{}^{\beta}\,\delta_{\alpha\beta}$
and the Riemann and Ricci tensors given in (2.18) have the following
components

$\operatorname{Riem}^{A}{}_{BCD}=\mathsf{F}_{\alpha}{}^{A}\,\mathsf{F}_{B}{}^{\beta}\,\mathsf{F}_{C}{}^{\xi}\,\mathsf{F}_{D}{}^{\eta}\,\operatorname{Riem}^{\alpha}{}_{\beta\xi\eta}\,,\qquad\operatorname{Ric}_{AB}=\mathsf{F}_{A}{}^{\alpha}\,\mathsf{F}_{B}{}^{\beta}\,\operatorname{Ric}_{\alpha\beta}\,,$
(2.19)

where $\mathsf{F}_{A}{}^{\gamma}\,\mathsf{F}_{\gamma}{}^{B}=\delta_{A}^{B}$.
The Ricci scalar reads $\operatorname{R}=\operatorname{Ric}_{AB}\,G^{AB}$,
where
$G^{AB}=\mathsf{F}_{\alpha}{}^{A}\,\mathsf{F}_{\beta}{}^{B}\,\delta^{\alpha\beta}$
is the inverse of the metric $G_{AB}$ in the coordinate frame. Since the Ricci
scalar is an invariant, its value is the same in any frame. As a matter of
fact,
$\operatorname{R}=\operatorname{Ric}_{AB}\,G^{AB}=\mathsf{F}_{A}{}^{\alpha}\,\mathsf{F}_{B}{}^{\beta}\operatorname{Ric}_{\alpha\beta}\,\mathsf{F}_{\gamma}{}^{A}\,\mathsf{F}_{\rho}{}^{B}\,\delta^{\gamma\rho}=(\mathsf{F}_{A}{}^{\alpha}\,\mathsf{F}_{\gamma}{}^{A})(\mathsf{F}_{B}{}^{\beta}\,\mathsf{F}_{\rho}{}^{B})\operatorname{Ric}_{\alpha\beta}\delta^{\alpha\rho}=\delta_{\gamma}^{\alpha}\,\delta^{\beta}_{\rho}\,\operatorname{Ric}_{\alpha\beta}\,\delta^{\gamma\rho}=\operatorname{Ric}_{\alpha\beta}\,\delta^{\alpha\beta}$.

###  2.4 Pseudo-Riemannian manifolds

For a pseudo-Riemannian manifold, in the Cartan’s moving frame the metric
$\mathbf{G}$ in Eq. (2.1) generalizes to (O’Neill, 2014; Sternberg, 2013)

$\mathbf{G}=\sum_{\alpha=1}^{N}\epsilon_{\alpha}\,\vartheta^{\alpha}\otimes\vartheta^{\alpha}\,,$
(2.20)

where $\epsilon_{\alpha}=\pm 1$, and $(\epsilon_{1},\ldots,\epsilon_{N})$ is
the signature of the manifold. The orthonormality of the moving frame field
implies that $\mathopen{\hbox{\set@color${\langle}$}\mkern
2.0mu\kern-3.49998pt\leavevmode\hbox{\set@color${\langle}$}}\mathbf{e}_{\alpha},\mathbf{e}_{\beta}\mathclose{\hbox{\set@color${\rangle}$}\mkern
2.0mu\kern-3.49998pt\leavevmode\hbox{\set@color${\rangle}$}}_{\mathbf{G}}=\delta_{\alpha\beta}\,\epsilon_{\alpha}$
(no summation on $\alpha$). If the connection is metric compatible, one has

$\omega^{\gamma}{}_{\alpha}\,\delta_{\gamma\beta}\,\epsilon_{\beta}+\omega^{\gamma}{}_{\beta}\,\delta_{\gamma\alpha}\,\epsilon_{\alpha}=0\quad(\text{no-
summation~{}on~{}}\alpha~{}\text{or~{}}\beta)\,,$ (2.21)

or

$\omega_{\alpha\beta}+\omega_{\beta\alpha}=0\,.$ (2.22)

Thus

$\epsilon_{\alpha}\,\omega^{\alpha}{}_{\beta}+\epsilon_{\beta}\,\omega^{\beta}{}_{\alpha}=0\quad(\text{no-
summation~{}on~{}}\alpha~{}\text{or~{}}\beta)\,,$ (2.23)

which is equivalent to

$\omega^{\alpha}{}_{\beta}=-\epsilon_{\alpha}\,\epsilon_{\beta}\,\omega^{\beta}{}_{\alpha}\quad(\text{no-
summation~{}on~{}}\alpha~{}\text{or~{}}\beta)\,.$ (2.24)

The first and the second structural equations remain unchanged. The
expressions for the Riemann and Ricci curvatures remain unaltered as well. The
Ricci scalar has the following expression

$\operatorname{R}=\operatorname{Ric}_{\alpha\beta}G^{\alpha\beta}=\sum_{\alpha=1}^{N}\operatorname{Ric}_{\alpha\alpha}\,\epsilon_{\alpha}\,.$
(2.25)

###  2.5 Riemannian product spaces

Let $(\mathcal{B}_{2},\mathbf{G}_{2})$, …, and
$(\mathcal{B}_{N},\mathbf{G}_{N})$ be Riemannian manifolds and
$\mathcal{B}_{2}\times\cdots\times\mathcal{B}_{N}$ be their product manifold.
At any point
$(X_{2},\cdots,X_{N})\in\mathcal{B}_{2}\times\cdots\times\mathcal{B}_{N}$, one
has the direct sum
$T_{(X_{2},\ldots,X_{N})}(\mathcal{B}_{2}\times\cdots\times\mathcal{B}_{N})\cong
T_{X_{2}}\mathcal{B}_{2}\oplus\cdots\oplus T_{X_{N}}\mathcal{B}_{N}$, where
$\cong$ means “isomorphic to”. The product metric
$\mathbf{G}_{2}\times\cdots\times\mathbf{G}_{N}$ on
$\mathcal{B}_{1}\times\cdots\times\mathcal{B}_{N}$ is defined as

$\mathbf{G}_{2}\times\cdots\times\mathbf{G}_{N}\big{|}_{(X_{2},\ldots,X_{N})}=\mathbf{G}_{2}\big{|}_{X_{2}}+\cdots+\mathbf{G}_{N}\big{|}_{X_{N}}\,,\quad\forall
X_{2}\in\mathcal{B}_{2},\cdots,X_{N}\in\mathcal{B}_{N}\,.$ (2.26)

The Riemannian manifold
$(\mathcal{B}_{2}\times\cdots\times\mathcal{B}_{N},\mathbf{G}_{2}\times\cdots\times\mathbf{G}_{N})$
is called a Riemannian product space (Joyce, 2007). If a Riemannian manifold
is isometric to a Riemannian product space, it is called reducible
(decomposable). Otherwise, it is irreducible (indecomposable). It should be
noted that for a Riemannian product space,

$\nabla^{\mathbf{G}_{2}\times\cdots\times\mathbf{G}_{N}}_{(\mathbf{U}_{2},\cdots,\mathbf{U}_{N})}(\mathbf{W}_{2},\cdots,\mathbf{W}_{N})=\left(\nabla^{\mathbf{G}_{2}}_{\mathbf{U}_{2}}\mathbf{W}_{2},\cdots,\nabla^{\mathbf{G}_{N}}_{\mathbf{U}_{N}}\mathbf{W}_{N}\right)\,.$
(2.27)

In particular, the Ricci curvature of the Riemannian product space is written
as

$\operatorname{Ric}\left((\mathbf{U}_{2},\cdots,\mathbf{U}_{N}),(\mathbf{W}_{2},\cdots,\mathbf{W}_{N})\right)=\operatorname{Ric}_{2}(\mathbf{U}_{2},\mathbf{W}_{2})+\cdots+\operatorname{Ric}_{N}(\mathbf{U}_{N},\mathbf{W}_{N})\,.$
(2.28)

###  2.6 Cartan’s curvature $2$-forms of an $N$-dimensional pseudo-Riemannian
manifold with a diagonal metric

Consider an $N$-dimensional pseudo-Riemannian manifold $\mathcal{B}$ with a
diagonal metric

$\mathbf{G}=\sum_{A=1}^{N}\epsilon_{A}\,\mathsf{G}_{A}\,dX^{A}\otimes
dX^{A}=\sum_{A=1}^{N}\epsilon_{A}\,\sqrt{\mathsf{G}_{A}}\,dX^{A}\otimes\sqrt{\mathsf{G}_{A}}\,dX^{A}\,.$
(2.29)

Let us define the co-frame field

$E^{*}=\left\\{\vartheta^{A}=\sqrt{\mathsf{G}_{A}}\,dX^{A}\right\\}\quad(\text{no~{}summation~{}on}~{}A)\,,$
(2.30)

and its dual moving frame field

$E=\left\\{\mathbf{e}_{A}=\frac{1}{\sqrt{\mathsf{G}_{A}}}\,\partial_{A}\right\\}\quad(\text{no~{}summation~{}on}~{}A)\,,$
(2.31)

which by construction $\vartheta^{A}(\mathbf{e}_{B})=\delta_{B}^{A}$. Then,
the metric in the moving frame $E$ is simply written as

$\mathbf{G}=\sum_{A=1}^{N}\epsilon_{A}\,\vartheta^{A}\otimes\vartheta^{A}\,.$
(2.32)

Note that

$d\vartheta^{A}=\sum_{B=1}^{N}\frac{\partial_{B}\mathsf{G}_{A}}{2\mathsf{G}_{A}\sqrt{\mathsf{G}_{B}}}\,\vartheta^{B}\wedge\vartheta^{A}\quad(\text{no~{}summation~{}on}~{}A)\,.$
(2.33)

We next calculate the Levi-Civita connection $1$-forms for which
$\mathcal{T}^{A}=0$. Note that $\omega_{BA}=-\omega_{AB}$ and there are
$N(N-1)/2$ connection $1$-forms to be determined. Cartan’s first structural
equations read $d\vartheta^{A}+\omega^{A}{}_{B}\,\wedge\vartheta^{B}=0$. Note
that one can use (2.17). However, there is an easier approach for calculating
the connection forms. Recalling that
$\omega^{A}{}_{B}=\omega^{A}{}_{CB}\,\vartheta^{C}$, we have

$\sum_{B=1}^{N}\frac{\mathsf{G}_{A,B}}{2\mathsf{G}_{A}\sqrt{\mathsf{G}_{B}}}\,\vartheta^{B}\wedge\vartheta^{A}+\omega^{A}{}_{CB}\,\vartheta^{C}\wedge\vartheta^{B}=0\quad(\text{no~{}summation~{}on}~{}A)\,,$
(2.34)

where $\mathsf{G}_{A,B}=\partial_{B}\mathsf{G}_{A}$. Thus

$\sum_{B=1}^{N}\left(\omega^{A}{}_{CB}\,\vartheta^{C}-\frac{\mathsf{G}_{A,B}}{2\mathsf{G}_{A}\sqrt{\mathsf{G}_{B}}}\,\vartheta^{A}\right)\wedge\vartheta^{B}=0\quad(\text{no~{}summation~{}on}~{}A)\,.$
(2.35)

Cartan’s lemma implies that (Sternberg, 1999)

$\omega^{A}{}_{CB}\,\vartheta^{C}-\frac{\mathsf{G}_{A,B}}{2\mathsf{G}_{A}\sqrt{\mathsf{G}_{B}}}\,\vartheta^{A}=\xi^{A}{}_{BC}\,\vartheta^{C}\quad(\text{no~{}summation~{}on}~{}A~{}\text{or}~{}B)\,,$
(2.36)

where $\xi^{A}{}_{BC}(X)=\xi^{A}{}_{CB}(X)$ are $\frac{N^{2}(N+1)}{2}$
arbitrary functions. Thus

$\omega^{A}{}_{B}=\frac{\mathsf{G}_{A,B}}{2\mathsf{G}_{A}\sqrt{\mathsf{G}_{B}}}\,\vartheta^{A}+\xi^{A}{}_{BC}\,\vartheta^{C}\quad(\text{no~{}summation~{}on}~{}A~{}\text{or}~{}B)\,.$
(2.37)

Hence

$\omega_{AB}=\epsilon_{A}\,\frac{\mathsf{G}_{A,B}}{2\mathsf{G}_{A}\sqrt{\mathsf{G}_{B}}}\,\vartheta^{A}+\epsilon_{A}\,\xi^{A}{}_{BC}\,\vartheta^{C}\quad(\text{no~{}summation~{}on}~{}A~{}\text{or}~{}B)\,.$
(2.38)

Knowing that $\omega_{AB}+\omega_{BA}=0$, one can guess that

$\omega_{AB}=\epsilon_{A}\,\frac{\mathsf{G}_{A,B}}{2\mathsf{G}_{A}\sqrt{\mathsf{G}_{B}}}\,\vartheta^{A}-\epsilon_{B}\,\frac{\mathsf{G}_{B,A}}{2\mathsf{G}_{B}\sqrt{\mathsf{G}_{A}}}\,\vartheta^{B}\quad(\text{no~{}summation~{}on}~{}A~{}\text{or}~{}B)\,.$
(2.39)

Thus

$\omega^{A}{}_{B}=\mathsf{L}_{AB}\,\vartheta^{A}-\epsilon_{A}\,\epsilon_{B}\,\mathsf{L}_{BA}\,\vartheta^{B}\quad(\text{no~{}summation~{}on}~{}A~{}\text{or}~{}B)\,,$
(2.40)

where

$\mathsf{L}_{AB}=\frac{\mathsf{G}_{A,B}}{2\mathsf{G}_{A}\sqrt{\mathsf{G}_{B}}}\,.$
(2.41)

It is straightforward to check that the $1$-forms given in (2.40) satisfy
Cartan’s first structural equations, and hence, are the unique Levi-Civita
connection $1$-forms. Note that
$\omega^{A}{}_{B}=-\epsilon_{A}\,\epsilon_{B}\,\omega^{B}{}_{A}$, and

$\displaystyle d\omega^{A}{}_{B}$
$\displaystyle=\sum_{C}(\mathsf{L}_{AB,C}+\mathsf{L}_{AB}\mathsf{L}_{AC})\,\vartheta^{C}\wedge\vartheta^{A}$
(2.42)
$\displaystyle\quad-\sum_{C}\epsilon_{A}\,\epsilon_{B}\,(\mathsf{L}_{BA,C}+\mathsf{L}_{BA}\mathsf{L}_{BC})\,\vartheta^{C}\wedge\vartheta^{B}\quad(\text{no~{}summation~{}on}~{}A~{}\text{or}~{}B)\,,$

where from Eq. (2.33) the relation
$d\vartheta^{A}=\mathsf{L}_{AC}\,\vartheta^{C}\wedge\vartheta^{A}$ (no
summation on $A$) has been used.

Cartan’s second structural equations read

$\mathcal{R}^{A}{}_{B}=d\omega^{A}{}_{B}+\omega^{A}{}_{C}\wedge\omega^{C}{}_{B}\,,$
(2.43)

where curvature $2$-forms are (pseudo) anti-symmetric, i.e.,
$\mathcal{R}^{A}{}_{B}+\epsilon_{A}\,\epsilon_{B}\,\mathcal{R}^{B}{}_{A}=0$.
More explicitly,

$\displaystyle\mathcal{R}^{A}{}_{B}$
$\displaystyle=\sum_{C}(\mathsf{L}_{AB,C}+\mathsf{L}_{AB}\mathsf{L}_{AC}-\mathsf{L}_{AC}\mathsf{L}_{CB})\,\vartheta^{C}\wedge\vartheta^{A}$
(2.44)
$\displaystyle-\sum_{C}\epsilon_{A}\,\epsilon_{B}\,(\mathsf{L}_{BA,C}+\mathsf{L}_{BA}\mathsf{L}_{BC}-\mathsf{L}_{BC}\mathsf{L}_{CA})\,\vartheta^{C}\wedge\vartheta^{B}$
$\displaystyle-\sum_{C}\epsilon_{C}\,\epsilon_{B}\,\mathsf{L}_{AC}\,\mathsf{L}_{BC}\,\vartheta^{A}\wedge\vartheta^{B}\qquad\qquad(\text{no~{}summation~{}on}~{}A~{}\text{or}~{}B)\,.$

###  2.7 Curvature $2$-forms of a $2$-dimensional pseudo-Riemannian manifold
with a diagonal metric

Consider a two-dimensional pseudo-Riemannian manifold and a coordinate chart
$U=\\{X_{1},X_{2}\\}$. Assume a diagonal metric in the coordinate frame

$ds^{2}=\epsilon_{1}\,\mathsf{G}_{1}(dX^{1})^{2}+\epsilon_{2}\,\mathsf{G}_{2}(dX^{1})^{2}\,.$
(2.45)

From Eq. (2.40), there is only one connection $1$-form $\omega^{1}{}_{2}$
given as

$\omega^{1}{}_{2}=\frac{\mathsf{G}_{1,2}}{2\sqrt{\mathsf{G}_{1}\mathsf{G}_{2}}}\,dX^{1}-\epsilon_{1}\epsilon_{2}\,\frac{\mathsf{G}_{2,1}}{2\sqrt{\mathsf{G}_{1}\mathsf{G}_{2}}}\,dX^{2}\,,$
(2.46)

where $\mathsf{G}_{1,2}=\partial_{X_{2}}\mathsf{G}_{1}$. Alternatively,

$\omega^{1}{}_{2}=\frac{\mathsf{G}_{1,2}}{2\mathsf{G}_{1}\sqrt{\mathsf{G}_{2}}}\,\vartheta^{1}-\epsilon_{1}\epsilon_{2}\,\frac{\mathsf{G}_{2,1}}{2\mathsf{G}_{2}\sqrt{\mathsf{G}_{1}}}\,\vartheta^{2}\,.$
(2.47)

From Cartan’s second structural equations (2.43), there is only one curvature
$2$-form $\mathcal{R}^{1}{}_{2}$, which reads

$\mathcal{R}^{1}{}_{2}=d\omega^{1}{}_{2}=-\frac{1}{2}\left[\epsilon_{1}\epsilon_{2}\left(\frac{\mathsf{G}_{2,1}}{\sqrt{\mathsf{G}_{1}\mathsf{G}_{2}}}\right)_{,1}+\left(\frac{\mathsf{G}_{1,2}}{\sqrt{\mathsf{G}_{1}\mathsf{G}_{2}}}\right)_{,2}\right]dX^{1}\wedge
dX^{2}\,.$ (2.48)

Alternatively,

$\mathcal{R}^{1}{}_{2}=K\,\vartheta^{1}\wedge\vartheta^{2}\,,$ (2.49)

where the Gaussian curvature
$K=\mathcal{R}^{1}{}_{2}(\mathbf{e}_{1},\mathbf{e}_{2})$ is written as

$K=-\frac{1}{2\sqrt{\mathsf{G}_{1}\mathsf{G}_{2}}}\left[\epsilon_{1}\epsilon_{2}\left(\frac{\mathsf{G}_{2,1}}{\sqrt{\mathsf{G}_{1}\mathsf{G}_{2}}}\right)_{,1}+\left(\frac{\mathsf{G}_{1,2}}{\sqrt{\mathsf{G}_{1}\mathsf{G}_{2}}}\right)_{,2}\right]\,.$
(2.50)

From Eq. (2.24) it follows that
$d\omega^{1}{}_{2}=-\epsilon_{1}\epsilon_{2}\,d\omega^{2}{}_{1}$ and hence

$\mathcal{R}^{2}{}_{1}=-\epsilon_{1}\epsilon_{2}\,\mathcal{R}^{1}{}_{2}\,.$
(2.51)

The Ricci tensor is calculated using (2.18)2 as

$\operatorname{Ric}_{\alpha\beta}=\mathcal{R}^{\gamma}{}_{\alpha}(\mathbf{e}_{\gamma},\mathbf{e}_{\beta})=\mathcal{R}^{1}{}_{\alpha}(\mathbf{e}_{1},\mathbf{e}_{\beta})+\mathcal{R}^{2}{}_{\alpha}(\mathbf{e}_{2},\mathbf{e}_{\beta})\,.$
(2.52)

In particular,

$\operatorname{Ric}_{11}=\mathcal{R}^{2}{}_{1}(\mathbf{e}_{2},\mathbf{e}_{1})=\epsilon_{1}\epsilon_{2}\,\mathcal{R}^{1}{}_{2}(\mathbf{e}_{1},\mathbf{e}_{2})=\epsilon_{1}\epsilon_{2}\,K\,,\qquad\operatorname{Ric}_{22}=\mathcal{R}^{1}{}_{2}(\mathbf{e}_{1},\mathbf{e}_{2})=K\,,$
(2.53)

and $\operatorname{Ric}_{12}=\operatorname{Ric}_{21}=0$. The Ricci scalar is
calculated as

$\operatorname{R}=\epsilon_{1}\operatorname{Ric}_{11}+\epsilon_{2}\operatorname{Ric}_{22}=\epsilon_{1}^{2}\epsilon_{2}K+\epsilon_{2}K=2\epsilon_{2}K=-\frac{1}{\sqrt{\mathsf{G}_{1}\mathsf{G}_{2}}}\left[\epsilon_{1}\left(\frac{\mathsf{G}_{2,1}}{\sqrt{\mathsf{G}_{1}\mathsf{G}_{2}}}\right)_{,1}+\epsilon_{2}\left(\frac{\mathsf{G}_{1,2}}{\sqrt{\mathsf{G}_{1}\mathsf{G}_{2}}}\right)_{,2}\right]\,.$
(2.54)

##  3 Dynamics of (free) nonlinear planar double pendula

In this section we will study the geometric phase of a coupled planar double
pendulum, which conserves total angular momentum. This problem was discussed
by Marsden et al. (1990) to introduce the Hamiltonian reduction technique for
mechanical systems with symmetries. The continuous symmetry implies that the
associated phase space has the structure of a principal fiber bundle, i.e., a
shape manifold and transversal fibers attached to it. Marsden et al. (1990)
defined the associated geometric phases and related them to the curvature form
of the shape manifold. Hereafter, we present a new analysis exploiting
Cartan’s structural equations with zero torsion and find the Riemannian
structure of the shape manifold, which was not investigated in (Marsden et
al., 1990).

Consider the planar double pendulum depicted in Fig. 1. The associated
Lagrangian is written as

$\mathcal{L}=\frac{1}{2}I_{1}\,\dot{\theta}_{1}^{2}+\frac{1}{2}I_{2}\,\dot{\theta}_{2}^{2}-\Pi(\theta_{1},\theta_{2})\,,$
(3.1)

where the Lagrangian coordinates $\theta_{j}$ are the angular positions of the
two bobs of masses $m_{1}$ and $m_{2}$ as indicated in Fig. 1.
$I_{j}=m_{j}L_{j}^{2}$ are the mass moments of inertia of the two bobs, where
$L_{j}$ is the length of the $j$th rigid bar. The potential
$\Pi(\theta_{1},\theta_{2})$ describes conservative moments
$M_{j}=-\partial_{\theta_{j}}\Pi$, which are in equilibrium, that is

$M_{1}+M_{2}=-\frac{\partial\Pi}{\partial\theta_{1}}-\frac{\partial\Pi}{\partial\theta_{2}}=0\,.$
(3.2)

Thus, the potential must be a function of the Lagrangian coordinate
difference, i.e., $\Pi(\theta_{2}-\theta_{1})$, which is the potential of a
nonlinear spring, see Fig. 1. Minimizing the action $\int\mathcal{L}dt$ yields
the following dynamical equations

$\frac{d}{dt}\left(\frac{\partial\mathcal{L}}{\partial\dot{\theta}_{j}}\right)-\frac{\partial\mathcal{L}}{\partial\theta_{j}}=I_{j}\ddot{\theta}_{j}+\frac{\partial\Pi}{\partial\theta_{j}}=0\,,\quad
j=1,2\,.$ (3.3)

From (3.2) potential moments are in equilibrium and summing up equations (3.3)
yields

$I_{1}\ddot{\theta}_{1}+I_{2}\ddot{\theta}_{2}=M_{1}+M_{2}=0\,.$ (3.4)

Thus, the total angular momentum

$\mathsf{A}=I_{1}\dot{\theta}_{1}+I_{2}\dot{\theta}_{2}\,,$ (3.5)

is conserved. In the following, we assume that $\mathsf{A}\neq 0$. Such an
invariant endows the system with a continuous Lie symmetry: if the pair
$\mathbf{Z}=(\theta_{1}(t),\theta_{2}(t))$ is a solution of the Lagrangian
equations, so is

$G_{\alpha}(\mathbf{Z})=(\theta_{1}(t)+\alpha,\theta_{2}(t)+\alpha)\,,$ (3.6)

for any angle $\alpha\in\mathbb{R}$. In the following, we will use this
symmetry in the Hamiltonian setting to reveal the geometric structure of the
phase space as that of a principal fiber bundle. Then, the associated
Riemannian structure follows from Cartan’s structural equations as described
in §2.

Figure 1: A planar double pendulum with a nonlinear spring. The mass moments
of inertia are given as $I_{j}=m_{j}L_{j}^{2}$, $j=1,2$, where $m_{j}$ is the
mass of the $j$th bob and $L_{j}$ is the length of the $j$th rigid bar.

###  3.1 The Hamiltonian structure

The conjugate momenta follow from the Lagrangian (3.56) as

$p_{j}=\frac{\partial\mathcal{L}}{\partial\dot{\theta}_{j}}=I_{j}\dot{\theta}_{j}\,,\quad
j=1,2\,,$ (3.7)

and $\theta_{j}=p_{j}/I_{j}$. Then, the Legendre transform of $\mathcal{L}$
gives the Hamiltonian

$\mathcal{H}=p_{1}\,\dot{\theta}_{1}+p_{2}\,\dot{\theta}_{2}-\mathcal{L}=\frac{1}{2}\frac{p_{1}^{2}}{I_{1}}+\frac{1}{2}\frac{p_{2}^{2}}{I_{2}}+\Pi\left(\theta_{2}-\theta_{1}\right)\,.$
(3.8)

The configuration space is a $2$-torus $Q=\mathbb{T}^{2}$, which has the local
chart $\\{\theta_{1},\theta_{2}\\}$, and the phase space is $T^{*}Q$ with
local coordinates $\\{\theta_{1},\theta_{2},p_{1},p_{2}\\}$, where $T^{*}Q$ is
the cotangent bundle of $Q$. Let us define the vector

$\mathbf{X}=\begin{bmatrix}\theta_{1}\\\ \theta_{2}\\\ p_{1}\\\
p_{2}\end{bmatrix}\,.$ (3.9)

The dynamics is governed by

$\dot{\mathbf{X}}=\mathbf{J}\nabla_{\mathbf{X}}\mathcal{H}\,,$ (3.10)

where

$\nabla_{\mathbf{X}}=\begin{bmatrix}\partial_{\theta_{1}}\\\
\partial_{\theta_{2}}\\\ \partial_{p_{1}}\\\ \partial_{p_{2}}\end{bmatrix}\,,$
(3.11)

and $\mathbf{J}$ is the following $4\times 4$ symplectic matrix

$\mathbf{J}=\left[\begin{array}[]{cc}\mathbf{O}_{2}&\mathbf{I}_{2}\\\
-\mathbf{I}_{2}&\mathbf{O}_{2}\end{array}\right]=\left[\begin{array}[]{cccc}0&0&1&0\\\
0&0&0&1\\\ -1&0&0&0\\\ 0&-1&0&0\end{array}\right]\,.$ (3.12)

$\mathbf{I}_{2}=[\delta_{ij}]$ is the $2\times 2$ identity matrix,
$\mathbf{O}_{2}$ is the $2\times 2$ null matrix, and $\delta_{ij}$ is the
Kronecker tensor. From Eq. (3.10),

$\dot{\theta}_{j}=\frac{\partial\mathcal{H}}{\partial
p_{j}}\,,\quad\dot{p}_{j}=-\frac{\partial\mathcal{H}}{\partial\theta_{j}}\,,\quad
j=1,2\,,$ (3.13)

or

$\dot{\theta}_{j}=\frac{p_{j}}{I_{j}}\,,\quad\dot{p}_{j}=-\frac{\partial\Pi}{\partial\theta_{j}}\,,\quad
j=1,2\,.$ (3.14)

The Hamiltonian $\mathcal{H}$ and the total angular momentum
$\mathsf{A}=p_{1}+p_{2}$ given in (3.5) are invariants of motion. The
Hamiltonian system inherits the continuous Lie-group symmetry in (3.6), that
is

$G_{\alpha}(\mathbf{X})=(\theta_{1}+\alpha,\theta_{2}+\alpha,p_{1},p_{2}),$
(3.15)

for any angle $\alpha\in\mathbb{R}$. The associated $1$-form is

$\alpha=p_{1}d\theta_{1}+p_{2}d\theta_{2}\,,$ (3.16)

and the symplectic $2$-form is defined as

$d\alpha=dp_{1}\wedge d\theta_{1}+dp_{2}\wedge d\theta_{2}\,.$ (3.17)

Figure 2: Fiber bundle structure of the state space
$\mathcal{P}=T^{*}Q/\mathsf{A}$. $\mathcal{B}$ is the base (shape) manifold
and $\mathcal{F}$ is a generic fiber.

###  3.2 Hamiltonian reduction and geometric phases

To reveal the geometric nature of the dynamics, we consider another
configuration space $Q_{s}$ with Lagrangian coordinates
$\\{\theta_{1},\psi_{2}\\}$, where the shape parameter
$\psi_{2}=\theta_{2}-\theta_{1}$ represents the relative angular displacement
of the two bobs. Since the total angular momentum $\mathsf{A}=p_{1}+p_{2}$
must be conserved, $p_{1}=\mathsf{A}-p_{2}$ and the motion must occur on the
subspace $T^{*}Q_{s}/\mathsf{A}$ with the coordinate chart
$\\{\theta_{1},\psi_{2},p_{2}\\}$.

The $1$-form in Eq. (3.16) reduces to

$\alpha=p_{1}\,d\theta_{1}+p_{2}\,d\theta_{2}=(\mathsf{A}-p_{2})\,d\theta_{1}+p_{2}\,d\theta_{2}\,,$
(3.18)

and since $\psi_{2}=\theta_{2}-\theta_{1}$, one has

$\alpha=\mathsf{A}\,d\theta_{1}+p_{2}\,d\psi_{2}\,.$ (3.19)

The associated symplectic $2$-form reads

$d\alpha=dp_{2}\wedge d\psi_{2}\,.$ (3.20)

The reduced state space $\mathcal{P}=T^{*}Q_{s}/\mathsf{A}$ has the geometric
structure of a principal fiber bundle characterized by the quadruplet
$(\mathcal{P},\mathcal{B},G_{\alpha},\pi)$. The flow $\dot{\mathbf{X}}_{s}\in
TQ_{s}$ in the state space $\mathcal{P}$ can be decomposed as the sum
$\dot{\mathbf{X}}_{s}=\dot{\mathbf{X}}_{\mathcal{B}}+\dot{\mathbf{X}}_{\mathcal{F}}$
of a Hamiltonian flow

$\mathbf{\dot{X}}_{\mathcal{B}}=\begin{bmatrix}\dot{p}_{2}\\\
\dot{\psi_{2}}\end{bmatrix}\in T\mathcal{B}\,,$ (3.21)

on a two-dimensional shape, or base manifold $\mathcal{B}$ with the coordinate
chart $\\{p_{2},\psi_{2}\\}$, and a drift flow
$\mathbf{\dot{X}}_{\mathcal{F}}=\big{[}\dot{\theta}_{1}\big{]}\in
T\mathcal{F}$ along the one-dimensional fibers $\mathcal{F}$ with the
coordinate chart $\\{\theta_{1}\\}$ as depicted in Fig. 2. The map
$\pi:\mathcal{P}\overset{}{\rightarrow}\mathcal{S}$ projects an element
$\mathbf{X}_{s}$ of the state space $\mathcal{P}$ and all the elements of the
fiber, or group orbit $G_{\alpha}(\mathbf{X}_{s})$, into the same point
$\pi(\mathbf{X}_{s})$ of the base manifold $\mathcal{B}$, viz.
$\pi(\mathbf{X}_{s})=\pi(G_{\alpha}(\mathbf{X}_{s}))$, with
$\alpha\in\mathbb{R}$. In particular,

$\pi(\mathbf{X}_{s})=\mathbf{X}_{\mathcal{B}}=\begin{bmatrix}p_{2}\\\
\psi_{2}\end{bmatrix}\,.$ (3.22)

The reduced Hamiltonian flow $\dot{\mathbf{X}}_{\mathcal{B}}$ on $\mathcal{B}$
is governed by the following equation

$\dot{\mathbf{X}}_{\mathcal{B}}=\mathbf{J}_{R}\nabla_{\mathbf{X}_{\mathcal{B}}}\mathcal{H}_{R}\,,$
(3.23)

where the reduced Hamiltonian is written as

$\mathcal{H}_{R}=\frac{1}{2}\frac{\left(p_{2}-\mathsf{A}\right)^{2}}{I_{1}}+\frac{1}{2}\frac{p_{2}^{2}}{I_{2}}+\Pi\left(\psi_{2}\right)\,.$
(3.24)

$\mathbf{J}_{R}$ is the canonical symplectic matrix

$\mathbf{J}_{R}=\begin{bmatrix}0&1\\\
-1&0\end{bmatrix}\,,\qquad\nabla_{\mathbf{X}_{\mathcal{B}}}=\begin{bmatrix}\partial_{p_{2}}\\\
\partial_{\psi_{2}}\end{bmatrix}\,.$ (3.25)

The flow $\dot{\mathbf{X}}_{\mathcal{B}}$ on the shape manifold $\mathcal{B}$
is independent from the flow $\dot{\mathbf{X}}_{\mathcal{F}}$ along the fiber.
Indeed, from Eq. (3.23)

$\dot{\psi}_{2}=\left(\frac{1}{I_{1}}+\frac{1}{I_{2}}\right)p_{2}-\frac{\mathsf{A}}{I_{1}}\,,\qquad\dot{p}_{2}=-\frac{\partial\Pi}{\partial\psi_{2}}\,.$
(3.26)

On the contrary, $\dot{\mathbf{X}}_{\mathcal{F}}$ depends on
$\dot{\mathbf{X}}_{\mathcal{B}}$ since

$\dot{\theta}_{1}=\frac{\mathsf{A}-p_{2}}{I_{1}}\,.$ (3.27)

In simple words, the flow $\dot{\mathbf{X}}_{s}$ in the state space
$\mathcal{P}$ decouples in a symmetry-free Hamiltonian flow
$\dot{\mathbf{X}}_{\mathcal{B}}$ on the shape manifold $\mathcal{B}$, which
induces a rotation drift $\dot{\mathbf{X}}_{\mathcal{F}}$ along the fibers.
Physically, the reduced flow on $\mathcal{B}$ is the shape-changing evolution
of the connected bobs induced by the internal elastic moments. Such a shape
dynamics makes the two bobs to rigidly rotate together by the time-dependent
angle $\theta_{1}$. Thus, given a symmetry-free motion
$\mathbf{X}_{\mathcal{B}}$ on $\mathcal{B}$. The full motion in $\mathcal{P}$
follows by shifting $\mathbf{X}_{\mathcal{B}}$ along the fibers by
$\theta_{1}$, that is
$\mathbf{X}_{s}=G_{\theta_{1}}(\mathbf{X}_{\mathcal{B}})$, as depicted in Fig.
2. To evaluate such a rotation drift, from Eq. (3.19), we define the $1$-form

$\widetilde{\alpha}=\frac{\alpha}{\mathsf{A}}=d\theta_{1}+\frac{p_{2}}{\mathsf{A}}\,d\psi_{2}\,.$
(3.28)

Then the total rotation drift $\theta_{1}$ along the fiber follows by
integrating the form
$d\theta_{1}=\widetilde{\alpha}-\frac{p_{2}}{\mathsf{A}}d\psi_{2}$, i.e.,

$\theta_{1}=\int
d\theta_{1}=\int_{0}^{t}\widetilde{\alpha}\,dt-\frac{1}{\mathsf{A}}\int_{\gamma}p_{2}\,d\psi_{2}\,,$
(3.29)

where $\gamma$ is a closed trajectory of the motion up to time $t$ in the
shape manifold $\mathcal{B}$. Thus,

$\theta_{1}=\theta_{\text{dyn}}+\theta_{\text{geom}}\,,$ (3.30)

where the dynamical and geometric rotation drifts are defined as

$\theta_{\text{dyn}}(t)=\int_{0}^{t}\widetilde{\alpha}\,dt\,,\qquad\theta_{\text{geom}}=-\frac{1}{\mathsf{A}}\int_{\gamma}p_{2}\,d\psi_{2}\,.$
(3.31)

Here, the dynamical rotation drift $\theta_{\text{dyn}}(t)$ depends on the
inertia of the two bobs. Using Eqs. (3.26)-(3.27) it can written as

$\theta_{\text{dyn}}(t)=\frac{2}{\mathsf{A}}\int_{0}^{t}\mathsf{K}(t)\,dt\,,\qquad\mathsf{K}=\frac{1}{2}\left(\frac{p_{1}^{2}}{I_{1}}+\frac{p_{2}^{2}}{I_{2}}\right)\,,$
(3.32)

where $\mathsf{K}$ is the total kinetic energy and $\mathsf{A}$ is the non-
zero total angular momentum, which is conserved. If the two bobs are rigidly
connected and cannot change their ‘shape’, i.e., $\dot{\psi}_{2}=0$ (no flow
on the base manifold $\mathcal{B}$), then the rotation drift
$\theta_{\text{dyn}}$ is simply the manifestation of the inertia of the entire
system treated as a whole with angular momentum $\mathsf{A}$ and total kinetic
energy $\mathsf{K}$.

The two bobs can also undergo a change in shape due to the internal elastic
moments. As a result, the angle $\psi_{2}$ varies over time and the flow on
$\mathcal{B}$ induces also the geometric rotation drift, which from Eq.(3.40)
can be written as

$\theta_{\text{geom}}=\int_{S(\gamma)}d\,\widetilde{\alpha}=-\frac{1}{\mathsf{A}}\int_{S(\gamma)}dp_{2}\wedge
d\psi_{2}\,.$ (3.33)

Thus, the geometric rotation drift is proportional to the area $S(\gamma)$
enclosed by the trajectory of the motion $\gamma$ in the shape manifold
$\mathcal{B}$. Such a rotation drift is purely geometric since it does not
depend on the time it takes for the two bobs to undergo a cyclic shape change.

One can define an effective moment of inertia $I_{\text{eff}}$ for an
equivalent system with the total angular momentum $\mathsf{A}=p_{1}+p_{2}$ as

$\frac{1}{I_{\text{eff}}}=\frac{\dot{\theta}_{1}}{p_{1}+p_{2}}=\frac{\dot{\theta}_{\text{dyn}}+\dot{\theta}_{\text{geom}}}{p_{1}+p_{2}}=\frac{2\mathsf{K}}{\mathsf{A}^{2}}-\frac{p_{2}\,\dot{\psi}_{2}}{\mathsf{A}^{2}}\,.$
(3.34)

Using Eq. (3.26),

$\frac{1}{I_{\text{eff}}}=\frac{2\mathsf{K}}{\mathsf{A}^{2}}-\frac{p_{2}^{2}}{\mathsf{A}^{2}}+\frac{p_{2}}{\mathsf{A}\,I_{1}}\,.$
(3.35)

Thus, the shape-changing motion of the two bobs can slow down or speed up the
rotation drift. In particular, if the two bobs tend to expand in opposite
directions ($\dot{\psi}_{2}>0$) the effective moment of inertia increases
slowing down the rotation as the angular speed $\dot{\theta_{1}}$ reduces. On
the contrary, if the two bobs tend to contract ($\dot{\psi}_{2}<0$) the
angular speed increases as $I_{\text{eff}}$ reduces. This is the analogue of
spinning dancers that can increase their spinning rate by pulling their arms
close to their bodies, and to decrease it by letting their arms out. Thus, the
geometric phase component in Eq. (3.35) can be interpreted as the moment of
inertia of an added mass in analogy with the fish self-propulsion (Shapere and
Wilczek, 1987, 1989).

Figure 3: (Left) Fiber bundle structure of the state space
$\mathcal{P}=T^{*}Q/\mathsf{A}$ of a double pendulum with with parameters
$I_{1}=I_{2}=1\,\mathrm{mass}\times\mathrm{length}^{2}$,
$\mathsf{A}=\frac{1}{2}\,\mathrm{mass}\times\mathrm{length}^{2}\mathrm{time}^{-1}$,
and potential $\Pi(\psi)=\psi^{4}$. Full path $\mathbf{X}_{s}(t)$ (black
curve) and the reduced path $\mathbf{X}_{\mathcal{B}}(t)$ on the base manifold
$\mathcal{B}$ (green curve). The lifted path
$G_{\theta_{\text{dyn}}}(\mathbf{X}_{\mathcal{B}})$ by the dynamical rotation
drift $\theta_{\text{dyn}}$ (red curve) does not match with with the path
$\mathbf{X}_{s}$ because the total rotation drift
$\theta_{1}=\theta_{\text{dyn}}+\theta_{\text{geom}}$ includes also the
geometric component $\theta_{\text{geom}}$. (Right) The total rotation drift
$theta_{1}$ and the drifts $\theta_{\text{geom}}$ and $\theta_{\text{dyn}}$ as
functions of time.

As an example, consider the double pendulum with parameters
$I_{1}=I_{2}=1\,\mathrm{mass}\times\mathrm{length}^{2}$,
$\mathsf{A}=\frac{1}{2}\,\mathrm{mass}\times\mathrm{length}^{2}\mathrm{time}^{-1}$,
and potential $\Pi(\psi)=\psi^{4}$. Note that the total angular momentum is
positive and the initial conditions are chosen so that both bobs have positive
(counter-clockwise) rotational speed. The left panel of Fig. 3 depicts the
fiber bundle structure of the state space $\mathcal{P}=T^{*}Q/\mathsf{A}$ of
the double pendulum, the full path $\mathbf{X}_{s}(t)$ (black curve) and the
reduced path $\mathbf{X}_{\mathcal{B}}$ (green curve) on the base manifold
$\mathcal{B}$ are shown. The lifted path
$G_{\theta_{\text{dyn}}}(\mathbf{X}_{s})$ by the positive dynamical rotation
drift $\theta_{\text{dyn}}$ (red curve) does not coincide with the path
$\mathbf{X}_{s}$ (black curve). This is because the total rotational drift
$\theta_{1}=\theta_{\text{dyn}}+\theta_{\text{geom}}$ includes also a negative
geometric component $\theta_{\text{geom}}$, as is seen in the right panel of
the same figure. The positive dynamical rotation drift is induced by the
inertia of the system that has a positive angular momentum. However, the two
bobs undergo changes in shape inducing a clockwise rotation that balances the
counter-clockwise dynamical drift rotation.

######  Remark 3.1.

Given the total angular momentum $\mathsf{A}$, the associated dynamical
variables $\\{\theta_{1}(t),\psi_{2}(t),p_{2}(t)\\}$ satisfy Eqs. (3.26) and
(3.27). Let $\\{\bar{\theta}_{1}(t),\bar{\psi}_{2}(t),\bar{p}_{2}(t)\\}$ be
the dynamical variables that correspond to $-\mathsf{A}$, where
$\bar{\psi}_{2}=\bar{\theta}_{2}-\bar{\theta}_{1}$. Then, these satisfy

$\dot{\bar{\theta}}_{1}=\frac{-\mathsf{A}-\bar{p}_{2}}{I_{1}}\,,\quad\dot{\bar{\psi}}_{2}=\left(\frac{1}{I_{1}}+\frac{1}{I_{2}}\right)\bar{p}_{2}+\frac{\mathsf{A}}{I_{1}}\,,\qquad\dot{\bar{p}}_{2}=-\frac{\partial\Pi}{\partial\bar{\psi}_{2}}\,.$
(3.36)

Assume the initial conditions $\bar{\theta}_{1}(0)=-\theta_{1}(0)$,
$\bar{\psi}_{2}(0)=-\psi_{2}(0)$, and $\bar{p}_{2}(0)=-p_{2}(0)$. Then, from
Eqs. (3.26) and (3.27) we have that

$\\{\bar{\theta}_{1}(t),\bar{\psi}_{2}(t),\bar{p}_{2}(t)\\}=\\{-\theta_{1}(t),-\psi_{2}(t),-p_{2}(t)\\}\,,$
(3.37)

is a solution of the above system of first-order ODEs. The associated
symplectic form follows from Eq. (3.19) as

$\bar{\alpha}=-\mathsf{A}\,d\bar{\theta}_{1}+\bar{p}_{2}\,d\bar{\psi}_{2}\,,\qquad
d\bar{\alpha}=d\bar{p}_{2}\wedge d\bar{\psi}_{2}\,.$ (3.38)

Moreover,

$\widetilde{\bar{\alpha}}=\frac{\bar{\alpha}}{-\mathsf{A}}=-\widetilde{\alpha}\,.$
(3.39)

Therefore

$\bar{\theta}_{\text{dyn}}(t)=\int_{0}^{t}\widetilde{\bar{\alpha}}\,dt=-\theta_{\text{dyn}}(t)\,,\qquad\bar{\theta}_{\text{geom}}=\frac{1}{\mathsf{A}}\int_{\gamma}\bar{p}_{2}\,d\bar{\psi}_{2}=-\theta_{\text{geom}}\,.$
(3.40)

Thus, when changing the sign of the angular momentum (and the initial
conditions), both the dynamic and geometric phases change sign.

###  3.3 Curvature and intrinsic metric of the shape manifold

One can interpret the geometric drift as the curvature of the shape manifold
$\mathcal{B}$ equipped with a specific metric. As a matter of fact, drawing on
Cartan’s structural equations the $2$-form $d\,\widetilde{\alpha}$ in (3.33)
can be interpreted as the curvature form of a connection on $\mathcal{B}$. We
further require that the symplectic form be compatible with the volume 2-form
$\mathsf{vol}_{\mathbf{G}}$ of the metric $\mathbf{G}$ as is shown next.

The geometric drift $\theta_{\text{geom}}$ given in Eq. (3.33) is associated
to the symplectic $1$-form

$\widetilde{\alpha}=-\frac{p_{2}}{\mathsf{A}}\,d\psi_{2}\,,$ (3.41)

since $\theta_{\text{geom}}=\int d\,\widetilde{\alpha}$. The $1$-form
$\widetilde{\alpha}$ can be interpreted as the connection $1$-form
$\omega^{1}{}_{2}$ of a $2$-manifold represented by the coordinate charts
$\\{X^{1},X^{2}\\}=\\{p_{2},\psi_{2}\\}$ and with the metric

$ds^{2}=\epsilon_{1}\,\mathsf{G}_{1}\,dp_{2}^{2}+\epsilon_{2}\,\mathsf{G}_{2}\,d\psi_{2}^{2}\,.$
(3.42)

We want to find the metric coefficients $\mathsf{G}_{1}$ and $\mathsf{G}_{2}$
so that111We can add to $\omega^{1}{}_{2}$ an arbitrary closed $1$-form $\xi$,
i.e., $d\xi=0$. This form can be neglected because it does not contribute to
the geometric phase as its integral over any closed curve vanishes. Thus, the
freedom to add an arbitrary closed form is physically inconsequential.

$\omega^{1}{}_{2}=\widetilde{\alpha}\,.$ (3.43)

From Eq. (2.46) we then have

$\frac{\mathsf{G}_{1,2}}{2\sqrt{\mathsf{G}_{1}\mathsf{G}_{2}}}\,dp_{2}-\epsilon_{1}\epsilon_{2}\,\frac{\mathsf{G}_{2,1}}{2\sqrt{\mathsf{G}_{1}\mathsf{G}_{2}}}\,d\psi_{2}=-\frac{p_{2}}{\mathsf{A}}\,d\psi_{2}\,.$
(3.44)

This implies that

$\frac{\mathsf{G}_{1,\psi_{2}}}{2\sqrt{\mathsf{G}_{1}\mathsf{G}_{2}}}=0\,,\qquad\epsilon_{1}\epsilon_{2}\,\frac{\mathsf{G}_{2,p_{2}}}{2\sqrt{\mathsf{G}_{1}\mathsf{G}_{2}}}=\frac{p_{2}}{\mathsf{A}}\,.$
(3.45)

The first equation implies that $\mathsf{G}_{1,\psi_{2}}=0$, and hence
$\mathsf{G}_{1}=\mathsf{G}_{1}(p_{2})$. The Gaussian curvature is calculated
from Eq. (2.50) as

$K=-\frac{1}{\mathsf{A}\sqrt{\mathsf{G}_{1}\mathsf{G}_{2}}}\,.$ (3.46)

Eqs. (3.45) also imply that the symplectic $2$-form $d\,\widetilde{\alpha}$
equals the curvature $2$-form $d\omega^{1}{}_{2}$, that is
$d\,\widetilde{\alpha}=d\omega^{1}{}_{2}$, or explicitly

$d\,\widetilde{\alpha}=-\frac{1}{\mathsf{A}}\,dp_{2}\wedge
d\psi_{2}=K\sqrt{\mathsf{G}_{1}\mathsf{G}_{2}}\,dp_{2}\wedge d\psi_{2}\,.$
(3.47)

We now further require that the symplectic $2$-form $d\,\widetilde{\alpha}$ is
compatible with the (pseudo) Riemannian volume (area) $2$-form
$\mathsf{vol}_{\mathbf{G}}=\sqrt{\mathsf{G}_{1}\mathsf{G}_{2}}\,dp_{2}\wedge
d\psi_{2}$ in the sense that the absolute value of the geometric rotational
drift over a closed trajectory $\gamma$ in the shape manifold is equal to the
volume (area) of the region $S(\gamma)$ it encloses using
$\mathsf{vol}_{\mathbf{G}}$, that is,

$\frac{1}{|\mathsf{A}|}\,dp_{2}\wedge
d\psi_{2}=\sqrt{\mathsf{G}_{1}\mathsf{G}_{2}}\,dp_{2}\wedge d\psi_{2}\,,$
(3.48)

which is equivalent to

$\sqrt{\mathsf{G}_{1}\mathsf{G}_{2}}=\frac{1}{|\mathsf{A}|}\,,$ (3.49)

or $\mathsf{A}^{2}\,\mathsf{G}_{1}\,\mathsf{G}_{2}=1$. This, in particular,
implies that $G_{2}=G_{2}(p_{2})$. We can now solve for $G_{2}$. Substituting
(3.49) into (3.45)2 yields

$\mathsf{G}_{2,p_{2}}=\epsilon_{1}\epsilon_{2}\,\frac{2\,p_{2}}{\mathsf{A}|\mathsf{A}|}\,,$
(3.50)

and hence

$\mathsf{G}_{2}=\epsilon_{1}\epsilon_{2}\,\frac{p_{2}^{2}+C_{2}}{\mathsf{A}|\mathsf{A}|}\,,$
(3.51)

where $C_{2}$ is an arbitrary constant, and $G_{1}$ follows from Eq. (3.49).
Since we must have $G_{2}\geq 0$, we set $C_{2}=\mu^{2}$, $\mu\in\mathbb{R}$,
and $\mathrm{sgn}(\epsilon_{1}\epsilon_{2})=\mathrm{sgn}(\mathsf{A})$. We
choose $\epsilon_{1}=\mathrm{sgn}(\mathsf{A})$ and $\epsilon_{2}=1$ so that

$\mathsf{G}_{1}=\frac{1}{p_{2}^{2}+\mu^{2}}\,,\qquad\mathsf{G}_{2}=\frac{p_{2}^{2}+\mu^{2}}{\mathsf{A}^{2}}\,,$
(3.52)

and the family of metrics (3.42) specializes to

$\mathbf{G}=\frac{\mathrm{sgn}(\mathsf{A})}{p_{2}^{2}+\mu^{2}}\,dp_{2}\otimes
dp_{2}+\frac{p_{2}^{2}+\mu^{2}}{\mathsf{A}^{2}}\,d\psi_{2}\otimes
d\psi_{2}\,.$ (3.53)

From Eqs. (3.46) and (3.49), the corresponding Gaussian curvature
$K=-\mathrm{sgn}(\mathsf{A})$ and the Ricci scalar
$R=2K=-2\,\mathrm{sign}(\mathsf{A})$ depend on the sign of $\mathsf{A}$. The
metric structure and curvature of the shape manifold depends on the sign of
$\mathsf{A}$. In the following we will show that for $\mathsf{A}<0$ the shape
manifold is an expanding spacetime with positive curvature equipped with the
Robertson-Walker metric, whereas for $\mathsf{A}>0$ the shape manifold is the
hyperbolic plane with negative curvature. Clearly, the Riemannian structure of
the shape manifold depends on the sign of the total momentum. Nevertheless,
the geometric phase is evaluated by the same $2$-form given by the sectional
curvature form of $\mathcal{B}$ (see Eq. (3.33)).

Hernández-Garduño and Shashikanth (2018) studied the geometric phases of three
inviscid point vortices and also found that the curvature of the associated
shape manifold depends on the sign of a parameter related to the strengths of
the three vortices. In particular, their shape manifold is a sphere or the
hyperbolic plane. We note that the authors do not explain why the character of
the manifold changes with the sign of vortex strength parameter (see also
Shashikanth and Marsden (2003)).

######  Remark 3.2.

The metric $\mathbf{G}$ depends on the sign of $\mathsf{A}$. This reflects a
symmetry of the Hamiltonian dynamics. As a matter of fact, in Remark 3.1 It
was shown that the dynamics on the shape manifold $\mathcal{B}$ for given
$\mathsf{A}$ is the same as the dynamics with opposite momentum $-\mathsf{A}$
if one makes the coordinate change
$\\{\theta_{1},\psi_{2},p_{2}\\}\to\\{-\theta_{1},-\psi_{2},-p_{2}\\}$.
However, the geometric rotation drift (3.33) depends on the sign of
$\mathsf{A}$ since
$\theta_{\text{geom}}(\mathsf{A})=-\theta_{\text{geom}}(-\mathsf{A})$. Thus,
the geometric phase of the double pendulum with an initial imparted angular
momentum $\mathsf{A}>0$ on $\mathcal{B}=(\psi_{2},p_{2})$ endowed with the
Riemannian metric (3.53) is equal and opposite to the geometric phase of a
double pendulum with $\mathsf{A}<0$ on the transformed shape manifold
$\mathcal{B}^{\prime}=(-\psi_{2},-p_{2})$ with the corresponding pseudo-
Riemannian metric. This suggests that we can always consider either of the two
metrics.

######  Remark 3.3 (Physical significance of the metric).

The metric (3.53) defined on the shape, or base manifold $\mathcal{B}$
characterizes the kinematically admissible shape deformations of the double
pendulum. An orbit on $\mathcal{B}$ is a succession of infinitesimal changes
in the shape of the pendulum from an initial configuration to another. If the
pendulum returns to its initial shape, the orbit is closed and the area (or
curvature) spanned by it measures the induced rotation drift. Any curve, or
orbit on the base manifold is a kinematically admissible shape evolution,
i.e., a sequence of changing shapes. The orbit is also dynamically admissible
if it is consistent with the Hamiltonian flow (3.26). The metric allows
quantifying the similarity of a shape $S_{1}$ to another shape $S_{2}$, by
measuring the intrinsic distance between the corresponding points on the shape
manifold $\mathcal{B}$. Other non-intrinsic distances are misleading as they
do not account for the curvature, or induced geometric drift.

###  3.4 Geodesics of the metric

####  3.4.1 Negative angular momentum: Robertson-Walker spacetime

For $\mathsf{A}<0$, the metric in Eq. (3.53) is pseudo-Riemannian with $p_{2}$
as a time-like coordinate and $\psi_{2}$ as space-like,

$ds^{2}=-\frac{1}{p_{2}^{2}+\mu^{2}}\,(dp_{2})^{2}+\frac{p_{2}^{2}+\mu^{2}}{\mathsf{A}^{2}}\,(d\psi_{2})^{2}\,,$
(3.54)

and positive Gaussian curvature $\mathsf{K}=1$. The geodesic equations follow
by minimizing the action $\int ds^{2}=\int L\,d\lambda$ with Lagrangian
density

$L(p_{2}(\lambda),\psi_{2}(\lambda))=-\frac{(p^{\prime}_{2})^{2}}{p_{2}^{2}+\mu^{2}}+\frac{p_{2}^{2}+\mu^{2}}{\mathsf{A}^{2}}(\psi^{\prime}_{2})^{2}\,,$
(3.55)

where $f^{\prime}$ denotes derivative with respect to $\lambda$, which
parameterizes the geodesics. Trivial geodesics are the straight lines
$\psi_{2}=\mathrm{const}$ and $p_{2}=\mathrm{const}$ for which $L$ is
stationary. A family of non-trivial geodesics can be easily found by choosing
the parametrization $\lambda=p_{2}$. Then,

$L(\psi_{2}(p_{2}))=-\frac{1}{p_{2}^{2}+\mu^{2}}+\frac{p_{2}^{2}+\mu^{2}}{\mathsf{A}^{2}}\left(\frac{d\psi_{2}}{dp_{2}}\right)^{2}\,.$
(3.56)

Variational differentation yields

$\frac{d}{dp_{2}}\left(\frac{p_{2}^{2}+\mu^{2}}{\mathsf{A}^{2}}\,\frac{d\psi_{2}}{dp_{2}}\right)=0\,,$
(3.57)

from which

$\frac{d\psi_{2}}{dp_{2}}=\frac{\mathsf{A}^{2}C_{1}}{p_{2}^{2}+\mu^{2}}\,,$
(3.58)

where $C_{1}$ is an arbitrary constant. Integration yields

$\psi_{2}=\frac{\mathsf{A}^{2}C_{1}}{\mu}\,\tan^{-1}\left(\frac{p_{2}}{\mu}\right)+C_{2}\,,$
(3.59)

where $C_{2}$ is another arbitrary constant, which together with $C_{1}$
parameterize the family of geodesics. The Lagrangian density in Eq. (3.56)
simplifies to read

$L=\frac{-1+\mathsf{A}^{2}C_{1}^{2}}{p_{2}^{2}+\mu^{2}}\,,$ (3.60)

and the null-geodesics are given by Eq. (3.59) with $C_{1}=1/\mathsf{A}$ since
$L=0$. Fig. 4 depicts the null-geodesics (thin curves) and a few geodesics
(bold curves) for the metric with $\mathsf{A}=-5,\mu=2$.

######  Remark 3.4.

Drawing on General Relativity (Misner et al., 1973; Carroll, 2003), $p_{2}$ is
a time-like coordinate and $\psi_{2}$ is space-like, and the metric
$\mathbf{G}$ represents the analogue of a space-time where null-geodesics
(thin curves of Fig. 4) are the trajectories of massless light photons. The
associated light cones tend to close up as $p_{2}\rightarrow\pm\infty$ and
light slows down. In the same figure, the depicted geodesics (bold curves) are
always inside the light cones they intersect along their path. Thus, they are
the ’time-like’ trajectories of a massive particle traveling at a speed less
than the speed of light. Moreover, geodesics tend to converge in as an
indication of the positive Gaussian curvature.

######  Remark 3.5.

The metric (3.54) describes the analogue of a disguised spacetime of an
expanding universe (Misner et al., 1973; Carroll, 2003). As a matter of fact,
for $\mu\neq 0$, one can define new coordinate chart

$t=\tanh^{-1}\left[\frac{p_{2}}{\sqrt{p_{2}^{2}+\mu^{2}}}\right],\qquad
x=\psi_{2}\,.$ (3.61)

Then one has $dt=dp_{2}/\sqrt{p_{2}^{2}+\mu^{2}}$, $dx=d\psi_{2}$, and the
metric (3.54) transforms to

$ds^{2}=-dt^{2}+\frac{\mu^{2}(\cosh{t})^{2}}{\mathsf{A}^{2}}\,dx^{2}\,,$
(3.62)

which is the $2$-dimensional analogue of the Robertson-Walker (RW) metric in
General Relativity (Misner et al., 1973; Carroll, 2003). The metric describes
an expanding universe with the scale factor $a(t)=\cosh{t}$ and Hubble
constant $H=\dot{a}/a=\tanh(t)$ (Carroll, 2003). For $\mu=0$, the new
coordinates are

$t=\mathrm{e}^{p_{2}},\qquad x=\psi_{2}\,,$ (3.63)

where $dt=dp_{2}/p_{2}$ and $dx=d\psi_{2}$, and the metric transforms to

$ds^{2}=-dt^{2}+\frac{\mathrm{e}^{2t}}{\mathsf{A}^{2}}\,dx^{2}\,,$ (3.64)

which is still a Robertson-Walker metric with the scale factor
$a(t)=\mathrm{e}^{t}$ and Hubble constant $H=\dot{a}/a=1$ (Carroll, 2003).

######  Remark 3.6.

The analogy of the shape manifold $\mathcal{B}$ being like an expanding
universe implies that a point on $\mathcal{B}$, or shape $S_{1}$, appears
‘red-shifted’ by another point, or shape $S_{2}$, as the momentum $p_{2}$
(time-like coordinate) increases. Thus, the low-momentum shapes with small
geometric rotation drift are far apart from the high-momentum shapes with
large geometric drift. So the analogy with the expanding universe impllies
that different shapes, or points, can be very far away from each other on the
shape manifold and correspond to very different geometric rotation drifts. The
extrinsic Euclidean metric would give a smaller distance between the two
points misleading them as similar shapes.

####  3.4.2 Positive angular momentum: The hyperbolic plane $\mathbb{H}^{2}$

For $\mathsf{A}>0$, the shape manifold has negative Gaussian curvature $K=-1$
and the metric in Eq. (3.53) is Riemannian

$ds^{2}=\frac{1}{p_{2}^{2}+\mu^{2}}\,(dp_{2})^{2}+\frac{p_{2}^{2}+\mu^{2}}{\mathsf{A}^{2}}\,(d\psi_{2})^{2}\,.$
(3.65)

To reveal the nature of the geodesics, we can still use the coordinate
transformations (3.61), (3.63). As an example, for $\mu=0$ the metric (3.65)
transforms to

$ds^{2}=dt^{2}+R^{2}(t)\,dx^{2}\,,$ (3.66)

where $R(t)=\mathrm{e}^{2t}/\mathsf{A}^{2}$. This is a disguised metric of the
hyperbolic plane as the change of coordinates $\tilde{x}=x/\sqrt{\mathsf{A}}$
and $\tilde{y}=\mathsf{A}\exp(-t)$ transforms it to

$ds^{2}=\frac{d\tilde{x}^{2}+d\tilde{y}^{2}}{\tilde{y}^{2}}\,.$ (3.67)

######  Remark 3.7.

A point on $\mathcal{B}$, or shape $S_{1}$, appears far away from another
point, or shape $S_{2}$, as the momentum $p_{2}$ reduces because of the
hyperbolic character of the metric. Thus, the low-momentum shapes with small
geometric rotation drift are far apart from the high-momentum shapes with
large geometric drift. If one uses the Euclidean metric instead, the two
points would appear closer than they are. The Euclidean metric is misleading
in the sense that far away shapes appear as similar shapes when looking at
them through Euclidean lens.

####  3.4.3 The set of all geodesics

More generally, let us assume that the geodesics of the metric in Eq. (3.53)
are parameterized by $\lambda$. Minimizing the action $\int ds^{2}=\int
L\,d\lambda$ with Lagrangian density

$L(p_{2}(\lambda),\psi_{2}(\lambda))=\mathrm{sgn}(\mathsf{A})\frac{(p^{\prime}_{2})^{2}}{p_{2}^{2}+\mu^{2}}+\frac{p_{2}^{2}+\mu^{2}}{\mathsf{A}^{2}}(\psi^{\prime}_{2})^{2}\,,$
(3.68)

gives

$\left(\frac{p_{2}^{\prime}}{p_{2}^{2}+\mu^{2}}\right)^{\prime}+\mathrm{sgn}(\mathsf{A})\frac{p_{2}}{\mathsf{A}^{2}}\,{\psi^{\prime}_{2}}^{2}-\frac{p_{2}p_{2}^{\prime}}{p_{2}^{2}+\mu^{2}}=0\,,\qquad\psi^{\prime}_{2}=\frac{\mathsf{A}^{2}c_{1}}{p_{2}^{2}+\mu^{2}}\,,$
(3.69)

where $f^{\prime}$ denotes derivative with respect to $\lambda$ and $c_{1}$ is
a constant. Then, the first equation for $p_{2}$ can be written as

$p_{2}^{\prime\prime}=p_{2}\frac{-\mathrm{sgn}(\mathsf{A})\mathsf{A}^{2}c_{1}^{2}+p_{2}^{\prime}}{p_{2}^{2}+\mu^{2}}\,.$
(3.70)

This ODE can be solved for by using the substitution
$p_{2}^{\prime}=F(p_{2})$. Notice that
$p_{2}^{\prime\prime}=\frac{dF}{dp_{2}}p_{2}^{\prime}=F\,\frac{dF}{dp_{2}}$
and

$\frac{dF}{dp_{2}}F=p_{2}\frac{-\mathrm{sgn}(\mathsf{A})\mathsf{A}^{2}c_{1}^{2}+F}{p_{2}^{2}+\mu^{2}}\,,$
(3.71)

which can be easily integrated to solve for $F$:

$p_{2}^{\prime}=F(p_{2})=\pm\sqrt{c_{2}\,(p_{2}^{2}+\mu^{2})+\mathrm{sgn}(\mathsf{A})\mathsf{A}^{2}c_{1}^{2}}\,,$
(3.72)

where $c_{2}$ is another constant. Thus, the geodesic equations (3.69) are
reduced to the following first order system

$p_{2}^{\prime}=\pm\sqrt{c_{2}\,(p_{2}^{2}+\mu^{2})+\mathrm{sgn}(\mathsf{A})\mathsf{A}^{2}c_{1}^{2}}\,,\qquad\quad\psi^{\prime}_{2}=\frac{\mathsf{A}^{2}c_{1}}{p_{2}^{2}+\mu^{2}}\,.$
(3.73)

Integrating the first equation yields

$-\frac{1}{\sqrt{c_{2}}}\,\log\left(-p_{2}\sqrt{c_{2}}+\sqrt{c_{2}\,(p_{2}^{2}+\mu^{2})+\mathrm{sgn}(\mathsf{A})\mathsf{A}^{2}c_{1}^{2}}\right)=\pm\lambda+\lambda_{0}\,,$
(3.74)

which is valid in the range of values of $\lambda$ for which the argument
under the square root is non-negative, $\lambda_{0}$ is a constant, and
$c_{2}\geq 0$. Thus, the geodesics are parameterized by

$p_{2}(\lambda)=\frac{\mathrm{e}^{-\sqrt{c_{2}}(\pm\lambda+\lambda_{0})}\left[-1+\left(c_{2}\mu^{2}+\mathrm{sgn}(\mathsf{A})\,\mathsf{A}^{2}c_{1}^{2}\right)\mathrm{e}^{2\sqrt{c_{2}}(\pm\lambda+\lambda_{0})}\right]}{2\sqrt{c_{2}}}\,,$
(3.75)

and

$\psi_{2}(\lambda)=\begin{cases}\begin{array}[]{c}\frac{\mathsf{A}c_{1}}{\mu}\tanh^{-1}\left[\frac{2\mathsf{A}\,\mu\,c_{1}\sqrt{c_{2}}\,\mathrm{e}^{2\sqrt{c_{2}}(\pm\lambda+\lambda_{0})}}{1+\left(\mathsf{A}^{2}\,c_{1}^{2}+c_{2}\,\mu^{2}\right)\mathrm{e}^{2\sqrt{c_{2}}(\pm\lambda+\lambda_{0})}}\right],\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\mathsf{A}<0\,,\\\
\\\
\frac{\mathsf{A}\,c_{1}}{\mu}\Bigg{\\{}\tanh^{-1}\left[\frac{A\,c_{1}+\left(A^{2}c_{1}^{2}+c_{2}\,\mu^{2}\right)\mathrm{e}^{\sqrt{c_{2}}(\pm\lambda+\lambda_{0})}}{\sqrt{c_{2}}\,\mu}\right]-\tanh^{-1}\left[\frac{Ac_{1}-\left(A^{2}c_{1}^{2}+c_{2}\,\mu^{2}\right)\mathrm{e}^{\sqrt{c_{2}}(\pm\lambda+\lambda_{0})}}{\sqrt{c_{2}}\mu}\right]\Bigg{\\}}\,,\quad\mathsf{A}>0\,.\end{array}\end{cases}$
(3.76)

Figure 4: Geodesics of the pseudo Riemannian metric
$-\frac{1}{p_{2}^{2}+\mu^{2}}\,dp_{2}\otimes
dp_{2}+\frac{p_{2}^{2}+\mu^{2}}{\mathsf{A}^{2}}\,d\psi_{2}\otimes d\psi_{2}$.
Null-geodesics are the thin black curves and geodesics are the bold blue
curves.
Figure 5: A planar $N$-pendulum with $N-1$ nonlinear springs. Note that the
moments of inertia $I_{j}=m_{j}L_{j}^{2}$, $j=1,\cdots,N$, where $m_{j}$ is
the mass of the $j$th bob and $L_{j}$ is the length if the $j$th rigid bar.
The applied time-dependent moments are self equilibrated, i.e.,
$\sum_{j=1}^{N}\mathsf{M}^{e}_{j}(t)=0$.

##  4 Dynamics of (free) nonlinear planar $N$-pendula

We next generalize the double pendulum system described above to an
$N$-pendulum with $N$ bobs with mass moments of inertia
$(I_{1},I_{2},...I_{N})$. The Lagrangian coordinates $\theta_{j}$ are the
angular positions of the rigid bars as depicted in Fig. 5. For the specific
problem we consider the action of $N-1$ nonlinear springs on the bars as
depicted in Figure 5. The associated potential depends on the angle
differences of adjacent bars

$\Pi\left(\theta_{2}-\theta_{1},\theta_{3}-\theta_{2},...,\theta_{N}-\theta_{N-1}\right)=\Pi_{2}(\theta_{2}-\theta_{1})+\ldots+\Pi_{N}(\theta_{N}-\theta_{N-1})\,,$
(4.1)

and describes internal conservative moments $M_{j}=-\partial_{\theta_{j}}\Pi$
acting on the bars, which are in equilibrium, that is

$M_{1}+M_{2}+\cdots+M_{N}=-\sum_{j=1}^{N}\frac{\partial\Pi}{\partial\theta_{j}}=0\,.$
(4.2)

The associated Lagrangian is written as

$\mathcal{L}=\sum_{n=1}^{N}\frac{1}{2}I_{n}\,\dot{\theta}_{n}^{2}-\Pi\left(\theta_{2}-\theta_{1},\theta_{3}-\theta_{2},...,\theta_{N}-\theta_{N-1}\right)\,.$
(4.3)

Minimizing the action $\int\mathcal{L}dt$ yields the following dynamical
equations

$\frac{d}{dt}\left(\frac{\partial\mathcal{L}}{\partial\dot{\theta}_{j}}\right)-\frac{\partial\mathcal{L}}{\partial\theta_{j}}=I_{j}\ddot{\theta}_{j}+\frac{\partial\Pi}{\partial\theta_{j}}=0\,,\qquad
j=1,\cdots N\,.$ (4.4)

From (4.2) potential moments are in equilibrium and summing up equations (5.3)
yields

$\frac{d}{dt}\left(\sum_{j=1}^{N}I_{j}\dot{\theta}_{j}(t)\right)=0\,.$ (4.5)

Thus, the total angular momentum of the N-pendulum

$I_{1}\dot{\theta}_{1}(t)+I_{2}\dot{\theta}_{2}(t)+\cdots+I_{N}\dot{\theta_{N}}(t)=\mathsf{A}\,,$
(4.6)

is conserved over time and
$\mathsf{A}=I_{1}\dot{\theta_{1}}(0)+\cdots+I_{N}\dot{\theta_{N}}(0)$ is the
initial momentum imparted by the angular velocities $\dot{\theta_{j}}$ at time
$t=0$. We can associate a Hamiltonian system on the cotangent space $T^{*}Q$
of the configuration space $Q=\mathbb{T}^{N}$, that is the $N$-torus with a
coordinate chart $\\{\theta_{1},\theta_{2},\cdots,\theta_{N}\\}$. The
conjugate momenta of the angles $\theta_{j}$ are
$p_{j}=\partial_{\dot{\theta}_{j}}\mathcal{L}=I_{j}\dot{\theta}_{j}$. Thus,
the phase space $T^{*}Q$ has the coordinate chart
$\\{\theta_{1},\theta_{2},...,\theta_{N},p_{1},p_{2},...,p_{N}\\}$ and the
Hamiltonian is given by

$\mathcal{H}=\frac{1}{2}\sum_{j=1}^{N}\frac{p_{j}^{2}}{I_{j}}+\Pi\left(\theta_{2}-\theta_{1},\theta_{3}-\theta_{2},...,\theta_{N}-\theta_{N-1}\right)\,.$
(4.7)

The dynamical equations follow from the Hamiltonian and read
$\dot{\mathbf{X}}=\mathbf{J}\nabla_{\mathbf{X}}\mathcal{H}$, where

$\mathbf{X}=\begin{bmatrix}\theta_{1}\\\ \theta_{2}\\\ \vdots\\\ p_{1}\\\
p_{2}\\\ \vdots\end{bmatrix}\,,$ (4.8)

and $\mathbf{J}$ is the following symplectic matrix

$\mathbf{J}=\begin{bmatrix}\mathbf{O}_{N}&\mathbf{I}_{N}\\\
-\mathbf{I}_{N}&\mathbf{O}_{N}\end{bmatrix}\,.$ (4.9)

$\mathbf{I}_{N}=[\delta_{ij}]$ is the $N\times N$ identity matrix,
$\mathbf{O}_{N}$ is the $N\times N$ null matrix, and $\delta_{ij}$ is the
Kronecker tensor. In particular,

$\dot{\theta}_{j}=\frac{p_{j}}{I_{j}}\,,\qquad\dot{p}_{j}=-\frac{\partial\Pi}{\partial\theta_{j}}\,,\quad
j=1,\ldots N\,,$ (4.10)

and from Eq. (5.5) the conserved angular momentum is written as

$\mathsf{A}=\sum_{j=1}^{N}p_{j}(t)\,.$ (4.11)

The associated symplectic $1$ and $2$-forms are written as

$\alpha=\sum_{j=1}^{N}p_{j}\,d\theta_{j},\qquad
d\alpha=\sum_{j=1}^{N}dp_{j}\wedge d\theta_{j}\,.$ (4.12)

The total kinetic energy of the N-pendulum is given by integrating the 1-form
$\alpha$:

$\mathsf{E}(t)=\int_{0}^{t}\,\alpha\,d\tau=\sum_{j=1}^{N}\int_{0}^{t}\,p_{j}(\tau)\,\dot{\theta}_{j}(\tau)\,d\tau=\sum_{j=1}^{N}\frac{1}{2}I_{j}\dot{\theta}^{2}_{j}(t)\,.$
(4.13)

To reveal the geometric nature of the dynamics, we consider the shape
configuration space $Q_{s}$, which has the coordinate chart
$\\{\theta_{1},\psi_{2},\psi_{3},...\psi_{N}\\}$, where the shape parameters
$\psi_{j}=\theta_{j}-\theta_{1}$ represent the relative angular displacement
of the $N-1$ bars with respect to the first bar. Since the total angular
momentum $p_{1}+p_{2}+...+p_{N}=\mathsf{A}$ is known a priori, then
$p_{1}=\mathsf{A}-p_{2}-p_{3}-\ldots-p_{N}$ and the motion must occur on the
subspace $T^{*}Q/\mathsf{A}$, which has the coordinate chart
$\\{\theta_{1},\psi_{2},p_{2},\psi_{3},p_{3},\ldots,\psi_{N},p_{N}\\}$, where
$(p_{j},\psi_{j})$ are a pair of conjugate variables. The $1$-form in Eq.
(4.12) reduces to read

$\alpha=\mathsf{A}\,d\theta_{1}+\sum_{j=2}^{N}p_{j}\,d\psi_{j}\,,$ (4.14)

and the associated symplectic $2$-form is written as

$d\alpha=\sum_{j=2}^{N}dp_{j}\wedge d\psi_{j}\,.$ (4.15)

The reduced phase space $\mathcal{P}=T^{*}Q/\mathsf{A}$ has the geometric
structure of a principal fiber bundle: the $2(N-1)$-dimensional shape manifold
$\mathcal{B}$ with a coordinate chart
$\\{\psi_{2},p_{2},\cdots,\psi_{N},p_{N}\\}$ and transversal one-dimensional
fibers $\mathcal{F}$ with coordinate chart $\\{\theta_{1}\\}$.

The vector field
$\dot{\mathbf{X}}_{s}=\dot{\mathbf{X}}_{\mathcal{B}}+\dot{\mathbf{X}}_{\mathcal{F}}$
can be decomposed as the sum of the flow

$\dot{\mathbf{X}}_{\mathcal{B}}=\begin{bmatrix}\dot{\psi}_{2}\\\
\dot{p}_{2}\\\ \vdots\\\ \dot{\psi}_{N}\\\ \dot{p}_{N}\\\ \end{bmatrix}\,,$
(4.16)

on the shape manifold $\mathcal{B}$ and the flow
$\dot{\mathbf{X}}_{\mathcal{F}}=\dot{\theta_{1}}$ along the fiber
$\mathcal{F}$. Note that the motion on the shape manifold $\mathcal{B}$ is
independent from that along the fiber. As a matter of fact,
$\dot{\mathbf{X}}_{\mathcal{B}}$ does not depend on $\dot{\theta}_{1}$:

$\dot{\psi_{j}}=p_{j}\left(\frac{1}{I_{1}}+\frac{1}{I_{j}}\right)-\frac{1}{I_{1}}\left(\mathsf{A}-\sum_{k=2}^{N}p_{k}\right)\,,\qquad\dot{p}_{j}=\frac{\partial\hat{\Pi}}{\partial\psi_{j}}\,,$
(4.17)

and the associated reduced Hamiltonian is given by

$\mathcal{H}_{R}(\psi_{2},\cdots,\psi_{N},p_{2},\cdots,p_{N})=\frac{1}{2I_{1}}\left(\mathsf{A}-\sum_{k=2}^{N}p_{k}\right)^{2}+\sum_{j=2}^{N}\frac{p_{j}^{2}}{2I_{j}}+\Pi\left(\psi_{2},\psi_{3},...,\psi_{N}\right)\,,$
(4.18)

where the potential is now given by

$\hat{\Pi}\left(\psi_{2},\psi_{3},\cdots,\psi_{N}\right)=\Pi_{2}(\psi_{2})+\Pi_{3}(\psi_{2}-\psi_{1})+\cdots+\Pi_{N}(\psi_{N}-\psi_{N-1})\,.$
(4.19)

On the contrary, the motion along the fiber depends on $\dot{X}_{\mathcal{B}}$
since

$\dot{\theta}_{1}=\frac{1}{I_{1}}\left(\mathsf{A}-\sum_{k=2}^{N}p_{k}\right)\,.$
(4.20)

The motion in the reduced state space $\mathcal{P}=T^{*}Q/\mathsf{A}$
decouples in a reduced motion $\dot{\mathbf{X}}_{\mathcal{B}}$ on the shape
manifold $\mathcal{B}$ and a drift $\dot{\mathbf{X}}_{\mathcal{F}}$ along the
fibers. The reduced motion on $\mathcal{B}$ is the shape-changing evolution of
the connected bobs. Such a shape dynamics induces the bobs to rigidly rotate
together by the varying angle $\theta_{1}$. From Eq. (4.14), we define the
$1$-form $\widetilde{\alpha}=\alpha/\mathsf{A}$ and the total drift
$\theta_{1}$ along the fiber follows by integrating the 1-form

$d\theta_{1}=\widetilde{\alpha}-\sum_{k=2}^{N}\frac{p_{k}}{\mathsf{A}}\,d\psi_{k}\,,$
(4.21)

that is

$\theta_{1}=\int
d\theta_{1}=\int_{0}^{t}\widetilde{\alpha}\,dt-\sum_{k=2}^{N}\int_{\gamma}\frac{p_{k}}{\mathsf{A}}\,d\psi_{k}\,,$
(4.22)

where $\gamma$ is a closed trajectory of the motion up to time $t$ in the
shape manifold $\mathcal{B}$. Thus,

$\theta_{1}(t)=\theta_{\text{dyn}}(t)+\theta_{\text{geom}}(t)\,,$ (4.23)

where the dynamical and geometric rotation drifts are defined as

$\theta_{\text{dyn}}(t)=\int_{0}^{t}\widetilde{\alpha}\,d\tau,\qquad\theta_{\text{geom}}(t)=-\sum_{k=2}^{N}\int_{\gamma}\frac{p_{k}}{\mathsf{A}}\,d\psi_{k}\,.$
(4.24)

Here, the dynamical rotation drift $\theta_{\text{dyn}}$ depends on the
inertia of the bobs and can be written as

$\theta_{\text{dyn}}(t)=2\int_{0}^{t}\frac{\mathsf{K}(\tau)}{\mathsf{A}}\,d\tau\,,\qquad\mathsf{K}(t)=\frac{1}{2}\sum_{j=1}^{N}\frac{p_{j}^{2}}{I_{j}}\,,$
(4.25)

where $\mathsf{K}(t)$ is the total kinetic energy and $\mathsf{A}$ is the
total angular momentum. If all the bobs are rigidly connected and cannot
change their shape, i.e., $\dot{\psi_{j}}=0$ and so no motion on the base
manifold $\mathcal{B}$, then the rotation drift is solely due to the inertia
of the system measured by the total angular momentum. If the bobs undergo
changes in shape, i.e., the angles $\psi_{j}$ vary over time, then the motion
on $\mathcal{B}$ induces the geometric rotation drift, which from Eq.(4.24)
can be written as

$\theta_{\text{geom}}=-\sum_{k=2}^{N}\int_{\gamma}\frac{p_{k}}{\mathsf{A}}\,d\psi_{k}=-\sum_{k=2}^{N}\int_{S(\gamma)}\frac{1}{\mathsf{A}}\,dp_{k}\wedge\,d\psi_{k}\,.$
(4.26)

Such a rotation drift is proportional to the area $S(\gamma)$ enclosed by the
path $\gamma$ spanned by the motion on the shape manifold $\mathcal{B}$. Thus,
it is purely geometric since it does not depend on the time it takes for the
bobs to undergo a cyclic shape change, or to span the closed path $\gamma$ on
$\mathcal{B}$. The part of kinetic energy that arises from the same cyclic
shape change

$\mathsf{E}_{\text{geom}}(t)=-\int_{0}^{t}\mathsf{A}\,\dot{\theta}_{\text{geom}}(\tau)\,d\tau=\sum_{k=2}^{N}\int_{\gamma}p_{k}\,d\psi_{k}=\sum_{k=2}^{N}\int_{S(\gamma)}dp_{k}\wedge\,d\psi_{k}\,,$
(4.27)

does not depend on the duration of the cyclic change either. Note that the
energy difference

$\mathsf{E}(t)-\mathsf{E}_{\text{geom}}(t)=\int_{0}^{t}\mathsf{A}\,\dot{\theta}_{1}\,d\tau\,,$
(4.28)

is that relative to the total rotation drift $\theta_{1}$. In the following,
we will show that the base manifold can be endowed with a Riemannian
structure.

###  4.1 Curvature and intrinsic metric of the shape manifold

The geometric rotation drift can be interpreted as the curvature of the shape
manifold $\mathcal{B}$ equipped with a pseudo-Riemannian diagonal metric of
the form

$ds^{2}=\sum_{j=2}^{N}\left[\epsilon_{p_{j}}G_{p_{j}}(dp_{j})^{2}+\epsilon_{\psi_{j}}G_{\psi_{j}}(d\psi_{j})^{2}\right]\,,$
(4.29)

where the $2(N-1)$ positive metric coefficients depend on the coordinates
$\\{p_{2},\psi_{2},\cdots,p_{N},\psi_{N}\\}$, in general, and
$(\epsilon_{p_{2}},\epsilon_{\psi_{2}},\cdots)$ is the signature of the
manifold. The metric coefficients will be calculated using Cartan’s structural
equations as follows. From Eq. (4.26) the geometric drift follows by
integrating the $2$-form

$d\,\widetilde{\alpha}=\sum_{j=2}^{N}d\,\widetilde{\alpha}_{j}\,,$ (4.30)

where

$\widetilde{\alpha}_{j}=-\frac{p_{j}}{\mathsf{A}}\,d\psi_{j}\,,\qquad
d\,\widetilde{\alpha}_{j}=-\frac{1}{\mathsf{A}}\,dp_{j}\wedge\,d\psi_{j}\\\
,\quad j=2,\cdots,N\,.$ (4.31)

Drawing on Cartan’s second structural equations (2.43), the collection of the
$(N-1)$ $2$-forms $d\,\widetilde{\alpha}_{j}$ are interpreted as the non-zero
curvature $2$-forms of a $2(N-1)$-dimensional manifold, and the associated
connections 1-forms are $\widetilde{\alpha}_{j}$. For a metric-compatible
connection on the $M=2(N-1)$-dimensional shape manifold there are
$M(M-1)/2=(N-1)(2N-3)$ connection $1$-forms and as many curvature $2$-forms.
In particular,

$\omega^{p_{j}}{}_{p_{k}}\,,~{}\omega^{\psi_{j}}{}_{\psi_{k}}\,,\quad
j<k=2,\ldots,N\,,$ (4.32)

are $2\times\frac{(N-1)(N-2)}{2}=(N-1)(N-2)$ connection $1$-forms. The
remaining $(N-1)^{2}$ connection $1$-forms are

$\omega^{\psi_{k}}{}_{p_{j}}\,,\quad j,\,k=2,\ldots,N\,.$ (4.33)

Therefore, in total we have $(N-1)(N-2)+(N-1)^{2}=(N-1)(2N-3)$ connection
$1$-forms and as many curvature $2$-forms given by

$\displaystyle\mathcal{R}^{p_{j}}{}_{p_{k}}=\mathcal{R}^{\psi_{j}}{}_{\psi_{k}}=0\,,$
$\displaystyle j<k=2,\ldots,N\,,$ (4.34)
$\displaystyle\mathcal{R}^{\psi_{k}}{}_{p_{j}}=0\,,$ $\displaystyle j\neq
k\,,$
$\displaystyle\mathcal{R}^{\psi_{j}}{}_{p_{j}}=d\,\widetilde{\alpha}_{j}=-\frac{1}{\mathsf{A}}\,dp_{j}\wedge\,d\psi_{j}\,,$
$\displaystyle j=2,\ldots,N\,.$

The unknown connection $1$-forms satisfy Cartan’s second structural equations
(2.43)

$\displaystyle\mathcal{R}^{p_{j}}{}_{p_{k}}=0=d\omega^{p_{j}}{}_{p_{k}}+\omega^{p_{j}}{}_{\gamma}\wedge\omega^{\gamma}{}_{p_{k}}\,,$
$\displaystyle j<k=2,\ldots,N\,,$ (4.35)
$\displaystyle\mathcal{R}^{\psi_{j}}{}_{\psi_{k}}=0=d\omega^{\psi_{j}}{}_{\psi_{k}}+\omega^{\psi_{j}}{}_{\gamma}\wedge\omega^{\gamma}{}_{\psi_{k}}\,,$
$\displaystyle j<k=2,\ldots,N\,,$
$\displaystyle\mathcal{R}^{\psi_{k}}{}_{p_{j}}=0=d\omega^{\psi_{k}}{}_{p_{j}}+\omega^{\psi_{k}}{}_{\gamma}\wedge\omega^{\gamma}{}_{p_{j}}\,,$
$\displaystyle j\neq k\,,$
$\displaystyle\mathcal{R}^{\psi_{j}}{}_{p_{j}}=-\frac{1}{\mathsf{A}}\,dp_{j}\wedge\,d\psi_{j}=d\omega^{\psi_{j}}{}_{p_{j}}+\omega^{\psi_{j}}{}_{\gamma}\wedge\omega^{\gamma}{}_{p_{j}}\,,$
$\displaystyle j=2,\ldots,N\,.$

These are more explicitly written as

$\displaystyle
d\omega^{p_{j}}{}_{p_{k}}+\sum_{i=2}^{N}\omega^{p_{j}}{}_{\psi_{i}}\wedge\omega^{\psi_{i}}{}_{p_{k}}+\sum_{i=2}^{N}\omega^{p_{j}}{}_{p_{i}}\wedge\omega^{p_{i}}{}_{p_{k}}=0\,,$
$\displaystyle j<k=2,\ldots,N\,,$ (4.36) $\displaystyle
d\omega^{\psi_{j}}{}_{\psi_{k}}+\sum_{i=2}^{N}\omega^{\psi_{j}}{}_{\psi_{i}}\wedge\omega^{\psi_{i}}{}_{\psi_{k}}+\sum_{i=2}^{N}\omega^{\psi_{j}}{}_{p_{i}}\wedge\omega^{p_{i}}{}_{\psi_{k}}=0\,,$
$\displaystyle j<k=2,\ldots,N\,,$ $\displaystyle
d\omega^{\psi_{k}}{}_{p_{j}}+\sum_{i=2}^{N}\omega^{\psi_{k}}{}_{\psi_{i}}\wedge\omega^{\psi_{i}}{}_{p_{j}}+\sum_{i=2}^{N}\omega^{\psi_{k}}{}_{p_{i}}\wedge\omega^{p_{i}}{}_{p_{j}}=0\,,$
$\displaystyle j\neq k\,,$ $\displaystyle
d\omega^{\psi_{j}}{}_{p_{j}}+\sum_{i=2}^{N}\omega^{\psi_{j}}{}_{\psi_{i}}\wedge\omega^{\psi_{i}}{}_{p_{j}}+\sum_{i=2}^{N}\omega^{\psi_{j}}{}_{p_{i}}\wedge\omega^{p_{i}}{}_{p_{j}}=-\frac{1}{\mathsf{A}}\,dp_{j}\wedge\,d\psi_{j}\,,$
$\displaystyle j=2,\ldots,N\,.$

The case $N=2$ is trivial as there is a unique solution
$\omega^{\psi_{2}}{}_{p_{2}}=\frac{\psi_{2}}{\mathsf{A}}\,dp_{2}+\xi$ given in
Eq. (3.43), where $\xi$ is any closed $1$-form, which can be neglected as it
does not contribute to the geometric phase. For $N>2$ we have a system of
nonlinear equations to solve for the connection $1$-forms, and there may be
more than one solution. If we require the only non-zero connection forms to be
$\widetilde{\alpha}_{j},\,\,j=2,\ldots,N$ in Eq. (4.31) then we have a
solution222One can add arbitrary closed $1$-forms to each connection $1$-form,
but these do not yield new solutions since the difference is a closed
$1$-form. They can be neglected as they are not physically relevant. As a
matter of fact, they do not contribute to the geometric phase as their
integrals over any closed curve vanish. Thus, the freedom to add an arbitrary
closed $1$-form is physically inconsequential and it does not give any new
solutions. that follows from Eq. (4.32) and (4.33) as
$\omega^{\psi_{j}}{}_{p_{j}}=-\frac{p_{j}}{\mathsf{A}}\,d\psi_{j}$,
$j=2,\ldots,N$, and
$\omega^{p_{k}}{}_{\psi_{j}}=\omega^{p_{k}}{}_{p_{j}}=\omega^{\psi_{k}}{}_{\psi_{j}}=0$.
From Eq. (4.30), it follows that the non-zero curvature 2-forms are the
exterior derivatives of the $1$-forms $\widetilde{\alpha}_{j}$

$\mathcal{R}^{\psi_{j}}{}_{p_{j}}=d\omega^{\psi_{j}}{}_{p_{j}}=d\,\widetilde{\alpha}_{j}\,,\quad
j=2,\ldots,N\,.$ (4.37)

From Eq. (2.40) we then have

$\displaystyle\omega^{p_{k}}{}_{\psi_{j}}=\frac{\partial_{\psi_{j}}\mathsf{G}_{p_{k}}}{2\sqrt{\mathsf{G}_{p_{k}}\mathsf{G}_{\psi_{j}}}}\,dp_{k}-\epsilon_{p_{k}}\epsilon_{\psi_{j}}\,\frac{\partial_{p_{k}}\mathsf{G}_{\psi_{j}}}{2\sqrt{\mathsf{G}_{p_{k}}\mathsf{G}_{\psi_{j}}}}\,d\psi_{j}=0\,,$
$\displaystyle j\neq k\,,$ (4.38)
$\displaystyle\omega^{\psi_{k}}{}_{\psi_{j}}=\frac{\partial_{\psi_{j}}\mathsf{G}_{\psi_{k}}}{2\sqrt{\mathsf{G}_{\psi_{k}}\mathsf{G}_{\psi_{j}}}}\,d\psi_{k}-\epsilon_{\psi_{k}}\epsilon_{\psi_{j}}\,\frac{\partial_{\psi_{k}}\mathsf{G}_{\psi_{j}}}{2\sqrt{\mathsf{G}_{\psi_{k}}\mathsf{G}_{\psi_{j}}}}\,d\psi_{j}=0\,,$
$\displaystyle j<k=2,\ldots,N\,,$
$\displaystyle\omega^{p_{k}}{}_{p_{j}}=\frac{\partial_{p_{j}}\mathsf{G}_{p_{k}}}{2\sqrt{\mathsf{G}_{p_{k}}\mathsf{G}_{p_{j}}}}\,dp_{k}-\epsilon_{p_{k}}\epsilon_{p_{j}}\,\frac{\partial_{p_{k}}\mathsf{G}_{p_{j}}}{2\sqrt{\mathsf{G}_{p_{k}}\mathsf{G}_{p_{j}}}}\,dp_{j}=0\,,$
$\displaystyle j<k=2,\ldots,N\,,$
$\displaystyle\omega^{p_{j}}{}_{\psi_{j}}=\frac{\partial_{\psi_{j}}\mathsf{G}_{p_{j}}}{2\sqrt{\mathsf{G}_{p_{j}}\mathsf{G}_{\psi_{j}}}}\,dp_{j}-\epsilon_{p_{j}}\epsilon_{\psi_{j}}\,\frac{\partial_{p_{j}}\mathsf{G}_{\psi_{j}}}{2\sqrt{\mathsf{G}_{p_{j}}\mathsf{G}_{\psi_{j}}}}\,d\psi_{j}=-\frac{p_{j}}{\mathsf{A}}\,d\psi_{j}\,,$
$\displaystyle j=2,\ldots,N\,.$

Thus, we must have

$\partial_{\psi_{k}}\mathsf{G}_{p_{j}}=0,\qquad\partial_{p_{k}}\mathsf{G}_{p_{j}}=0,\qquad\partial_{\psi_{k}}\mathsf{G}_{\psi_{j}}=0,\qquad\partial_{p_{k}}\mathsf{G}_{\psi_{j}}=0,\quad
k\neq j\,,$ (4.39)

and

$\partial_{\psi_{j}}\mathsf{G}_{p_{j}}=0,\qquad-\epsilon_{p_{j}}\epsilon_{\psi_{j}}\,\frac{\partial_{p_{j}}\mathsf{G}_{\psi_{j}}}{2\sqrt{\mathsf{G}_{p_{j}}\mathsf{G}_{\psi_{j}}}}=-\frac{p_{j}}{\mathsf{A}}\,.$
(4.40)

The above relations imply that $\mathsf{G}_{p_{j}}=\mathsf{G}_{p_{j}}(p_{j})$
and $\mathsf{G}_{\psi_{j}}=\mathsf{G}_{\psi_{j}}(\psi_{j},p_{j})$. Thus, the
metric coefficients $G_{p_{j}}$ and $G_{\psi_{j}}$ depend only on the
coordinates $\\{\psi_{j},p_{j}\\}$ of the submanifold (hyper-plane)
$\mathcal{B}_{j}$. The curvature $2$-forms $\mathcal{R}^{\psi_{j}}{}_{p_{j}}$
in Eq. (4.37) are now expressed in terms of the metric coefficients using Eq.
(2.48) as

$\mathcal{R}^{\psi_{j}}{}_{p_{j}}=d\omega^{\psi_{j}}{}_{p_{j}}=K_{(p_{j},\psi_{j})}\sqrt{\mathsf{G}_{p_{j}}\mathsf{G}_{\psi_{j}}}\,dp_{j}\wedge
d\psi_{j}\,,$ (4.41)

where $K_{(p_{j},\psi_{j})}$ is the Gaussian curvature of the hyper-plane
$\mathcal{B}_{j}$ with coordinates $\\{\psi_{j},p_{j}\\}$ (see Eq. (2.50)).
Then, Eq. (4.37) imposes the equality of the $2$-forms
$d\,\widetilde{\alpha}_{j}=d\omega^{\psi_{j}}{}_{p_{j}}$, that is

$-\frac{1}{\mathsf{A}}\,dp_{j}\wedge
d\psi_{j}=K_{(p_{j},\psi_{j})}\sqrt{\mathsf{G}_{p_{j}}\mathsf{G}_{\psi_{j}}}\,dp_{j}\wedge
d\psi_{j}\,,$ (4.42)

and it follows that

$K_{(p_{j},\psi_{j})}=-\frac{\sqrt{\mathsf{G}_{p_{j}}\mathsf{G}_{\psi_{j}}}}{\mathsf{A}}\,.$
(4.43)

Similar to the $2$-pendulum, we further require that the symplectic $2$-form
$d\,\widetilde{\alpha}_{j}$ is compatible with the (pseudo) Riemannian volume
(area) $2$-form
$\mathsf{vol}_{\mathcal{B}_{j}}=\sqrt{\mathsf{G}_{p_{j}}\mathsf{G}_{\psi_{j}}}\,dp_{j}\wedge
d\psi_{j}$ of the submanifold $\mathcal{B}_{j}$, that is

$\frac{1}{|\mathsf{A}|}\,dp_{j}\wedge
d\psi_{j}=\sqrt{\mathsf{G}_{p_{j}}\mathsf{G}_{\psi_{j}}}\,dp_{j}\wedge
d\psi_{j}\,.$ (4.44)

This implies that

$\frac{1}{|\mathsf{A}|}=\sqrt{\mathsf{G}_{p_{j}}\mathsf{G}_{\psi_{j}}}\,,$
(4.45)

which together with the second equation of Eq. (4.40) gives us

$\epsilon_{p_{j}}\epsilon_{\psi_{j}}\,\partial_{p_{j}}\mathsf{G}_{\psi_{j}}=2\frac{p_{j}}{\mathsf{A}|\mathsf{A}|}\,.$
(4.46)

Solving for $\mathsf{G}_{\psi_{j}}$, and using Eq. (4.45) to solve for
$\mathsf{G}_{p_{j}}$ we get

$\mathsf{G}_{p_{j}}=\frac{1}{{p_{j}^{2}+\mu_{j}^{2}}},\qquad\mathsf{G}_{\psi_{j}}=\frac{p_{j}^{2}+\mu_{j}^{2}}{\mathsf{A}^{2}},\qquad\epsilon_{p_{j}}=\mathrm{sgn}(\mathsf{A}),\qquad\epsilon_{\psi_{j}}=1\,,$
(4.47)

where we have imposed that both metric coefficients are positive and $\mu_{j}$
are arbitrary constants. The shape manifold $\mathcal{B}$ is thus the product
manifold of $(N-1)$ shape submanifolds $\mathcal{B}_{j}$ with local coordinate
charts $\\{p_{j},\psi_{j}\\}$, or
$\mathcal{B}=\mathcal{B}_{2}\times\cdots\times\mathcal{B}_{N}$ (see §2.5).

Each submanifold is the shape space of two adjacent bobs, or a two-pendulum.
Thus, the intrinsic metric of each submanifold follows from Eq. (4.47), (or
from Eq. (3.53)) as

$\mathbf{G}_{j}=\frac{\mathrm{sgn}(\mathsf{A})}{p_{j}^{2}+\mu_{j}^{2}}\,dp_{j}\otimes
dp_{j}+\frac{p_{j}^{2}+\mu_{j}^{2}}{\mathsf{A}^{2}}\,d\psi_{j}\otimes
d\psi_{j}\,.$ (4.48)

Then the metric of $\mathcal{B}$ is the product of these metrics, i.e.,

$\mathbf{G}=\mathbf{G}_{2}\times\ldots\times\mathbf{G}_{N}=\sum_{j=2}^{N}\left[\frac{\mathrm{sgn}(\mathsf{A})}{p_{j}^{2}+\mu_{j}^{2}}\,dp_{j}\otimes
dp_{j}+\frac{p_{j}^{2}+\mu_{j}^{2}}{\mathsf{A}^{2}}\,d\psi_{j}\otimes
d\psi_{j}\right]\,.$ (4.49)

From Eq. (4.30) the geometric drift follows by integrating the $2$-form

$d\,\widetilde{\alpha}=\sum_{j=2}^{N}\,\mathcal{R}^{p_{j}}{}_{\psi_{j}}(\mathbf{e}_{p_{j}},\mathbf{e}_{\psi_{j}})\,.$
(4.50)

This is the sum of the curvature $2$-forms of each submanifold
$\mathcal{B}_{j}$, that is

$\theta_{\text{geom}}=-\sum_{j=2}^{N}\int_{S(\gamma)}\mathcal{R}^{p_{j}}{}_{\psi_{j}}(\mathbf{e}_{p_{j}},\mathbf{e}_{\psi_{j}})=-\sum_{j=2}^{N}\int_{S(\gamma)}\frac{1}{\mathsf{A}}\,dp_{j}\wedge\,d\psi_{j}\,,$
(4.51)

where each term is both the oriented area and curvature of the projected path
$\gamma$ on the hyper-plane $\mathcal{B}_{j}$ with coordinates
$\\{\psi_{j},p_{j}\\}$. The geodesics of the product manifold $\mathcal{B}$
are the Cartesian product of the geodesics of each submanifold
$\mathcal{B}_{j}$, which follow from Eq. (3.59).

###  4.2 Negative angular momentum: Multi-universe

For $\mathsf{A}<0$, the metric (4.49) describes a multi-universe of expanding
$2$D Robertson-Walker spacetime universes (Misner et al., 1973; Carroll,
2003). Indeed, for $\mu_{j}\neq 0$ we use the chart coordinate transformation
(3.61)

$t_{j}=\tanh^{-1}\left[\frac{p_{j}}{\sqrt{p_{j}^{2}+\mu_{j}^{2}}}\right],\qquad
x_{j}=\psi_{j}\,,\quad j=2,\cdots,N\,.$ (4.52)

Then $dt_{j}=dp_{j}/\sqrt{p_{j}^{2}+\mu_{j}^{2}}$, $dx_{j}=d\psi_{j}$, and the
metric (4.49) transforms into the sum of $(N-2)$ $2$-dimensional Robertson-
Walker metrics of each submanifold $\mathcal{B}_{j}$

$ds^{2}=\sum_{j=2}^{N}ds^{2}_{j},\quad\quad\,ds^{2}_{j}=-dt_{j}^{2}+\frac{\mu_{j}^{2}(\cosh{t_{j}})^{2}}{\mathsf{A}^{2}}\,dx_{j}^{2}\,,\quad
j=2,\cdots,N\,,$ (4.53)

with the scale factor $a(t_{j})\sim\cosh{t_{j}}$. The associated Hubble
constants of each spacetime is $H_{j}=\dot{a}_{j}/a_{j}=\tanh(t_{j})$,
indicating a matter-dominated universe for small $t_{j}$ and vacuum-dominated
for large $t_{j}$ (Carroll, 2003). Similarly, if $\mu_{j}=0$ we use the
coordinate transformation (3.63)

$t_{j}=\mathrm{e}^{p_{j}},\qquad x_{j}=\psi_{j}\,,\quad j=2,\cdots,N\,,$
(4.54)

where $dt_{j}=dp_{j}/p_{j}$ and $dx_{j}=d\psi_{j}$, and the metric transforms
to

$ds^{2}_{j}=-dt_{j}^{2}+\frac{\mathrm{e}^{2t_{j}}}{\mathsf{A}^{2}}dx_{j}^{2}\,,\quad
j=2,\cdots,N\,,$ (4.55)

which is still a Robertson-Walker metric with the scale factor
$a(t_{j})\sim\mathrm{e}^{t}_{j}$ and Hubble constant $H=\dot{a}_{j}/a_{j}=1$
(Carroll, 2003).

###  4.3 Positive angular momentum: The hyperbolic product space
$\mathbb{H}^{2(N-1)}$

For $\mathsf{A}>0$, each submanifold $\mathcal{B}_{j}$ is a hyperbolic plane
$\mathbb{H}^{2}$ and the shape manifold is the Cartesian product of $N-1$
hyperbolic planes $\mathbb{H}^{2}$, each with negative Gaussian curvature
$\mathsf{K}_{j}=-1$. So, $\mathcal{B}$ is the hyperbolic product space
$\mathbb{H}^{2(N-1)}$. As an example, for $\mu=0$ we use the coordinate
transformations (3.61), (3.63) and the metric (4.49) transforms into the sum
of $N-2$ metrics

$ds^{2}=\sum_{j=2}^{N}ds^{2}_{j},\quad\quad\,ds^{2}_{j}=dt_{j}^{2}+R_{j}^{2}(t)\,dx_{j}^{2}\,,\quad
j=2,\cdots,N\,,$ (4.56)

where $R_{j}(t)=\frac{\mathrm{e}^{2t_{j}}}{\mathsf{A}^{2}}$. The metrics
$ds^{2}_{j}$ of the submanifolds $\mathcal{B}_{j}$ are disguised metrics of
the hyperbolic plane as the change of coordinates
$\tilde{x}_{j}=x_{j}/\sqrt{\mathsf{A}}$ and
$\tilde{y}_{j}=\mathsf{A}\exp(-t_{j})$ transform each of them into

$ds^{2}_{j}=\frac{d\tilde{x}_{j}^{2}+d\tilde{y}_{j}^{2}}{\tilde{y}_{j}^{2}}\,,\quad
j=2,\cdots,N\,.$ (4.57)

##  5 Dynamics of nonlinear planar $N$-pendula under self-equilibrated
external moments

We next generalize the $N$-pendulum system described above by assuming that
time-dependent external moments $\mathsf{M}^{e}_{j}(t)$, $j=1,\cdots,N$ act on
the rigid bars (see Fig. 5). In order to preserve the invariance of the total
angular momentum we assume that

$\sum_{j=1}^{N}\mathsf{M}^{e}_{j}(t)=0\,.$ (5.1)

The associated Lagrangian is333The external moments appear in the Lagrange
d’Alembert principle. Equivalently, one can use Hamilton’s principle using the
modified Lagrangian given in (5.2).

$\mathcal{L}=\sum_{j=1}^{N}\frac{1}{2}I_{j}\,\dot{\theta}_{j}^{2}-\Pi\left(\theta_{2}-\theta_{1},\theta_{3}-\theta_{2},...,\theta_{N}-\theta_{N-1}\right)+\sum_{j=1}^{N}\mathsf{M}^{e}_{j}(t)\,\theta_{j}\,,$
(5.2)

and the associated dynamical equations follow by minimizing the action
$\int\mathcal{L}dt$ as

$\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial\dot{\theta}_{j}}-\frac{\partial\mathcal{L}}{\partial\theta_{j}}=I_{j}\ddot{\theta}_{j}+\frac{\partial\Pi}{\partial\theta_{j}}-\mathsf{M}^{e}_{j}(t)=0\,,\quad
j=1,\cdots N\,.$ (5.3)

From (4.2) potential moments are in equilibrium and summing up equations (5.3)
yields

$\frac{d}{dt}\sum_{j=1}^{N}I_{j}\dot{\theta}_{j}(t)=\sum_{j=1}^{N}\mathsf{M}^{e}_{j}(t)\,.$
(5.4)

From Eq. (5.1) the sum of the external moments on the right-hand side is null
and the total angular momentum is conserved, i.e.,

$I_{1}\dot{\theta}_{1}(t)+I_{2}\dot{\theta}_{2}(t)+\cdots+I_{N}\dot{\theta_{N}}(t)=\mathsf{A}\,.$
(5.5)

###  5.1 Extended autonomous Hamiltonian system

The $N$-pendulum is a non-autonomous system since the Lagrangian is explicitly
time-dependent. We can associate an extended autonomous Hamiltonian system on
the cotangent bundle $T^{*}Q_{t}$ of the extended configuration space
$Q_{t}=\mathbb{R}\times\mathbb{T}^{N}$, i.e., the Cartesian product of the
real line $\mathbb{R}$ and the $N$-torus. $Q_{t}$ has the coordinate chart
$\\{t,\theta_{1},\theta_{2},\cdots,\theta_{N}\\}$. The conjugate momentum of
time $t$ is the energy $E$ and $p_{j}=I_{j}\dot{\theta}_{j}$ are the conjugate
momenta of the angles $\theta_{j}$. Thus, the phase space $T^{*}Q_{t}$ is the
cotangent space of $Q_{t}$, and has the coordinate chart
$\\{t,\theta_{1},\theta_{2},...,\theta_{N},E,p_{1},p_{2},...,p_{N}\\}$. A
generic trajectory in the extended phase space is parameterized by the
parameter $\lambda$. The Hamiltonian is given by

$\mathcal{H}=E+\frac{1}{2}\sum_{j=1}^{N}\frac{p_{j}^{2}}{I_{j}}+\Pi\left(\theta_{2}-\theta_{1},\theta_{3}-\theta_{2},...,\theta_{N}-\theta_{N-1}\right)-\sum_{j=1}^{N}\mathsf{M}^{e}_{j}\,\theta_{j}\,.$
(5.6)

The dynamical equations follow from the Hamiltonian by
$\mathbf{X}^{\prime}=\mathbf{J}\nabla_{\mathbf{X}}\mathcal{H}$, where
$\mathbf{X}^{\prime}=\frac{d\mathbf{X}}{d\lambda}$ denotes differentiation
with respect to $\lambda$, and

$\mathbf{X}=\begin{bmatrix}t\\\ \theta_{1}\\\ \theta_{2}\\\ \vdots\\\ E\\\
p_{1}\\\ p_{2}\\\ \vdots\end{bmatrix}\,,$ (5.7)

and $\mathbf{J}$ is the symplectic matrix

$\mathbf{J}=\begin{bmatrix}\mathbf{O}_{N+1}&\mathbf{I}_{N+1}\\\
-\mathbf{I}_{N+1}&\mathbf{O}_{N+1}\end{bmatrix}\,.$ (5.8)

$\mathbf{I}_{N+1}=[\delta_{ij}]$ is the $N+1\times N+1$ identity matrix,
$\mathbf{O}_{N+1}$ is the $N+1\times N+1$ null matrix, and $\delta_{ij}$ is
the Kronecker tensor. In particular,

$t^{\prime}=\frac{\partial\mathcal{H}}{\partial E}=1\,,\quad
E^{\prime}=-\frac{\partial\mathcal{H}}{\partial
t}=\sum_{j=1}^{N}\frac{d\mathsf{M}^{e}_{j}}{dt}\theta_{j}\,,\quad\theta_{j}^{\prime}=\frac{p_{j}}{I_{j}}\,,\quad
p_{j}^{\prime}=-\frac{\partial\Pi}{\partial\theta_{j}}+\mathsf{M}^{e}_{j}(t)\,,\quad
j=1,\ldots N\,.$ (5.9)

From Eq. (5.5) the conserved total angular momentum is
$\mathsf{A}=\sum_{j=1}^{N}p_{j}$. The associated symplectic $1$ and $2$-forms
are

$\alpha=Edt+\sum_{j=1}^{N}p_{j}\,d\theta_{j},\qquad d\alpha=dE\wedge
dt+\sum_{j=1}^{N}dp_{j}\wedge d\theta_{j}\,.$ (5.10)

We now consider the shape configuration space $Q_{s}$, which has the
coordinate chart $\\{t,\theta_{1},\psi_{2},\psi_{3},...\psi_{N}\\}$, where the
shape parameters $\psi_{j}=\theta_{j}-\theta_{1}$ represent the relative
angular displacement of the $N-1$ rigid bars with respect to the first bar.
Since the total angular momentum $p_{1}+p_{2}+...+p_{N}=\mathsf{A}$ is known a
priori, then $p_{1}=\mathsf{A}-p_{2}-p_{3}-\ldots-p_{N}$ and the motion must
occur on the subspace $T^{*}Q_{t}/\mathsf{A}$, which has the coordinate chart
$\\{t,E,\theta_{1},\psi_{2},p_{2},\psi_{3},p_{3},\ldots,\psi_{N},p_{N}\\}$,
where $(t,E)$ and $(p_{j},\psi_{j})$ are pairs of conjugate variables. The
$1$-form in Eq. (4.12) reduces to

$\alpha=\mathsf{A}\,d\theta_{1}+Edt+\sum_{j=2}^{N}p_{j}\,d\psi_{j}\,,$ (5.11)

and the associated symplectic $2$-form is written as

$d\alpha=dE\wedge dt+\sum_{j=2}^{N}dp_{j}\wedge d\psi_{j}\,.$ (5.12)

The reduced phase space $\mathcal{P}=T^{*}Q/\mathsf{A}$ has the geometric
structure of a principal fiber bundle: the $2N$-dimensional shape manifold
$\mathcal{B}$ with coordinate chart
$\\{t,E,\psi_{2},p_{2},\cdots,\psi_{N},p_{N}\\}$ and transversal one-
dimensional fibers $\mathcal{F}$ with coordinate $\\{\theta_{1}\\}$. The
Hamiltonian vector field $\mathbf{X}_{s}^{\prime}$ in
$\mathcal{P}=T^{*}Q/\mathsf{A}$ can be decomposed as the sum of the flow

$\mathbf{X}^{\prime}_{\mathcal{B}}=\begin{bmatrix}t^{\prime}\\\ E^{\prime}\\\
\psi_{2}^{\prime}\\\ p_{2}^{\prime}\\\ \vdots\\\ \psi_{N}^{\prime}\\\
p_{N}^{\prime}\end{bmatrix}\,,$ (5.13)

on the shape manifold $\mathcal{B}$ and the flow
$\mathbf{X}^{\prime}_{\mathcal{F}}=\theta_{1}^{\prime}$ along the fiber
$\mathcal{F}$. The dynamics on the shape manifold $\mathcal{B}$ is governed by
the reduced (time-varying) Hamiltonian

$\mathcal{H}_{R}(t,E,\psi_{2},p_{2},\cdots,\psi_{N},p_{N})=E+\frac{1}{2I_{1}}\left[\mathsf{A}-\sum_{k=2}^{N}p_{k}\right]^{2}+\sum_{j=2}^{N}\left[\frac{p_{j}^{2}}{2I_{j}}-\mathsf{M}^{e}_{j}(t)\psi_{j}\right]+\Pi\left(\psi_{2},\psi_{3},...,\psi_{N}\right)\,,$
(5.14)

and the components of the Hamiltonian vector field
$\mathbf{X}^{\prime}_{\mathcal{B}}$ are

$t^{\prime}=1\,,\quad E^{\prime}=-\frac{\partial\mathcal{H}_{R}}{\partial
t}=\sum_{j=1}^{N}\frac{d\mathsf{M}^{e}_{j}}{dt}\psi_{j}\,,\quad\psi_{j}^{\prime}=p_{j}\left(\frac{1}{I_{1}}+\frac{1}{I_{j}}\right)-\frac{1}{I_{1}}\left(\mathsf{A}-\sum_{k=2}^{N}p_{k}\right)\,,\quad
p_{j}^{\prime}=\frac{\partial\hat{\Pi}}{\partial\psi_{j}}-\mathsf{M}^{e}_{j}(t)\,.$
(5.15)

Notice that the motion along the fiber depends on
$\mathbf{X}^{\prime}_{\mathcal{B}}$ since

$\theta_{1}^{\prime}=\frac{1}{I_{1}}\left(\mathsf{A}-\sum_{k=2}^{N}p_{k}\right)\,.$
(5.16)

From Eq. (4.14), we define the $1$-form $\widetilde{\alpha}=\alpha/\mathsf{A}$
and the total drift $\theta_{1}$ along the fiber follows by integrating the
form

$d\theta_{1}=\widetilde{\alpha}-\frac{E}{\mathsf{A}}dt-\sum_{k=2}^{N}\frac{p_{k}}{\mathsf{A}}\,d\psi_{k}\,,$
(5.17)

that is

$\theta_{1}=\int
d\theta_{1}=\int_{0}^{\lambda}\widetilde{\alpha}\,d\widetilde{\lambda}-\int_{\gamma}\left(\frac{E}{\mathsf{A}}dt+\sum_{k=2}^{N}\frac{p_{k}}{\mathsf{A}}\,d\psi_{k}\right)\,,$
(5.18)

where $\gamma$ is a closed trajectory of the motion on the shape manifold
$\mathcal{B}$ parameterized by $\lambda$. Thus,

$\theta_{1}=\theta_{\text{dyn}}+\theta_{\text{geom}}\,,$ (5.19)

where the dynamical and geometric rotation drifts are defined as

$\theta_{\text{dyn}}(\lambda)=\int_{0}^{\lambda}\widetilde{\alpha}\,d\widetilde{\lambda},\qquad\theta_{\text{geom}}(\lambda)=-\int_{\gamma}\left(\frac{E}{\mathsf{A}}dt+\sum_{k=2}^{N}\frac{p_{k}}{\mathsf{A}}\,d\psi_{k}\right)\,.$
(5.20)

Here, the dynamical rotation drift $\theta_{\text{dyn}}$ depends on the
inertia of the $N$-pendulum and can be written as

$\theta_{\text{dyn}}(\lambda)=2\int_{0}^{\lambda}\frac{\mathsf{K}(\widetilde{\lambda})+E(\widetilde{\lambda})}{\mathsf{A}}\,d\widetilde{\lambda}\,,\qquad\mathsf{K}=\frac{1}{2}\sum_{j=1}^{N}\frac{p_{j}^{2}}{I_{j}}\,,$
(5.21)

where $\mathsf{K}$ and $\mathsf{A}$ are the total kinetic energy and the total
angular momentum, respectively. If the bars of the $N$-pendulum are rigidly
connected and cannot change their shape, i.e., no motion on the shape manifold
as $\psi_{j}^{\prime}=0$, then the rotation drift is solely due to the inertia
of the system measured by the total angular momentum and it is measured by
$\theta_{\text{dyn}}$. If the $N$-pendulum changes its shape, i.e., the angles
$\psi_{j}$ vary over time, then the motion on $\mathcal{B}$ induces also the
geometric rotation drift $\theta_{\text{geom}}$. From Eq.(5.20), and using
Stokes’ theorem

$\theta_{\text{geom}}=-\int_{S(\gamma)}\left(\frac{1}{\mathsf{A}}dE\wedge\,dt+\sum_{k=2}^{N}\frac{1}{\mathsf{A}}dp_{k}\wedge\,d\psi_{k}\right)\,.$
(5.22)

The geometric drift is thus proportional to the area $S(\gamma)$ enclosed by
the path $\gamma$ spanned by the motion on the shape manifold $\mathcal{B}$.
The $2$-form $dE\wedge dt$ encodes the effects of the time-dependent external
moments on the geometric drift. The remaining $2$-forms are the same as those
of a free $N$-pendulum given in Eq. (4.51) and measure the effects of the
pendulum shape changes. In the following, we will show that the base manifold
$\mathcal{B}$ can be endowed with a Riemannian structure.

###  5.2 Curvature and intrinsic metric of the shape manifold

One can interpret the geometric rotation drift in Eq. (5.22) as the curvature
of the $2N$-dimensional shape manifold $\mathcal{B}$ equipped with a pseudo-
Riemannian metric of the following form

$ds^{2}=\epsilon_{t}\,G_{t}\,dt^{2}+\epsilon_{E}\,G_{E}\,dE^{2}+\sum_{j=2}^{N}\left[(\epsilon_{p_{j}}G_{p_{j}}(dp_{j})^{2}+\epsilon_{\psi_{j}}G_{\psi_{j}}(d\psi_{j})^{2}\right]\,,$
(5.23)

where the $2N$ positive metric coefficients depend on the coordinates
$\\{t,E,p_{2},\psi_{2},\cdots,p_{N},\psi_{N}\\}$. The signature of the
manifold is
$(\epsilon_{t},\epsilon_{E},\epsilon_{p_{2}},\epsilon_{\psi_{2}},\cdots)$. The
metric coefficients will be calculated using Cartan’s structural equations as
follows. From Eq. (5.22) the geometric drift follows by integrating the
$2$-form

$d\,\widetilde{\alpha}=d\,\widetilde{\alpha}_{t}+\sum_{j=2}^{N}d\,\widetilde{\alpha}_{j}\,,$
(5.24)

where $d\,\widetilde{\alpha}_{t}=-\frac{1}{\mathsf{A}}dE\wedge\,dt$, and
$d\,\widetilde{\alpha}_{j}=-\frac{1}{\mathsf{A}}dp_{j}\wedge\,d\psi_{j}$. The
associated $1$-forms are

$\widetilde{\alpha}_{t}=-\frac{E}{\mathsf{A}}dt\,,\quad\widetilde{\alpha}_{j}=-\frac{p_{j}}{\mathsf{A}}d\psi_{j}\,,\quad
j=2,\cdots,N\,.$ (5.25)

Drawing on Cartan’s structural equations, the $2$-forms
$d\,\widetilde{\alpha}_{t}$ and $d\,\widetilde{\alpha}_{j}$ are interpreted as
the only non-zero curvature $2$-forms of the $2N$-dimensional shape manifold
$\mathcal{B}$. We relabel the pair $(t,E)$ as $(p_{1},\psi_{1})$ so that
$d\,\widetilde{\alpha}$ is written as

$d\,\widetilde{\alpha}=\sum_{j=1}^{N}d\,\widetilde{\alpha}_{j}\,,$ (5.26)

where we set $\widetilde{\alpha}_{1}=\widetilde{\alpha}_{t}$. Comparing with
Eqs. (4.30) and (4.31), $\widetilde{\alpha}_{j}$ and
$d\,\widetilde{\alpha}_{j}$ can be interpreted as the connection and curvature
forms of a free $(N+1)$-pendulum. Thus, we can use the results we obtained in
§4.1. For the forced $N$-pendulum, the shape manifold $\mathcal{B}$ has
dimension $2N$. It is reducible since it is the product manifold of $N$
submanifolds (hyper-planes) $\mathcal{B}_{j}$ with coordinate charts
$\\{\psi_{j},p_{j}\\},\,j=1,\ldots N$. From Eq. (4.49), the metric of
$\mathcal{B}$ is written as

$\mathbf{G}=\mathbf{G}_{1}\times\ldots\times\mathbf{G}_{N}=\sum_{j=1}^{N}\left[\frac{\mathrm{sgn}(\mathsf{A})}{p_{j}^{2}+\mu_{j}^{2}}\,dp_{j}\otimes
dp_{j}+\frac{p_{j}^{2}+\mu_{j}^{2}}{\mathsf{A}^{2}}\,d\psi_{j}\otimes
d\psi_{j}\right]\,,$ (5.27)

where $\mu_{j}$ are arbitrary parameters. Since $p_{1}=t$ and $\psi_{1}=E$,
then $G_{p_{1}}=G_{t}$ and $G_{\psi_{1}}=G_{E}$ and the intrinsic metric of
each submanifold $\mathcal{B}_{j}$ follows from Eq. (4.48) as

$\mathbf{G}_{1}=\frac{\mathrm{sgn}(\mathsf{A})}{E^{2}+\mu_{t}^{2}}\,dE\otimes
dE+\frac{E^{2}+\mu_{t}^{2}}{\mathsf{A}^{2}}\,dt\otimes dt\,,$ (5.28)

and

$\mathbf{G}_{j}=\frac{\mathrm{sgn}(\mathsf{A})}{p_{j}^{2}+\mu_{j}^{2}}\,dp_{j}\otimes
dp_{j}+\frac{p_{j}^{2}+\mu_{j}^{2}}{\mathsf{A}^{2}}\,d\psi_{j}\otimes
d\psi_{j}\,,\quad j=2,\cdots,N\,.$ (5.29)

Then

$\mathbf{G}=\frac{\mathrm{sgn}(\mathsf{A})}{E^{2}+\mu_{1}^{2}}\,dE\otimes
dE+\frac{E^{2}+\mu_{1}^{2}}{\mathsf{A}^{2}}\,dt\otimes
dt+\sum_{j=2}^{N}\,\frac{\mathrm{sgn}(\mathsf{A})}{p_{j}^{2}+\mu_{j}^{2}}\,dp_{j}\otimes
dp_{j}+\frac{p_{j}^{2}+\mu_{j}^{2}}{\mathsf{A}^{2}}\,d\psi_{j}\otimes
d\psi_{j}\,.$ (5.30)

Similar to that of the free $N$-pendulum the shape manifold is the product
manifold of Robertson-Walker spacetime universes ($\mathsf{A}<0$) or
hyperbolic planes ($\mathsf{A}>0$).

The geometric drift follows by integrating the $2$-form

$d\,\widetilde{\alpha}=\mathcal{R}_{t}^{E}(\mathbf{e}_{E},\mathbf{e}_{t})+\sum_{j=2}^{N}\,\mathcal{R}_{\psi_{j}}^{p_{j}}(\mathbf{e}_{p_{j}},\mathbf{e}_{\psi_{j}})\,,$
(5.31)

which is the sum of the curvature $2$-forms of each submanifold
$\mathcal{B}_{j}$, that is

$\displaystyle\theta_{\text{geom}}$
$\displaystyle=\int_{S(\gamma)}d\,\widetilde{\alpha}=\int_{S(\gamma)}\left[\mathcal{R}_{t}^{E}(\mathbf{e}_{E},\mathbf{e}_{t})+\sum_{j=2}^{N}\,\mathcal{R}_{\psi_{j}}^{p_{j}}(\mathbf{e}_{p_{j}},\mathbf{e}_{\psi_{j}})\right]$
(5.32)
$\displaystyle=-\int_{S(\gamma)}\left[\frac{1}{\mathsf{A}}dE\wedge\,dt+\sum_{j=2}^{N}\frac{1}{\mathsf{A}}dp_{j}\wedge\,d\psi_{j}\right]\,,$

where each term is both the oriented area and curvature of the projected path
$\gamma$ on the hyper-planes $\mathcal{B}_{j}$ and $\mathcal{B}_{t}$ with
coordinates $\\{\psi_{j},p_{j}\\}$ and $\\{t,E\\}$, respectively.

##  6 Conclusions

We investigated the geometric phase of nonlinear planar $N$-pendula with
continuous rotational symmetry in the Hamiltonian framework. The geometric
structure of the phase space is a principal fiber bundle, i.e., a base, or
shape manifold $\mathcal{B}$, and fibers $\mathcal{F}$ along the symmetry
direction attached to it. The connection and curvature forms of the shape
manifold are defined by the symplectic structure of the Hamiltonian dynamics.
Then, Cartan’s moving frames provide the means to derive an intrinsic metric
structure for $\mathcal{B}$. This characterizes the kinematically admissible
shape deformations of the pendula. An orbit on $\mathcal{B}$ is a succession
of infinitesimal changes in shape of the pendulum from an initial
configuration to another. If the pendulum returns to its initial shape, the
orbit is closed and the area (or curvature) spanned by it measures the induced
geometric rotation drift. We first studied the geometric phase of a nonlinear
planar double pendulum that conserves the total angular momentum $\mathsf{A}$.
If $\mathsf{A}<0$ we found that the metric is pseudo-Riemannian and the shape
manifold is an expanding spacetime described by the Robertson-Walker metric
with positive curvature. For $\mathsf{A}>0$, the shape manifold is the
hyperbolic plane $\mathbb{H}^{2}$ with negative curvature. We then generalized
these results to nonlinear planar $N$-pendula. We found that the associated
shape manifold $\mathcal{B}$ is reducible since it is the product manifold of
$(N-1)$ hyperbolic planes $\mathbb{H}^{2}$ ($\mathsf{A}>0$), or Robertson-
Walker $2$D spacetimes ($\mathsf{A}<0$). We then considered $N$-pendula
subject to time-dependent self-equilibrated moments. The geometric phase is
studied in the extended autonomous Hamiltonian framework. The associated shape
manifold is still reducible as the product space of $N$ hyperbolic planes
$\mathbb{H}^{2}$ ($\mathsf{A}>0$), or Robertson-Walker $2$D spacetimes
($\mathsf{A}<0$).

We found that the Riemannian structure of the shape manifold depends on the
sign of the conserved total angular momentum. The geometric phase is evaluated
by the same $2$-form given by the sum of the sectional curvature forms of
$\mathcal{B}$. The intrinsic metric allows one to quantify the similarity of a
shape $S_{1}$, or point on $\mathcal{B}$, to another point, or shape $S_{2}$,
by measuring the intrinsic geodesic distance between the two points in terms
of curvature, or induced geometric phase. The Euclidean metric would give
misleading shorter distances between the two shapes. This is because it is not
an intrinsic structure that follows from the dynamics. Thus, low-momentum
shapes are far apart from high-momentum shapes. If $\mathsf{A}<0$, the shape
manifold is an expanding spacetime universe and the two different shapes are
‘red-shifted’ as very far away from each other. If $\mathsf{A}>0$, the shape
manifold has the character of the hyperbolic plane and the two shapes appear
far away from each other as the difference of their momenta becomes larger.
Furthermore, the intrinsic distance between shapes is relevant for measuring
how close an orbit is to the stable/unstable sub-manifolds of fixed points of
the dynamics on the shape manifold.

In future work, we will use Cartan’s moving frames to derive an intrinsic
metric for the shape manifold of Navier-Stokes turbulence with continuous
translational symmetry, or turbulent channel flows (Fedele et al., 2015). To
unveil the shape of turbulence one needs to quotient out the translation
symmetry of the Navier-Stokes equations. This can be achieved, for example, by
means of a physically meaningful slice or chart representation of the quotient
space or shape manifold (Budanur et al., 2015; Cvitanović et al., 2012; Fedele
et al., 2015). To measure how close one vortical shape is to another, the
standard Euclidean metric is typically used. An important conclusion of our
present study is that the similarities of shapes should be measured by a
metric intrinsic to the shape manifold. Other non-intrinsic distances are
misleading as they do not account for the curvature, or induced geometric
phase.

## Acknowledgement

This work was partially supported by NSF – Grant No. CMMI 1939901, and ARO
Grant No. W911NF-18-1-0003.

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