Any inaccuracies in this index may be explained by the fact that it has been prepared with the help of a computer.

Donald E. Knuth, Fundamental Algorithms (Volume 1 of The Art of Computer Programming)

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)
    for local tuple
    Lagrangian state path
L[q] (Hamiltonian state path)
(coordinate function)
(variation operator)
(configuration-path function)
(angular velocity)
(t) × operator
(symplectic 2-form)
(phase-space path)
C (canonical phase-space transformation)
C (local-tuple transformation)
D, see Derivative
vs. D
, see Partial derivative
Dt (total time derivative)
E (Euler-Lagrange operator)
(energy state function)
F1-F4, see also Generating functions
    F1(t, q, q')
    F2(t, q, p')
    F3(t, p, q')
    F4(t, p, p')
H (Hamiltonian)
Ii, Iz (selector), [2]
(shuffle function)
J, Jn (symplectic unit), [2]
L (Lie derivative)
L or L (Lagrangian)
P (momentum selector), [2]
(momentum state function)
Q (coordinate selector)
q (coordinate path)
S or S (action)
[ ]
' (quote in Scheme)
, in tuple
:, names starting with
; in tuple, [2]
# in Scheme
{ } for Poisson brackets
[ ] for down tuples
[ ] for functional arguments
( ) for up tuples
() in Scheme, [2], [3]

    coordinate representation of
    coordinate-independence of, [2]
    free particle
    generating functions and
    Hamilton-Jacobi equation and
    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
Antisymmetry of Poisson bracket
Area preservation
    by maps
    Liouville's theorem and
    Poincaré-Cartan integral invariant and
    of surfaces of section, [2]
    active vs. passive in Legendre transformation
    in Scheme
    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
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
    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
    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]
    banana-like behavior of
    rotation of, [2]
Brackets, see also Poisson brackets
    for down tuples
    for functional arguments
Bulirsch-Stoer integration method
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
    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
Cantori, [2]
Cartan, Élie, see Poincaré-Cartan integral invariant
Cauchy, Augustin Louis
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
    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
    of angular-momentum Lie operators
    Jacobi identity for
    of Lie derivative
    Poisson brackets and
Compatible shape
component, [2]
    of canonical transformations, [2], [3]
    of functions, [2], [3]
    of Lie transforms
    of linear transformations
    of operators
    of rotations
Compositional canonical
Compound data in Scheme
Conditionals in Scheme
Configuration manifold
Configuration path, see Path
Configuration space
Configuration-path function ( ), [2]
Conjugate momentum
    non-uniqueness of
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
    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
    augmented Lagrangian and, [2]
    configuration space and
    as coordinate transformations
    in extended bodies
    holonomic, [2]
    integrable, [2]
    linear in velocities
    nonholonomic (non-integrable)
    on coordinates
    as subsystem couplers
    total time derivative and
Constructors in Scheme
Contact transformation, see Canonical transformations
Continuation procedure
Continued-fraction approximation of irrational number
Contraction of tuples
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]
    ignorable (cyclic)
    polar, see Polar coordinates
    redundant, and initial conditions
    rotating, see Rotating coordinates
    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
Dt (total time derivative)
d'Alembert-Lagrange principle (Jean leRond d'Alembert)
Damped harmonic oscillator
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
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)
    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
Empty list
    as sum of kinetic and potential energies
    conservation of, [2], [3]
    dissipation of, see Dissipation of energy
Energy state function ()
    Hamiltonian and
Epicyclic motion
Equilibria, [2], [3], see also Fixed points
    for angular momentum
    inverted, for pendulum, [2], [3], [4]
    linear stability of
    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 )
Evolution, see Time evolution of state
evolve, [2], [3]
Exponential divergence, [2], [3], see also Lyapunov exponent
    homoclinic tangle and
    of differential operator
    of Lie derivative
    of noncommuting operators
Expressions in Scheme
Extended phase space
    generating functions in

F1-F4, see also Generating functions
    F1(t, q, q')
    F2(t, q, p')
    F3(t, p, q')
    F4(t, p, p')
F->C, [2]
Fermat, Pierre, [2]
    Fermat's principle (optics), [2]
Fermi, Enrico
Feynman, Richard P.
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
    Poincaré-Birkhoff fixed points
    Poincaré-Birkhoff theorem
    rational rotation number and
Floating-point numbers in Scheme
Floquet multiplier
Flow, defined by vector field
    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
Free libration of the Moon
Free particle
    Lagrange equations for
    Lagrangian for
Free rigid body, see Rigid body (free)
Freudenthal, Hans
Function definition
    arithmetic operations on, [2]
    composition of, [2], [3]
    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

    axisymmetric potential of
Gamma (Scheme procedure for )
    optional argument
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
    F2 and point transformations
    F2 for polar coordinate transformation
    F2 for rotating coordinates
    incorrect derivation of
    integral invariants and
    Lagrangian action and
    Legendre transformation between F1 and F2
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
    of galaxy
    multipole expansion of
Group properties
    of canonical transformations
    of rotations, see Euler's theorem on rotations

H (Hamiltonian)
Hamilton, Sir William Rowan, [2]
Hamilton's equations
    in action-angle variables
    computation of
    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
    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]
Harmonic oscillator
    decoupling via Lie transform
    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]
    Hénon-Heiles problem
    variational principles, [2], [3], [4]
Holonomic system, [2]
Homoclinic intersection
Homoclinic tangle
    chaotic regions and
    exponential divergence and
Homogeneous function, Euler's theorem
Huygens, Christiaan
Hyperion, chaotic tumbling of, [2]

Ii, Iz (selector), [2]
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-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, Jn (symplectic unit), [2]
J-func (shuffle function)
Jac (Jacobian of map)
Jacobi constant
Jacobi coordinates
Jacobi identity
    for commutators
    for Poisson brackets
Jacobi, Carl Gustav Jacob, see also Hamilton-Jacobi equation
Jeans, Sir James, ``theorem'' of

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)
L-central-polar, [2]
L-uniform-acceleration, [2]
Lagrange equations
    at a moment
    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 polynomial
Lagrange multiplier, see Lagrangian, augmented
Lagrange, Joseph Louis, [2]
    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]
Lambda calculus
Lambda expression
Lanczos, Cornelius
Least action, principle of, see Principle of stationary action
Legendre, Adrien Marie, see Legendre...
Legendre polynomials
Legendre transformation
    active arguments in
    passive arguments in
    of quadratic functions
Leibniz, Gottfried
Libration of the Moon, [2], [3]
Lie derivative
    commutator for
    Lie transform and
    operator LH
Lie series
    for central field
    for harmonic oscillator
    in perturbation theory
Lie transforms
    advantage of
    composition of
    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
Liouville equation
Liouville's theorem
    from canonical transformation
Liouville, Joseph, see Liouville...
Lipschitz condition (Rudolf Lipschitz)
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

MacCullagh's formula
make-path, [2]
    stable and unstable
    fixed points of, see also Fixed points
    Hénon's quadratic
    representation in programs
Mars, see Phobos
Mass point, see Point mass
Mathematical notation, see Notation
Mather, John N. (discoverer of sets named cantori by Ian Percival)
    inertia, see also Inertia tensor
    Pauli spin
    symplectic, [2], [3]
    as tuple
Maupertuis, Pierre-Louis Moreau de, [2]
Mean motion
    Newtonian vs. variational formulation, [2]
Mercury, resonant rotation of, [2]
    of action
    in Scmutils, [2]
Moment(s) of inertia
    about a line
    about a pivot point
    of top
Momentum, see also Angular momentum
    conjugate to coordinate, see Conjugate momentum
    conservation of
    generalized, see Generalized momentum
    variation of
Momentum path
Momentum state function ()
    history of
    libration of, [2], [3]
    rotation of, [2], [3], [4]
Moser, Jürgen, see Kolmogorov-Arnold-Moser theorem
    chaotic, see Chaotic motion
    constrained, see also Constraint(s)
    dense, on torus
    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]
    composition of functions
    definite integral
    derivative, partial: , [2], [3]
    derivative: D, [2]
    function of local tuple
    functional, [2]
    functional arguments
    selector function: Ii, Iz, [2]
    total time derivative: Dt
    traditional, [2], [3], [4], [5], [6]
Numbers in Scheme
Numerical integration
    of Hamilton's equations
    of Lagrange equations
    in Scmutils, [2], [3]
Numerical minimization in Scmutils, [2]
Nutation of top, [2]

omega (symplectic 2-form)
    arithmetic operations on, [2]
    exponential identities
    function vs., [2]
    derivative (D), see Derivative
    Euler-Lagrange ( E )
    Lie derivative (LH)
    Lie transform (E', W)
    partial derivative ( ), see Partial derivative
    variation ( )
Optical libration of the Moon
    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
Osculation of paths
Ostrogradsky, M. V.
Out-of-roundness parameter

P (momentum selector), [2]
(momentum state function)
p->r (polar-to-rectangular)
Pairs in Scheme
Parallel tuple-valued functions
Parametric path
    with graph
    in Scheme, [2]
    for up tuples
Partial derivative, [2], [3]
    chain rule, [2]
    notation: , [2], [3]
Particle, free, see Free particle
    coordinate path (q), see also Local tuple
    momentum path
    osculation of
    realizable, see Realizable path
    variation of, [2], [3]
Path functions, abstraction of
Path-distinguishing function, [2], see also Action
Pauli spin matrices (Wolfgang Pauli)
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]
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
    of pendulum, [2]
    qualitative features, [2], [3]
    regular regions
    volume, see Phase-volume conservation
Phase space reduction
    conserved quantities and
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
Planets, see also Earth; Jupiter; Mercury
    moment of inertia of
    rotation of
    rotational alignment of
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
    general canonical transformations vs.
    generating functions for
    polar-rectangular conversion, [2]
    to rotating coordinates, [2]
Poisson brackets
    commutator and
    of conserved quantities
    as derivations
    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 transformation
    generating function for
    harmonic oscillator and
Potential, see Central force; Gravitational potential
Potential energy
    for axisymmetric top
    in Lagrangian
    multipole expansion of
    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]
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
    arithmetic operations on
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
Quasiperiodic motion, [2]
    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
Recurrence theorem of Poincaré
Recursive procedures
Reduced mass
Reduced phase space
    of phase space, see Phase space reduction
    to quadrature
ref, [2]
Regular motion, [2], [3]
    linear separation of trajectories
Resonance, see also Commensurability
    islands, see Islands in surfaces of section
    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
Rigid constraints
    as coordinate transformations
Rotating coordinates
    in extended phase space
    generating function for
    point transformation for, [2]
Rotation number
    irrational, and invariant curves
    rational, and commensurability
    rational, and fixed and periodic points
Rotation(s), see also Orientation
    composition of, [2]
    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
    as tuples
    driven, [2]
    pendulum as perturbation of
    periodically driven pendulum as perturbation of
Routh, Edward John
    Routhian equations
Runge-Kutta integration method

S or S (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
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]
    for tuples, [2]
Stability, see Equilibria; Instability; Linear stability
Stable manifold
Standard map
Stars, see Galaxy
    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 vs. Lagrangian
State path
State tuple
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
    down and
    for down-tuple components
    for momentum components, [2]
    for selectors
Summation convention
    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
Symbolic values
Symbols in Scheme
    conserved quantities and, [2]
    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, Jn, [2]
Syntactic sugar
System derivative, see State derivative

[ ]
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 evolution of state
    action and, [2]
    as canonical transformation
    in phase space
    Poincaré-Cartan integral invariant and
Time-dependent canonical transformations
Time-independence, see also Extended phase space
    energy conservation and
Time-independent canonical transformations
    Axisymmetric, see Axisymmetric top
Top banana, see Non-axisymmetric top
    in Euler's equations
    in spin-orbit coupling
Total time derivative
    adding to Lagrangians
    affecting conjugate momentum
    commutativity of
    constraints and
    notation: Dt
Trajectory, see Path; Phase-space trajectory
    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
    arithmetic on, [2]
    commas and semicolons in
    component selector: Ii, Iz, [2]
    composition and
    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
    of Lagrangian -- not!
    of phase-space description -- not!
    of realizable path
    of solution to Lagrange equations
Unstable manifold
up, [2]
Up tuples

Vakonomic mechanics
    chain rule
    of a function
    of a path, [2], [3]
    of action
Variational equations
Variational formulation of mechanics, [2]
Variational principle, see Principle of stationary action
    body components of
    in Scheme
    square of
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
Velocity, see Angular velocity; Generalized 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]