Goldstein Notes
by Dennil Joby
A complete transcription to ”Classical Mechanics” by Goldstein et.al.,
June 20, 2024
,Contents
1 Survey of Elementary Particles 2
1.1 Single Particle Mechanics . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Conservation Theorems . . . . . . . . . . . . . . . . . . . 3
1.1.2 Work and Energy of a Particle . . . . . . . . . . . . . . . 4
1.2 Mechanics of a System of Particles . . . . . . . . . . . . . . . . . 6
1.2.1 Linear and Angular Momentum of a System . . . . . . . . 6
1.2.2 Weak and Strong Law of Action and Reaction . . . . . . 8
1.2.3 Angular Momentum in terms of Center of Mass . . . . . . 8
1.2.4 Work and Energy . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.1 Types of Constraints . . . . . . . . . . . . . . . . . . . . . 11
1.3.2 Disk on a Surface . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 D’Alembert’s Principle . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.1 ⃗ri → ⃗qj Transformation Equations . . . . . . . . . . . . . 14
1.5 Application of the Lagrangian to Electromagnetic Systems . . . . 18
1.5.1 Expressing the force in terms of U (q, q̇) . . . . . . . . . . 19
1.5.2 Gaining the Lorentz Force from the Lagrangian . . . . . . 21
1.5.3 Dealing with ”extra” Forces . . . . . . . . . . . . . . . . . 22
1
, 1 Survey of Elementary Particles
1.1 Single Particle Mechanics
For a given coordinate system for a particle of mass m, its position be defined
by its position/radius vector ⃗r. Following this we define the particle’s velocity
and consequently the momentum respectively as;
d
⃗v = ⃗r (1)
dt
p⃗ = m⃗v (2)
while equation (1) and its derivatives together establish the kinematics of the
particle, equation (2) and especially its first derivative establish the dynamics
of the particle.
Essentially,
d
F⃗ = p⃗ (3)
dt
which is the crux of Newton’s second Law, which states that for a given frame
of reference the force exerted on a particle is proportional (equal to be precise)
to the rate of change of momentum of the particle.
In most cases we’ll see that a particle’s mass remains constant, which leads to
alternative forms of equation (3) such as,
d
F⃗ = m ⃗v
dt
d2 (4)
= m 2 ⃗r
dt
= m⃗a
Thus our equation of motion is a second order differential equation1 .
1 assuming ⃗ does not depend on higher order derivatives
F
2
by Dennil Joby
A complete transcription to ”Classical Mechanics” by Goldstein et.al.,
June 20, 2024
,Contents
1 Survey of Elementary Particles 2
1.1 Single Particle Mechanics . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Conservation Theorems . . . . . . . . . . . . . . . . . . . 3
1.1.2 Work and Energy of a Particle . . . . . . . . . . . . . . . 4
1.2 Mechanics of a System of Particles . . . . . . . . . . . . . . . . . 6
1.2.1 Linear and Angular Momentum of a System . . . . . . . . 6
1.2.2 Weak and Strong Law of Action and Reaction . . . . . . 8
1.2.3 Angular Momentum in terms of Center of Mass . . . . . . 8
1.2.4 Work and Energy . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.1 Types of Constraints . . . . . . . . . . . . . . . . . . . . . 11
1.3.2 Disk on a Surface . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 D’Alembert’s Principle . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.1 ⃗ri → ⃗qj Transformation Equations . . . . . . . . . . . . . 14
1.5 Application of the Lagrangian to Electromagnetic Systems . . . . 18
1.5.1 Expressing the force in terms of U (q, q̇) . . . . . . . . . . 19
1.5.2 Gaining the Lorentz Force from the Lagrangian . . . . . . 21
1.5.3 Dealing with ”extra” Forces . . . . . . . . . . . . . . . . . 22
1
, 1 Survey of Elementary Particles
1.1 Single Particle Mechanics
For a given coordinate system for a particle of mass m, its position be defined
by its position/radius vector ⃗r. Following this we define the particle’s velocity
and consequently the momentum respectively as;
d
⃗v = ⃗r (1)
dt
p⃗ = m⃗v (2)
while equation (1) and its derivatives together establish the kinematics of the
particle, equation (2) and especially its first derivative establish the dynamics
of the particle.
Essentially,
d
F⃗ = p⃗ (3)
dt
which is the crux of Newton’s second Law, which states that for a given frame
of reference the force exerted on a particle is proportional (equal to be precise)
to the rate of change of momentum of the particle.
In most cases we’ll see that a particle’s mass remains constant, which leads to
alternative forms of equation (3) such as,
d
F⃗ = m ⃗v
dt
d2 (4)
= m 2 ⃗r
dt
= m⃗a
Thus our equation of motion is a second order differential equation1 .
1 assuming ⃗ does not depend on higher order derivatives
F
2
