Acceleration
of circular orbit, 45–46
due to gravity, 71–72
of particle, 40–41, 42, 43–44, 45–46
state-space of system of particles and, 88, 89
units of, 68
total, 112
See also Principle of least action
Active change of coordinates, 131
Addition, of vectors, 24–25, 26–27
Advanced mechanics, 105–107
Angle, measure of, 19–20
Angular frequency, 45
Angular momentum, 124–125
conservation of, 140, 143–144, 219, 225–226
Earth’s orbit and, 216–217
Poisson brackets and, 178–181, 182–187
Angular velocity, radial distance and, 219–221
Antisymmetry of Poisson Brackets, 174
Aristotle’s (false) law of motion, 58–63
Axes, 15–16
Basis vectors, 25–26
Brahe, Tycho, 223
Calculus
fundamental theorem of, 50–53
of variations, 110
See also Differential calculus; Integral calculus
Canonical momentum, 203–204, 205–207
Carat, 25
Cartesian coordinates, 15–19, 121–122
Central forces, 212–215
equations of motion and, 218–221
gravitational potential energy and, 215–216
planetary orbits and, 213–215
polar coordinates and, 217–218
Centrifugal force, 120
in effective potential energy diagrams, 221, 222
effective potential energy function and, 220, 221
Chain rule, 35–36
Chaos, 14
Charges, 85–86
Circle, right triangle drawn in, 21–22
Circular motion, 44–46
Circular orbit, 222–223, 226–227
velocity and acceleration of, 45–46
Classical mechanics, 1–2
Classical physics
assumptions about time in, 17–18
cycles and conservation laws and, 12–13
defined, 1–2
dynamical systems with infinite number of states and, 10–12
limits of precision and, 13–14
minus-first law and, 8–10
systems and state-space and, 2–8
Closed systems, 2
Commutator, 173
Components, of vectors, 26
Condensed matter physics, 210
Configuration space, 91
Conjugate momentum, 123–125, 126–127
Conservation, symmetry and, 178–181, 187–189
Conservation laws
cycles and, 12–13
for simple systems, 128–130
symmetry and, 139–144
Conservation of angular momentum, 140, 216–217, 219, 225–226
Conservation of energy, 101–103, 145–152
Conservation of information, 9–10
Conservation of momentum, 92–94, 139–140
Constant, derivative of a, 34
Continuous evolution, 3
Continuous transformations, 134, 137
Convergence, 165
Coordinates, 15–19
cyclic, 125–127
generalized, 121–125
Coordinate system, Cartesian, 15–17
Coordinate transformation, 117–120, 130–135
Curl, 193–194
Cycles, 5–6
conservation laws and, 12–13
Cyclic coordinates, 125–127
Definite integral, 49
Degrees of freedom, 4–5
Euler-Lagrange equation for a single, 110–111
Del, 191–194
Delta (Δ), 30
Derivatives
calculating, 31–36
defined, 30–31
of a constant, 34
integrals and, 50–53
of powers, 32–34
rules for, 33–36
second-order partial, 76
special cases, 33–34
time, 38–39
Determinant, 82
Deterministic, 2
dynamical laws and, 8–9
laws of classical physics and, 5
Differential calculus, 29–37
partial derivatives and multivariable, 74–84
first-order, 60–63
second-order, 69–72
Direction of time, 17
Distance
angular velocity and radial, 219–221
coordinates and, 16
electric and gravitational forces and, 85–86
Divergence, 165
flow and, 165–167
of magnetic fields, 195
curl and, 194
Dot product, 27
Double pendulum, angular
momentum and, 140–144
Drag force, 85
Dynamical laws, 3–8
reversible and deterministic, 8–10
Dynamical systems, 3
with infinite number of states, 10–12
Aristotle’s law of motion, 58–63
mass, acceleration, and force, 63–66
Newton’s equations, 69–73
Earth, angular velocity and distance from Sun, 219–220
Earth’s orbit, 224
central forces and, 213–215
equation of motion for, 215
gravitational force and, 214–215
as movement in a plane, 216–217
radius of, 222–223
Effective potential energy diagrams, 221–223
Electric field, 198
Electric forces, 85–86
Electromagnetic energy, 103–104
Electrostatic energy, 103, 104
Ellipse law, 223–225
Elliptical orbit, 223–225
Energy, 95–104
conservation of (see Energy conservation)
heat, 103–104
mechanical, 103
multiple dimensions and, 99–102
potential, 95–99
rate of change of, 97–99
total, 97
Energy conservation, 97–99, 147–152
motion in uniform magnetic field and, 208–210
phase space fluid and, 162–164
symmetry and, 145–152
Equations of motion, 203–205, 218–221
determining trajectories from, 105–108
for Earth’s orbit, 215
generalized coordinates and, 121–125
Lagrange and the vector potential, 203–205
Newton-Lorentz, 204–205, 207, 209
Euler-Lagrange equations, 110–114, 202
conjugate momentum and, 123–125
derivation of, 111–114
use of, 116–121
Fields
electric, 198
gauge, 210–211
magnetic (see Magnetic fields)
vector (see Vector fields)
First-order differential equations, 60–63
Flow
divergence and, 165–167
in phase space, 162–164
stationary, 166
central (see Central forces)
centrifugal (see Centrifugal force)
on charged particle, 198–199
defining, 63–66
electric, 85–86
fictitious, 199
fundamental, 85–86
gravitational (see Gravity/gravitational force)
Lorentz, 198–199
magnetic (see Magnetic forces)
mass, acceleration, and, 63–67, 70
nonconservative, 100
potential energy and, 95–99
types of, 85
units of, 68–69
velocity and, 58–60
velocity-dependent, 198
Functional, 110
Functions
graphing, 18–19
implicit, 35
minimizing, 76–79
trigonometric, 19–23
Fundamental forces, 85–86
Fundamental theorem of calculus, 50–53
Gauge fields, 210–211
Gauge invariance, 203, 210–211
Gauge invariant, 198, 203–205, 208
Gauge transformation, 197, 201, 202, 211
Generalized coordinate system, 121–125
Generalized coordinates, 121–125
Generalized momentum, 123–125
Gibbs, Josiah Willard, 169
Gibbs-Liouville theorem, 169
Global minimum, 77
Gradient, 191–192
Graph
of functions, 18–19
of harmonic oscillator in phase space, 163–164
of integration, 47–48
of local maxima, 78
of local minima, 77
of simple harmonic motion, 42
of trigonometric functions, 21
of vector, 24
of velocity field, 165
Gravitational potential energy, 215–216
Gravity/gravitational force, 85–86, 212–216
acceleration due to, 71–72
in effective potential energy
diagrams, 221–222
effective potential energy function
and, 220–222
inverse-square law of, 222–223
on orbiting Earth, 214–215, 226–227
properties of, 214
Gyroscope, 185–187
Hamiltonian, 149–152
charged particle in magnetic field
and, 205–208
gauge fields and, 210–211
harmonic oscillator, 156–159
phase space and, 152–155
symmetry of, 157
Hamilton’s equations, 153–155
derivation of, 160–161
Poisson brackets and, 178
Harmonic oscillator, 72–73
Liouville’s theorem and, 170
phase space fluid and, 162–164
time-translation symmetry and, 145–147
Harmonic oscillator Hamiltonian, 156–159
Heat energy, 103–104
Hessian matrix, 82–84
Implicit function, 34
Incompressibility of fluid, 166–167
Liouville’s theorem and, 167–170
Indefinite integral, 50, 51–54
Inertia, law of, 63
Infinitesimal transformations, 134–136, 137
Inflection, point of, 78–79
Information, conservation of, 9–10
Initial conditions, 13–14
Integral calculus, 47–57
integration by parts, 55–57
Integrals
definite, 49
derivatives and, 50–53
integration by parts and, 55–57
Integrand, 49
Integration
formulas, 53–54
graph of, 48
by parts, 55–57
Inverse-square law of gravity, 223
Irreversible, 8–9
Kepler, Johannes, 223
Kepler’s laws, 223–227
Kilogram (kg), 68
defined, 97
Lagrange equation of motion for a charged particle, 203
Lagrangian (L), 107–111
cyclic coordinates and, 125–127
energy conservation and, 147–152
gauge fields and, 210–211
in generalized coordinate system, 123–125
Hamiltonian and, 149–152
Lorentz forces and, 199–203
in polar coordinates, 218–219
symmetry and, 130–136, 137–139
time-translation symmetry and, 146–147
use of, 116–121
Laplace, Pierre-Simon, 1–2, 13, 85
Law of inertia, 63
Law of motion, 3–10
Length, units of, 67
Levi-Civita symbol, 181–182, 183–184, 193
Limits, 29–30
of integration, 47–48
of precision, 13–14
Linear combination, of basis vectors, 26
Linearity of Poisson Brackets, 174–175
Liouville, Joseph, 169
Liouville’s theorem, 167–170
Local maximum(a), 78
Hessian and, 82
Local minimum(a), 77–78
Hessian and, 82
in higher dimensions, 80–82
Lorentz, H.A., 198
Lorentz forces, 198–199
Lagrangian and, 199–203
Magnetic fields, 194–198
equations of motion and, 203–205
gauge invariance and, 210–211
Hamiltonian of charged particle in, 205–207
motion in uniform, 208–210
on charged particles, 198–199
Lagrangian and, 199–203
Mass, acceleration, force, and, 63–66, 70
Mathematical induction, 176
Matrix, Hessian, 82–84
Maximum, local. See Local Maximum(a)
Mechanical energy, 103
Mechanical momentum, 205–206
Mechanics, axiomatic formulation of, 174–178
Meter (m), 67
Meters per second (m/s), 68
Minimizing functions, 76–79
Minimum
global, 77
local (see Local minimum(a))
Minus-first law, 8–10
Mixed partial derivatives, 76
Momentum(a)
angular (see Angular momentum)
conservation of, 92–94, 139–140
defined, 90–91
generalized, 123–125
mechanical, 205–206
phase space and, 90–92
Momentum space, 91
Motion
Aristotle’s law of, 58–63
circular, 44–46
examples of, 41–46
oscillatory, 41–42
particle, 38–41
simple harmonic, 42–44
in uniform magnetic field, 208–210
See Dynamics; Equations of motion
Multiplication, of vectors, 24, 27
Multivariable differential calculus, partial derivatives, 74–76
N-dimensional space orbit through, 114–115
principle of least action and, 114–115
6N-dimensional space
conservation of momentum in, 94
motion of system through, 90–92
Newton, Isaac, Kepler’s laws and, 223
Newton-Lorentz equation of motion, 204–205, 207, 210
Newton (N), 69
Newton’s equations of motion, 86–88
solving, 69–73
Newton’s first law of motion, 63–64, 70
Newton’s law of gravitation, 214
Newton’s second law of motion, 66, 70
Newton’s third law of motion, 92–93, 139
Noether, Emmy, 130
Nonconservative forces, 99
Open systems, 2
Orbit, 114–115
elliptical, 223–225
Origin, 15
Orthogonal vectors, 28
Oscillatory motion, 41–42
Partial derivatives, 74–76
Hessian matrix and, 82
Partial differentiation
minimizing functions and, 79–84
Partial derivatives, 74–76
stationary points and, 76–84
Particle motion, 38–41
examples of, 41–46
Particles
force on charged, 198–199
gravitational force on, 214
Hamiltonian of charged, 205–208
on a line, Hamilton’s equations for, 155
rate of change of momentum of, 93–94
Passive change of coordinates, 130–131
Period of motion, 45
Phase space, 90–92
Hamiltonian and, 152–155
harmonic oscillator in, 158–159
infinitesimal transformation of, 188
Phase space fluid, 162–164
flow and divergence, 165–167
Liouville’s theorem and, 167–170
Phi (Φ), 19
Plane
Earth’s orbit in, 216–217
Planetary orbits
central forces and, 212–215
Kepler’s laws and, 223–227
See Earth’s orbit
Plotting points, 18
Point(s), 16
of inflection, 78–79
plotting, 18
stationary, 76–79
Poisson brackets, 171–173
angular momentum and, 179–181, 182–187
rotors and precession and, 183–187
rules for, 174–178
symmetry and conservation and, 187–189
Polar coordinates, 124, 217–218
Euler-Lagrange equations in, 124–125
Position
in phase space, momentum and, 91
representing, 38
velocity as rate of change of, 39–40
Potential energy
effective potential energy diagrams, 221–223
force and, 95–99
gravitational, 215–216
in more than one dimension, 99–103
Potential energy principle, 96
Pound (lb), 69
Powers, derivatives of, 33–34
Precession, 183–187
Principle of least action, 102, 105–127
advanced mechanics, 105–107
cyclic coordinates, 125–127
derivation of the Euler-Lagrange equation, 111–114
generalized coordinates and momenta, 121–125
Lagrangian and, 107–111
N-dimensional space and, 114–115
reasons for using, 116–121
Radiation energy, 103–104
Radius, of planet’s orbit, 222–223, 226–227
Rates of change, 30
of momentum of particles, 93–94
of potential energy, 98
of total energy, 98–99
See also Differential calculus
Reaction, action and, 93–94, 139
Reference frame, 18
Relative motion, 116–121
Resolving power of an experiment, 14
Reversed time, 61–62
dynamical laws and, 8–10
Right triangle
relation among three sides of, 20
trigonometric functions and, 20
Rotation
of frames of reference, 118–121
transformations of, 134–135
Rotation symmetry
about the origin, 216
conservation law and, 139, 140
conservation of angular momentum and, 178–179, 184
Rotors, 183–187
Hessian and, 82
Second derivatives, 82
stationary points and, 83–84
Second-derivative test, 79
Second-order partial derivative, 75–76
Sigma (σ), 5
Sigma (Σ), 49
Simple harmonic motion, 42
Speed, 40
Spring balance, 64–66
State-space
defined, 2–3
dynamical systems and, 3–8
infinite systems and, 10–12, 14
system of particles and, 88–90
Stationary flow, 166
Stationary points, 77–79
in higher dimensions, 79–84
Stroboscopic, 3
Subtraction, of vectors, 25
Summa (∫), 49
Sum rule, 34
Symmetry
antisymmetry of Poisson Brackets, 174
consequences of, 137–139
conservation and, 178–181, 187–189
conservation law and, 139–144
defined, 131
examples of, 130–136
general, 136–137
rotation, 136, 140, 143, 178–181, 184–185
time-translation, 145–147
translation, 132–133
Systems, 2–8
closed, 2
open, 2
Systems of particles, 85–88
action, reaction, and conservation of momentum and, 92–94
momentum and phase space and, 90–92
space of states of, 88–90
Theta (θ), 19
Time
Aristotle’s equations of motion and, 60–63
assumptions about, 17–18
direction of, 17
reversed, 62
Time derivative, 39–41
Time translation, 146
Time-translation invariance, 146–147
Time-translation symmetry, 145–147
Trace, 82
Trajectory
determining from equations of motion, 105–107
Lagrangians and, 107–111
of particle, 38
through N-dimensional space, 114–115
through phase space, 155
Transformations
coordinate, 130–135
Translation, transformations of, 134
Translation symmetry, 132–133
Trigonometric functions, 19–23
oscillatory motion and, 41–42
Unitarity, 170
Units
of acceleration, 68
of distance, 16
of force, 68–69
of mass, 68
of velocity, 67
Unit vectors, 25
Vector equations, 66
Vector fields, 190–191
magnetic fields and, 194–195
Vector potential, 195–197
equation of motion and, 204–205
Lagrangian and, 200
Lorentz force and, 201
Vectors, 23–28
dot product, 27
in component form, 25–26
linear combinations, 25–26
magnitude, 24
multiplication by a scalar, 24, 26
orthogonal, 28
subtraction of, 25
unit, 25
Velocity
acceleration and, 40–41, 42–46
of circular orbit, 44–45
Coriolis force and, 120
force and, 58–60
momentum and, in phase space, 90–92
representing, 39–40
state-space of system of particles and, 88–90
units of, 68
Velocity-dependent forces, 198
Velocity field, 165–166
Viscous drag coefficient, 63
x-axis, 15
x-coordinate, 16
y-axis, 15
y-coordinate, 16
z-axis, 15
z-coordinate, 16