Index

Each index entry links to the first occurrence of that entry in the book. Bracketed numbers (starting from 2) following the index entry link to the second and subsequent occurrences.

0, for all practical purposes,
*see also* Zero-based indexing

o (composition)

o (composition)

(``chart'' function)

[*q*]

for local tuple

Lagrangian state path

_{L}[*q*] (Hamiltonian state path)

(coordinate function)

function

_{} (variation operator)

(configuration-path function)

-calculus

-expression

-notation

matrix

(angular velocity)

(*t*) × operator

(symplectic 2-form)

(phase-space path)

*A*

*C* (canonical phase-space transformation)

*C* (local-tuple transformation)

*D*, *see* Derivative

vs. *D*

, *see* Partial derivative

*D*_{t} (total time derivative)

*E* (Euler-Lagrange operator)

(energy state function)

*F*_{1}-*F*_{4}, *see also* Generating functions

*F*_{1}(*t*, *q*, *q*')

*F*_{2}(*t*, *q*, *p*')

*F*_{3}(*t*, *p*, *q*')

*F*_{4}(*t*, *p*, *p*')

*H* (Hamiltonian)

*I*_{i}, *I*_{z} (selector),
[2]

(shuffle function)

** J**,

vs.

(Lagrangian)

(momentum state function)

vs.

(action)

[ ]

, in tuple

:, names starting with

; in tuple, [2]

{ } for Poisson brackets

[ ] for down tuples

[ ] for functional arguments

( ) for up tuples

*A*

Action

computing

coordinate representation of

coordinate-independence of,
[2]

free particle

generating functions and

Hamilton-Jacobi equation and

Lagrangian

minimizing

parametric

principles, *see* Principle of stationary action

*S* or *S*_{}

time evolution and,
[2]

variation of

Action principle, *see* Principle of stationary action

Action-angle coordinates

Hamilton's equations in

Hamilton-Jacobi equation and

Hamiltonian in

harmonic oscillator in

perturbation of Hamiltonian,
[2]

surfaces of section in

Action-like region

Alphabet, insufficient size of

Alternative in conditional

Angles, Euler, *see* Euler angles

Angular momentum, *see also* Vector angular momentum

conservation of,
[2], [3],
[4]

equilibrium points for

Euler's equations and

kinetic energy in terms of

Lie commutation relations for

as Lie generator of rotations

of free rigid body,
[2]

of rigid body

sphere of

*z* component of

Angular velocity vector (), [2]

Euler's equations for

kinetic energy in terms of,
[2]

representation of

Anomaly, true

`antisymmetric->column-matrix`

Antisymmetry of Poisson bracket

Area preservation

by maps

Liouville's theorem and

Poincaré-Cartan integral invariant and

of surfaces of section,
[2]

Arguments

active vs. passive in Legendre transformation

in Scheme

Arithmetic

generic, [2],
[3]

on functions,
[2]

on operators,
[2]

on procedures

on symbolic values

on tuples, [2]

Arnold, V. I., [2],
[3],
*see also* Kolmogorov-Arnold-Moser theorem

Associativity and non-associativity of tuple multiplication, [2]

Asteroids, rotational alignment of

Astronomy, *see* Celestial objects

Asymptotic trajectories,
[2], [3]

Atomic scale

Attractor

Autonomous systems,
*see also* Extended phase space

surfaces of section for

Awake top

Axes, principal

of this dense book

Axisymmetric potential of galaxy

Axisymmetric top

awake

behavior of,
[2]

conserved quantities for

degrees of freedom of

Euler angles for

Hamiltonian treatment of

Lagrangian treatment of

nutation of

potential energy for

precession of,
[2]

rotation of

sleeping

symmetries of

Baker, Henry, *see* Baker-Campbell-Hausdorff formula

Baker-Campbell-Hausdorff formula

Banana, *see* Book

Barrow-Green, June

Basin of attraction

Bicycle wheel

Birkhoff, George David, *see* Poincaré-Birkhoff theorem

`bisect` (bisection search),
[2]

Body components of vector

Boltzmann, Ludwig,
[2], [3]

Book

banana-like behavior of

rotation of,
[2]

Brackets, *see also* Poisson brackets

for down tuples

for functional arguments

Bulirsch-Stoer integration method

`bulirsch-stoer`

Butterfly effect

*C* (canonical phase-space transformation)

*C* (local-tuple transformation)

Campbell, John, *see* Baker-Campbell-Hausdorff formula

Canonical equations, *see* Hamilton's equations

Canonical heliocentric coordinates

Canonical perturbation theory, *see* Perturbation theory

Canonical plane

Canonical transformations,
*see also* Generating functions; Symplectic transformations

classical gauge

composition of,
[2], [3]

conditions for

for driven pendulum

general

group properties of

for harmonic oscillator

invariance of antisymmetric bilinear form under

invariance of phase volume under

invariance of Poisson brackets under

invariants of,
*see also* Integral invariants

as Lie series

Lie transforms, *see* Lie transforms

point transformations, *see* Point transformations

polar-canonical, *see* Polar-canonical transformation

to rotating coordinates,
[2]

symplectic-matrix test

time evolution as

time-dependent

time-independent

`canonical?`

Cantori, [2]

`car`

Cartan, Élie, *see* Poincaré-Cartan integral invariant

Cauchy, Augustin Louis

`cdr`

Celestial objects, *see also* Asteroids; Comets; Earth; Galaxy; Hyperion; Jupiter; Mercury; Moon; Phobos; Planets

rotation of,
[2], [3]

Center of mass

in two-body problem

Jacobi coordinates and

kinetic energy and

vector angular momentum and

Central force

collapsing orbits

epicyclic motion

gravitational

in 2 dimensions,
[2], [3]

in 3 dimensions

Lie series for motion in

reduced phase space for motion in

Central potential, *see* Central force

Chain rule

for derivatives,
[2]

for partial derivatives,
[2]

for total time derivatives

in traditional notation

for variations

Chaotic motion,
*see also* Exponential divergence

homoclinic tangle and

in Hénon-Heiles problem

in spin-orbit coupling,
[2]

near separatrices,
[2], [3]

of Hyperion,
[2]

of non-axisymmetric top

of periodically driven pendulum,
[2]

overlapping resonances and

Characteristic exponent

Characteristic multiplier

Chart function (
)

Chirikov, Boris V.

Chirikov-Taylor map

Church, Alonzo

Classical gauge transformations

Colon, names starting with

Comets, rotation of

Comma in tuple

Commensurability,
*see also* Resonance

islands and

of pendulum period with drive,
[2]

periodic orbits and,
[2]

rational rotation number and

small denominators and

Commutativity, *see also* Non-commutativity

of some tuple multiplication

of variation ( ) with differentiation and integration

Commutator

of angular-momentum Lie operators

Jacobi identity for

of Lie derivative

Poisson brackets and

Compatible shape

`compatible-shape`

`component`, [2]

`compose`

Composition

of canonical transformations,
[2], [3]

of functions,
[2], [3]

of Lie transforms

of linear transformations

of operators

of rotations

Compositional canonical

`compositional-canonical?`

Compound data in Scheme

`cond`

Conditionals in Scheme

Configuration

Configuration manifold

Configuration path, *see* Path

Configuration space

Configuration-path function ( ),
[2]

Conjugate momentum

non-uniqueness of

`cons`

Consequent in conditional

Conserved quantities,
[2],
*see also* Hénon-Heiles problem, integrals of motion

angular momentum,
[2], [3],
[4]

coordinate choice and

cyclic coordinates and

energy, [2],
[3]

Jacobi constant

Lyapunov exponents and

momentum

Noether's theorem

phase space reduction and

phase volume, *see* Phase-volume conservation

Poisson brackets of

symmetry and, [2]

for top

Constant of motion (integral of motion),
*see also* Conserved quantities; Hénon-Heiles problem

Constraint force

Constraint(s)

augmented Lagrangian and,
[2]

configuration space and

as coordinate transformations

explicit

in extended bodies

holonomic, [2]

integrable, [2]

linear in velocities

nonholonomic (non-integrable)

on coordinates

rigid

as subsystem couplers

total time derivative and

velocity-dependent

velocity-independent

Constructors in Scheme

Contact transformation, *see* Canonical transformations

Continuation procedure

Continued-fraction approximation of irrational number

Contraction of tuples

`coordinate`

Coordinate function ( )

Coordinate path (*q*),
*see also* Local tuple

Coordinate singularity

Coordinate transformations

constraints as

Coordinate(s), *see also* Generalized coordinates

action-angle, *see* Action-angle coordinates

conserved quantities and choice of

constraints on

cyclic, [2]

heliocentric

ignorable (cyclic)

Jacobi

polar, *see* Polar coordinates

redundant, and initial conditions

rotating, *see* Rotating coordinates

spherical

Coordinate-independence

of action, [2]

of Lagrange equations,
[2]

of variational formulation,
[2]

Correction fluid

Cotangent space, bundle

Coupling systems

Coupling, spin-orbit, *see* Spin-orbit coupling

Curves, invariant, *see* Invariant curves

Cyclic coordinate,
[2]

*D*, *see* Derivative

vs. *D*

`D` (Scheme procedure for derivative), [2]

, *see* Partial derivative

*D*_{t} (total time derivative)

d'Alembert-Lagrange principle (Jean leRond d'Alembert)

Damped harmonic oscillator

`define`

Definite integral

`definite-integral`

Definitions in Scheme

Degrees of freedom

Delta function

Derivative, [2],
*see also* Total time derivative

as operator

as Poisson bracket

chain rule, [2]

in Scheme programs: `D`,
[2]

notation: *D*,
[2]

of function of multiple arguments,
[2]

of function with structured arguments

of function with structured inputs and outputs

of state

partial, *see* Partial derivative

precedence of,
[2]

with respect to a tuple

`determinant`

Differentiable manifold

Dimension of configuration space

Dirac, Paul Adrien Maurice

Dissipation of energy

in free-body rotation

tidal friction

Dissipative system, phase-volume conservation

Dissolution of invariant curves,
[2]

Distribution functions

Divided phase space,
[2], [3]

Division in Scmutils

by a structure

generic character of

of vector by matrix

Dot notation

`down`, [2]

Down tuples

Driven harmonic oscillator

Driven pendulum, *see* Pendulum (driven)

Driven rotor, [2]

`Dt` (total time derivative)

Dynamical state, *see* State

*E* (Euler-Lagrange operator)

(energy state function)

Earth

precession of

rotational alignment of

Effective Hamiltonian

Eigenvalues and eigenvectors

for equilibria

for fixed points

for Hamiltonian systems

of inertia tensor

for unstable fixed point

Einstein summation convention

Einstein, Albert

`else`

Empty list

Energy

as sum of kinetic and potential energies

conservation of,
[2], [3]

dissipation of, *see* Dissipation of energy

Energy state function ()

Hamiltonian and

Epicyclic motion

`eq?`

Equilibria, [2],
[3], *see also* Fixed points

for angular momentum

inverted, for pendulum,
[2], [3],
[4]

linear stability of

relative

stable and unstable

Equinox, precession of

Ergodic motion

Ergodic theorem

Euler, Leonhard

Euler angles

for axisymmetric top

kinetic energy in terms of

singularities and,
[2]

`Euler-Lagrange-operator` ( *E* )

Euler's equations

singularities in

Euler's theorem on homogeneous functions

Euler's theorem on rotations

Euler angles and

Euler-Lagrange equations, *see* Lagrange equations

Euler-Lagrange operator ( *E* )

`Euler->M`

`Euler-state->L-space`

`Euler-state->omega-body`

Evolution, *see* Time evolution of state

`evolve`, [2],
[3]

`explore-map`

Exponential divergence,
[2], [3],
*see also* Lyapunov exponent

homoclinic tangle and

Exponential(s)

of differential operator

of Lie derivative

of noncommuting operators

Expressions in Scheme

Extended phase space

generating functions in

*F*_{1}-*F*_{4}, *see also* Generating functions

*F*_{1}(*t*, *q*, *q*')

*F*_{2}(*t*, *q*, *p*')

*F*_{3}(*t*, *p*, *q*')

*F*_{4}(*t*, *p*, *p*')

`F->C`, [2]

`F->CT`

Fermat, Pierre, [2]

Fermat's principle (optics),
[2]

Fermi, Enrico

Feynman, Richard P.

`find-path`

First amendment, *see* Degrees of freedom

First integral

Fixed points,
*see also* Equilibria

elliptic, [2]

equilibria or periodic motion and,
[2]

for Hamiltonian systems

hyperbolic, [2]

linear stability of

manifolds for

parabolic

Poincaré-Birkhoff fixed points

Poincaré-Birkhoff theorem

rational rotation number and

Floating-point numbers in Scheme

Floquet multiplier

Flow, defined by vector field

Force

central, *see* Central force

exerted by constraint

Forced libration of the Moon

Forced rigid body, *see* Rigid body, forced

Formal parameters

of a function

of a procedure

`frame`

Free libration of the Moon

Free particle

action

Lagrange equations for

Lagrangian for

Free rigid body, *see* Rigid body (free)

Freudenthal, Hans

Friction

internal

tidal

Function definition

Function(s)

arithmetic operations on,
[2]

composition of,
[2], [3]

homogeneous

operators vs.,
[2]

orthogonal, tuple-valued

parallel, tuple-valued

selector, [2]

tuple of, [2]

vs. value when applied,
[2]

with multiple arguments,
[2], [3]

with structured arguments,
[2], [3]

with structured output,
[2]

Functional arguments

Functional mathematical notation,
[2]

Fundamental Poisson brackets

Galaxy

axisymmetric potential of

`Gamma` (Scheme procedure for )

optional argument

`Gamma-bar`

Gas in corner of room

Gauge transformations, classical

Generalized coordinates,
[2], [3]

Euler angles as,
*see also* Euler angles

Lagrangian in

Generalized momentum

transformation of

Generalized velocity

transformation of

Generating functions

classical gauge transformations and

in extended phase space

*F*_{1}-*F*_{4}

*F*_{1}

*F*_{2}

*F*_{2} and point transformations

*F*_{2} for polar coordinate transformation

*F*_{2} for rotating coordinates

incorrect derivation of

integral invariants and

Lagrangian action and

Legendre transformation between *F*_{1} and *F*_{2}

Generic arithmetic,
[2], [3]

Gibbs, Josiah Willard,
[2]

Golden number

Golden ratio, a most irrational number

Golden rotation number

Goldstein, Herbert

Goldstein's hoop

Golf ball, tiny

Grand Old Duke of York, *see* neither `up` nor `down`

Graphing, [2],
[3]

Gravitational potential

central

of galaxy

multipole expansion of

rigid-body

Group properties

of canonical transformations

of rotations, *see* Euler's theorem on rotations

*H* (Hamiltonian)

`H-central`

`H-harmonic`

`H-pend-sysder`

Hamilton, Sir William Rowan,
[2]

Hamilton's equations

in action-angle variables

computation of

dynamical

for central potential

for damped harmonic oscillator

for harmonic oscillator

from action principle

from Legendre transformation

numerical integration of

Poisson bracket form

Hamilton's principal function

Hamilton's principle

for systems with rigid constraints

Hamilton-Jacobi equation

action at endpoints and

action-angle coordinates and

for harmonic oscillator

for Kepler problem

separation in spherical coordinates

time-independent

Hamiltonian

in action-angle variables

computing, *see* `H-...`

cyclic in coordinate

energy state function and

for axisymmetric potential

for central potential,
[2], [3],
[4]

for damped harmonic oscillator

for driven pendulum,
[2]

for driven rotor

for harmonic oscillator

for harmonic oscillator, in action-angle coordinates

for Kepler problem

for pendulum

for periodically driven pendulum,
[2]

for restricted three-body problem,
[2]

for top

for two-body problem

Hénon-Heiles,
[2]

Lagrangian and,
[2]

perturbation of action-angle,
[2]

time-dependent, and dissipation

Hamiltonian flow

Hamiltonian formulation

Lagrangian formulation and

Hamiltonian state

Hamiltonian state derivative,
[2]

Hamiltonian state path _{L}[*q*]

`Hamiltonian->Lagrangian`

`Hamilton-equations`

Harmonic oscillator

coupled

damped

decoupling via Lie transform

driven

first-order equations for

Hamilton's equations for

Hamiltonian for

Hamiltonian in action-angle coordinates

Lagrange equations for,
[2]

Lagrangian for

Lie series for

solution of, [2]

solution via canonical transformation

solution via Hamilton-Jacobi

Hausdorff, Felix, *see* Baker-Campbell-Hausdorff formula

Heiles, Carl, [2],
*see also* Hénon

Heisenberg, Werner,
[2]

Heliocentric coordinates

Hénon, Michel, [2],
[3]

Hénon-Heiles problem

computing surfaces of section

history of

integrals of motion,
[2], [3]

interpretation of model

model of

potential energy

surface of section

Hénon's quadratic map

Heteroclinic intersection

Higher-order perturbation theory,
[2]

History

Hénon-Heiles problem

variational principles,
[2], [3],
[4]

Holonomic system, [2]

Homoclinic intersection

Homoclinic tangle

chaotic regions and

computing

exponential divergence and

Homogeneous function, Euler's theorem

Huygens, Christiaan

Hyperion, chaotic tumbling of,
[2]

*I*_{i}, *I*_{z} (selector),
[2]

`if`

Ignorable coordinate, *see* Cyclic coordinate

Indexing, zero-based, *see* Zero-based indexing

Inertia matrix,
*see also* Inertia tensor

Inertia tensor

diagonalization of

kinetic energy in terms of

principal axes of

transformation of

Inertia, moments of, *see* Moment(s) of inertia

Initial conditions, *see* Sensitivity to initial conditions; State

Inner product of tuples

Instability, *see also* Equilibria; Linear stability

free-body rotation

Integers in Scheme

Integrable constraints,
[2]

Integrable systems

periodic orbits of near-integrable systems

perturbation of,
[2], [3]

reduction to quadrature and,
*see also* Quadrature

surfaces of section for

Integral invariants

generating functions and

Poincaré

Poincaré-Cartan,
[2]

Integral of motion,
*see also* Conserved quantities; Hénon-Heiles problem

Integral, definite

Integration, *see* Numerical integration

Invariant curves,
[2]

dissolution of,
[2]

finding (computing)

finding (strategy)

irrational rotation number and

Kolmogorov-Arnold-Moser theorem

Invariants of canonical transformations,
*see also* Integral invariants

Irrational number, continued-fraction approximation

Islands in surfaces of section, *see also* Resonance

for Hénon-Heiles problem

for periodically driven pendulum,
[2], [3]

for standard map

perturbative vs. actual

in Poincaré-Birkhoff construction

Poisson series and

secondary, [2]

size of, [2]

small denominators and,
[2]

Iteration in Scheme

(shuffle function)

** J**,

Jac (Jacobian of map)

Jacobi constant

Jacobi coordinates

Jacobi identity

for commutators

for Poisson brackets

Jacobi, Carl Gustav Jacob,

Jacobian

Jeans, Sir James, ``theorem'' of

Jupiter

KAM theorem, *see* Kolmogorov-Arnold-Moser theorem

Kepler problem

in reduced phase space

solution via Hamilton-Jacobi equation

Kepler's third law

Kepler, Johannes, *see* Kepler...

Kinematics of rotation

Kinetic energy

ellipsoid of

in Lagrangian

as Lagrangian for free body

as Lagrangian for free particle

of axisymmetric top

of free rigid body

of rigid body,
*see also* Rigid body, kinetic energy...

rotational and translational

in spherical coordinates

Knuth, Donald E.

Kolmogorov, A. N., *see* Kolmogorov-Arnold-Moser theorem

Kolmogorov-Arnold-Moser theorem,
[2]

*L* (Lie derivative)

*L* or *L*_{} (Lagrangian)

vs.

(Lagrangian)

`L-axisymmetric-top`

`L-central-polar`,
[2]

`L-central-rectangular`

`L-free-particle`

`L-harmonic`

`L-pend`

`L-rectangular`

`L-uniform-acceleration`,
[2]

Lagrange equations

at a moment

computing

coordinate-independence of,
[2]

derivation of

as first-order system

for central potential (polar)

for central potential (rectangular)

for damped harmonic oscillator

for driven pendulum

for free rigid body

for gravitational potential

for harmonic oscillator,
[2]

for periodically driven pendulum

for spin-orbit coupling

from Newton's equations,
[2]

vs. Newton's equations

numerical integration of

off the beaten path

singularities in

traditional notation for,
[2]

uniqueness of solution

`Lagrange-interpolation-function`

Lagrange interpolation polynomial

Lagrange multiplier, *see* Lagrangian, augmented

Lagrange, Joseph Louis,
[2]

`Lagrange-equations`

Lagrangian

adding total time derivatives to

augmented, [2]

computing,
*see also* `L-...`

coordinate transformations of

cyclic in coordinate

energy and

for axisymmetric top

for central potential (polar),
[2]

for central potential (rectangular)

for central potential (spherical)

for constant acceleration

for damped harmonic oscillator

for driven pendulum,
[2]

for free particle

for free rigid body,
[2]

for gravitational potential

for harmonic oscillator

for spin-orbit coupling

for systems with rigid constraints

in generalized coordinates

generating functions and

Hamiltonian and,
[2]

kinetic energy as,
[2], [3]

kinetic minus potential energy as,
*see also* Hamilton's principle

non-uniqueness of

parameter names in

rotational and translational

symmetry of

Lagrangian action

Lagrangian formulation

Hamiltonian formulation and

Lagrangian reduction

Lagrangian state, *see* State tuple

Lagrangian state derivative

Lagrangian state path [*q*]

`Lagrangian->energy`

`Lagrangian->Hamiltonian`

`Lagrangian->state-derivative`

`Lagrangian-action`

`lambda`

Lambda calculus

Lambda expression

Lanczos, Cornelius

Least action, principle of, *see* Principle of stationary action

Legendre, Adrien Marie, *see* Legendre...

Legendre polynomials

`Legendre-transform`

Legendre transformation

active arguments in

passive arguments in

of quadratic functions

Leibniz, Gottfried

`let`

Libration of the Moon,
[2], [3]

Lie derivative

commutator for

Lie transform and

operator *L*_{H}

Lie series

computing

for central field

for harmonic oscillator

in perturbation theory

`Lie-transform`

Lie transforms

advantage of

composition of

computing

exponential identities

for finding normal modes

Lie derivative and

in perturbation theory

Lie, Sophus, *see* Lie...

`Lie-derivative`,
[2]

Lindstedt, A.

Linear momentum

Linear separation of regular trajectories

Linear stability

equilibria and fixed points

nonlinear stability and

of equilibria

of fixed points

of inverted equilibrium of pendulum

Linear transformations as tuples

`linear-interpolants`

Liouville equation

Liouville's theorem

from canonical transformation

Liouville, Joseph, *see* Liouville...

Lipschitz condition (Rudolf Lipschitz)

Lisp

`list`

`list-ref`

Lists in Scheme

Literal symbol in Scheme

`literal-function`,
[2], [3],
[4]

Local names in Scheme

Local state tuple

Local tuple, [2]

(chart) and

component names

functions of

in Scheme programs

transformation of (*C*)

Loops in Scheme

Lorentz, Hendrik Antoon, *see* Lorentz transformations

Lorentz transformations as point transformations

Lorenz, Edward

Lyapunov exponent

conserved quantities and

exponential divergence and

Hamiltonian constraints

linear stability and

Lyapunov, Alexey M., *see* Lyapunov exponent

`M->omega-body`

`M-of-q->omega-of-t`

MacCullagh's formula

`make-path`, [2]

Manifold

differentiable

stable and unstable

Map

area-preserving

Chirikov-Taylor

fixed points of,
*see also* Fixed points

Hénon's quadratic

Poincaré

representation in programs

standard

symplectic

twist

Mars, *see* Phobos

Mass point, *see* Point mass

Mathematical notation, *see* Notation

Mather, John N. (discoverer of sets named *cantori* by Ian Percival)

Matrix

inertia,
*see also* Inertia tensor

orthogonal

Pauli spin

symplectic, [2],
[3]

as tuple

Maupertuis, Pierre-Louis Moreau de,
[2]

Mean motion

Mechanics

Newtonian vs. variational formulation, [2]

Mercury, resonant rotation of,
[2]

Minimization

of action

in Scmutils, [2]

`minimize`

Moment(s) of inertia

about a line

about a pivot point

principal

of top

Momentum, *see also* Angular momentum

conjugate to coordinate, *see* Conjugate momentum

conservation of

generalized, *see* Generalized momentum

variation of

`momentum`

Momentum path

Momentum state function ()

Moon

head-shaking

history of

libration of,
[2], [3]

rotation of,
[2], [3],
[4]

Moser, Jürgen, *see* Kolmogorov-Arnold-Moser theorem

Motion

atomic-scale

chaotic, *see* Chaotic motion

constrained,
*see also* Constraint(s)

dense, on torus

deterministic

epicyclic

ergodic

periodic, *see* Periodic motion

quasiperiodic, *see* Quasiperiodic motion

realizable vs. conceivable

regular vs. chaotic,
*see also* Regular motion

smoothness of

tumbling, *see* Chaotic motion, of Hyperion; Rotation(s), (in)stability of

`multidimensional-minimize`,
[2]

Multiplication of operators as composition

Multiplication of tuples

as composition

as contraction

Multiply-periodic functions, Poisson series for

Multipole expansion of potential energy

*n*-body problem

Nelder-Mead minimization method

Newton's equations

as Lagrange equations,
[2]

vs. Lagrange equations

Newton, Sir Isaac

Newtonian formulation of mechanics,
[2]

Noether, Emmy

Noether's theorem

angular momentum and

Non-associativity and associativity of tuple multiplication, [2]

Non-axisymmetric top

Non-commutativity, *see also* Commutativity

exponential(s) of noncommuting operators

of some partial derivatives,
[2]

of some tuple multiplication

Nonholonomic system

Nonsingular generalized coordinates

Nonsingular structure

Notation,
*see also* Subscripts; Superscripts; Tuples

{ }

( )

[ ], [2]

ambiguous

composition of functions

definite integral

derivative, partial: ,
[2], [3]

derivative: *D*,
[2]

derivative:

function of local tuple

functional, [2]

functional arguments

selector function: *I*_{i}, *I*_{z},
[2]

total time derivative: *D*_{t}

traditional, [2],
[3], [4],
[5], [6]

Numbers in Scheme

Numerical integration

of Hamilton's equations

of Lagrange equations

in Scmutils, [2],
[3]

symplectic

Numerical minimization in Scmutils,
[2]

Nutation of top, [2]

Oblateness

`*ode-integration-method*`

`omega` (symplectic 2-form)

`omega-cross`

Operator

arithmetic operations on,
[2]

exponential identities

function vs.,
[2]

generic

Operators

derivative (*D*), *see* Derivative

Euler-Lagrange ( *E* )

Lie derivative (*L*_{H})

Lie transform (*E*'_{, W})

partial derivative ( ), *see* Partial derivative

variation ( _{} )

Optical libration of the Moon

Optics

Fermat

Snell's law

Orbit, *see* Phase-space trajectory

Orbital elements

Orbital motion, *see also* Epicyclic motion; Kepler problem

Lagrange equations for

retrodiction of

Orientation, *see also* Rotation(s)

Euler's equations and

nonsingular coordinates for

specified by Euler angles

specified by rotations

Orientation vector

Orthogonal matrix

Orthogonal transformation, *see* Orthogonal matrix

Orthogonal tuple-valued functions

Oscillator, *see* Harmonic oscillator

`osculating-path`

Osculation of paths

Ostrogradsky, M. V.

Out-of-roundness parameter

*P* (momentum selector),
[2]

(momentum state function)

`p->r` (polar-to-rectangular)

`pair?`

Pairs in Scheme

Parallel tuple-valued functions

Parametric path

`parametric-path-action`

with graph

Parentheses

in Scheme, [2]

for up tuples

`partial`

Partial derivative,
[2], [3]

chain rule, [2]

notation: ,
[2], [3]

Particle, free, *see* Free particle

Path

coordinate path (*q*),
*see also* Local tuple

finding

momentum path

osculation of

parametric

realizable, *see* Realizable path

variation of, [2],
[3]

Path functions, abstraction of

Path-distinguishing function,
[2], *see also* Action

Pauli spin matrices (Wolfgang Pauli)

Peak

Pendulum, *see also* Pendulum (driven); Periodically driven pendulum

behavior of,
[2]

constraints and

degrees of freedom of

double (planar),
[2]

double (spherical)

equilibria, stable and unstable

Hamiltonian for

Lagrangian for

periodically driven pendulum vs.

as perturbed rotor

phase plane of,
[2]

phase-volume conservation for

spherical, [2]

width of oscillation region

Pendulum (driven),
*see also* Pendulum; Periodically driven pendulum

canonical gauge transformation and

drive as modification of gravity

Hamiltonian for,
[2]

Lagrange equations for

Lagrangian for,
[2]

Pericenter

Period doubling

Periodic motion

fixed points and

integrable systems and,
[2]

Periodic points

Poincaré-Birkhoff theorem

rational rotation number and

resonance islands and

Periodically driven pendulum, *see also* Pendulum (driven); Pendulum

behavior of,
[2]

chaotic behavior of

emergence of divided phase space

Hamiltonian for,
[2]

inverted equilibrium,
[2], [3],
[4]

islands in sections for,
[2], [3]

Lagrange equations for

linear stability analysis

as perturbed rotor

phase space evolution of

phase-space descriptions for

resonances for

spin-orbit coupling and

surface of section for,
[2], [3],
[4]

undriven pendulum vs.

with zero-amplitude drive

Periodically driven systems, surfaces of section

Perturbation of action-angle Hamiltonian, [2]

Perturbation theory

for many degrees of freedom

for pendulum

for periodically driven pendulum

for spin-orbit coupling

higher-order,
[2]

Lie series in

nonlinear resonance

secular terms in

secular-term elimination

small denominators in

Phase portrait, [2]

Phase space,
*see also* Surface of section

chaotic regions

divided, [2],
[3]

evolution in,
*see also* Time evolution of state

extended

non-uniqueness

of pendulum,
[2]

qualitative features,
[2], [3]

reduced

regular regions

two-dimensional

volume, *see* Phase-volume conservation

`phase-space-derivative`

Phase space reduction

conserved quantities and

Lagrangian

Phase-space state function

in Scheme

Phase-space trajectory (orbit)

asymptotic, [2],
[3]

chaotic, [2]

periodic, [2],
[3]

quasiperiodic,
[2]

regular, [2]

regular vs. chaotic

Phase-volume conservation,
[2]

for damped harmonic oscillator

for pendulum

under canonical transformations

Phobos, rotation of

Pit

Planets, *see also* Earth; Jupiter; Mercury

moment of inertia of

rotation of

rotational alignment of

`plot-parametric-fill`

`plot-point`

Plotting, [2],
[3]

Poe, Edgar Allan, *see* Pit; Pendulum

Poincaré, Henri,
[2], [3],
[4], [5]

Poincaré integral invariants

generating functions and

Poincaré map

Poincaré recurrence

Poincaré section, *see* Surface of section

Poincaré-Birkhoff theorem

computing fixed points

recursive nature of

Poincaré-Cartan integral invariant

time evolution and

Point mass,
*see also* Golf ball, tiny

Point transformations,
*see also* Canonical transformations

computing

general canonical transformations vs.

generating functions for

polar-rectangular conversion,
[2]

to rotating coordinates,
[2]

time-independent

Poisson brackets

commutator and

of conserved quantities

as derivations

fundamental

Hamilton's equations in terms of

in terms of

in terms of symplectic 2-form,

invariance under canonical transformations

Jacobi identity for

Lie derivative and

symplectic transformations and

Poisson series

for multiply periodic function

resonance islands and

Poisson, Siméon Denis

Polar coordinates

Lagrangian in

point transformation to rectangular, [2]

transformation to rectangular

`polar-canonical`

Polar-canonical transformation

generating function for

harmonic oscillator and

Potential, *see* Central force; Gravitational potential

Potential energy

for axisymmetric top

Hénon-Heiles

in Lagrangian

multipole expansion of

Precession

of equinox

of top, [2],
[3]

Predicate in conditional

Predicting the past

Principal axes

of this dense book

Principal moments of inertia

kinetic energy in terms of,
[2], [3]

`principal-value`

Principle of d'Alembert-Lagrange

Principle of least action, *see* Principle of stationary action

Principle of stationary action (action principle)

Hamilton's equations and

principle of least action,
[2]

statement of

used to find paths

`print-expression`,
[2]

Probability density in phase space

Procedure calls

Procedures

arithmetic operations on

generic

Products of inertia

*Q* (coordinate selector)

*q* (coordinate path)

`qcrk4` (quality-controlled Runge-Kutta)

Quadratic functions, Legendre transformation of

Quadrature, [2],
*see also* Integrable systems

integrable systems and

reduction to

Quartet

Quasiperiodic motion,
[2]

Quaternions

Hamilton's discovery of

Quotation in Scheme

Radial momentum

Reaction force, *see* Constraint force

Realizable path

conserved quantities and

as solution of Hamilton's equations

as solution of Lagrange equations

stationary action and

uniqueness

Recurrence theorem of Poincaré

Recursive procedures

Reduced mass

Reduced phase space

Reduction

Lagrangian

of phase space, *see* Phase space reduction

to quadrature

`ref`, [2]

Regular motion, [2],
[3]

linear separation of trajectories

Renormalization

Resonance, *see also* Commensurability

center

islands, *see* Islands in surfaces of section

nonlinear

of Mercury's rotation,
[2]

overlap criterion,
[2]

for periodically driven pendulum

width, [2]

Rigid body

forced,
*see also* Spin-orbit coupling; Top

free, *see* Rigid body (free)

kinetic energy

kinetic energy in terms of inertia tensor and angular velocity, [2]

kinetic energy in terms of principal moments and angular momentum

kinetic energy in terms of principal moments and angular velocity

kinetic energy in terms of principal moments and Euler angles

vector angular momentum

Rigid body (free)

angular momentum

angular momentum and kinetic energy

computing motion of

Euler's equations and

(in)stability

orientation

Rigid constraints

as coordinate transformations

Rotating coordinates

in extended phase space

generating function for

point transformation for,
[2]

Rotation number

golden

irrational, and invariant curves

rational, and commensurability

rational, and fixed and periodic points

Rotation(s), *see also* Orientation

active

composition of,
[2]

computing

group property of

(in)stability of

kinematics of

kinetic energy of, *see* Rigid body, kinetic energy...

Lie generator for

matrices for

of celestial objects,
[2], [3]

of Hyperion

of Mercury, [2]

of Moon, [2],
[3], [4]

of top, book, and Moon

orientation as

orientation vector and

passive

as tuples

Rotor

driven, [2]

pendulum as perturbation of

periodically driven pendulum as perturbation of

Routh, Edward John

Routhian

Routhian equations

Runge-Kutta integration method

`qcrk4`

*S* or *S*_{} (action)

vs.

(action)

`s->m` (structure to matrix)

`s->r` (spherical-to-rectangular)

Saddle point

Salam, Abdus

Saturn, *see* Hyperion

Scheme, [2],
[3], *see also* Scmutils

for Gnu/Linux, where to get it

Schrödinger, Erwin,
[2]

Scmutils, [2],
*see also* Scheme

division by a structure

division of vector by matrix

generic arithmetic,
[2], [3]

minimization, [2]

numerical integration,
[2], [3]

operations on operators

simplification of expressions,
[2]

where to get it

Second law of thermodynamics

Section, surface of, *see* Surface of section

Secular terms in perturbation theory

elimination of

Selector function,
[2]

Selectors in Scheme

Semicolon in tuple,
[2]

Sensitivity to initial conditions,
[2], [3],
*see also* Chaotic motion

Separatrix, [2],
*see also* Asymptotic trajectories

chaos near, [2],
[3]

motion near

`series`

`series:for-each`

`show-expression`

Shuffle function

Simplification of expressions,
[2]

Singularities, [2],
[3]

nonsingular generalized coordinates

Sleeping top

Small denominators

for periodically driven pendulum

in perturbation theory,
[2]

resonance islands and,
[2]

Small divisors, *see* Small denominators

Snell's law

Solvable systems, *see* Integrable systems

Spherical coordinates

kinetic energy in

Lagrangian in

Spin matrices, Pauli

Spin-orbit coupling

chaotic motion,
[2]

Lagrange equations for

Lagrangian for

periodically driven pendulum and

perturbation theory for

surface of section for

Spivak, Michael, [2]

`square`

for tuples, [2]

Stability, *see* Equilibria; Instability; Linear stability

Stable manifold

computing

Standard map

`standard-map`

Stars, *see* Galaxy

State

evolution of, *see* Time evolution of state

Hamiltonian vs. Lagrangian

in terms of coordinates and momenta (Hamiltonian)

in terms of coordinates and velocities (Lagrangian)

State derivative

Hamiltonian

Hamiltonian vs. Lagrangian

Lagrangian

State path

Hamiltonian

Lagrangian

State tuple

`state-advancer`

Stationarity condition

Stationary action, *see* Principle of stationary action

Stationary point

Steiner's theorem

String theory, [2],
*see also* Quartet

Stroboscopic surface of section,
*see also* Surface of section

computing

Subscripts

`down` and

for down-tuple components

for momentum components,
[2]

for selectors

Summation convention

Superscripts

for coordinate components,
[2], [3]

for up-tuple components

for velocity components,
[2]

`up` and

Surface of section

in action-angle coordinates

area preservation of,
[2]

computing (Hénon-Heiles)

computing (stroboscopic)

fixed points, *see* Fixed points

for autonomous systems

for Hénon-Heiles problem

for integrable system

for non-axisymmetric top

for periodically driven pendulum,
[2], [3],
[4]

for spin-orbit coupling

invariant curves, *see* Invariant curves

islands, *see* Islands in surfaces of section

stroboscopic

Symbolic values

Symbols in Scheme

Symmetry

conserved quantities and,
[2]

continuous

of Lagrangian

of top

Symplectic bilinear form (2-form)

invariance under canonical transformations

Symplectic condition

Symplectic integration

Symplectic map

Symplectic matrix,
[2], [3]

Symplectic transformations,
*see also* Canonical transformations

antisymmetric bilinear form and

Poisson brackets and

Symplectic unit ** J**,

Syntactic sugar

System derivative,

[ ]

`T-rigid-body`

Taylor, J. B.

Tensor arithmetic

notation and,
[2]

summation convention

tuple arithmetic vs.,
[2]

Theology and principle of least action

Thermodynamics, second law

Three-body problem, restricted

Tidal friction

`time`

Time evolution of state

action and, [2]

as canonical transformation

in phase space

Poincaré-Cartan integral invariant and

`time-independent-canonical?`

Time-dependent canonical transformations

Time-independence, *see also* Extended phase space

energy conservation and

Time-independent canonical transformations

Top

Axisymmetric, *see* Axisymmetric top

non-axisymmetric

Top banana, *see* Non-axisymmetric top

Torque

in Euler's equations

in spin-orbit coupling

Total time derivative

adding to Lagrangians

affecting conjugate momentum

commutativity of

computing

constraints and

identifying

notation: *D*_{t}

Trajectory, *see* Path; Phase-space trajectory

Transformation

canonical, *see* Canonical transformations

coordinate, *see* Coordinate transformations

Legendre, *see* Legendre transformation

Lie, *see* Lie transforms

orthogonal, *see* Orthogonal matrix

point, *see* Point transformations

symplectic, *see* Symplectic transformations

True anomaly

Tumbling, *see* Chaotic motion, of Hyperion; Rotation(s), (in)stability of

Tuples

arithmetic on,
[2]

commas and semicolons in

component selector: *I*_{i}, *I*_{z},
[2]

composition and

contraction

of coordinates

down and up

of functions,
[2]

inner product

linear transformations as

local, *see* Local tuple

matrices as

multiplication of

rotations as

squaring, [2]

state tuple

up and down

Twist map

Two-body problem

Two-trajectory method

Undriven pendulum, *see* Pendulum

Uniform circle map

Uniqueness

of Lagrangian -- not!

of phase-space description -- not!

of realizable path

of solution to Lagrange equations

Unstable manifold

computing

`up`, [2]

Up tuples

Vakonomic mechanics

Variation

chain rule

of a function

of a path, [2],
[3]

of action

operator: _{}

Variational equations

Variational formulation of mechanics,
[2]

Variational principle, *see* Principle of stationary action

Vector

body components of

in Scheme

square of

`vector`

Vector angular momentum,
*see also* Angular momentum

center-of-mass decomposition

in terms of angular velocity and inertia tensor

in terms of principal moments and angular velocity

in terms of principal moments and Euler angles

Vector space of tuples

Vector torque, *see* Torque

`vector-ref`

`vector?`

Velocity, *see* Angular velocity; Generalized velocity

`velocity`

Velocity dispersion in galaxy

Web site for this book

Whittaker transform (Sir Edmund Whittaker)

Zero-amplitude drive for pendulum

Zero-based indexing,
[2], [3],
[4], [5]