Goldstein Classical Mechanics Notes
Michael Good
May 30, 2004
1 Chapter 1: Elementary Principles
1.1 Mechanics of a Single Particle
Classical mechanics incorporates special relativity. ‘Classical’ refers to the con-
tradistinction to ‘quantum’ mechanics.
Velocity:
dr
v= .
dt
Linear momentum:
p = mv.
Force:
dp
F= .
dt
In most cases, mass is constant and force is simplified:
d dv
F= (mv) = m = ma.
dt dt
Acceleration:
d2 r
a= .
dt2
Newton’s second law of motion holds in a reference frame that is inertial or
Galilean.
Angular Momentum:
L = r × p.
Torque:
T = r × F.
Torque is the time derivative of angular momentum:
1
, dL
T= .
dt
Work:
Z 2
W12 = F · dr.
1
In most cases, mass is constant and work simplifies to:
Z 2 Z 2 Z 2
dv dv
W12 = m · vdt = m v· dt = m v · dv
1 dt 1 dt 1
m
W12 = (v22 − v12 ) = T2 − T1
2
Kinetic Energy:
mv 2
T =
2
The work is the change in kinetic energy.
A force is considered conservative if the work is the same for any physically
possible path. Independence of W12 on the particular path implies that the
work done around a closed ciruit is zero:
I
F · dr = 0
If friction is present, a system is non-conservative.
Potential Energy:
F = −∇V (r).
The capacity to do work that a body or system has by viture of is position
is called its potential energy. V above is the potential energy. To express work
in a way that is independent of the path taken, a change in a quantity that
depends on only the end points is needed. This quantity is potential energy.
Work is now V1 − V2 . The change is -V.
Energy Conservation Theorem for a Particle: If forces acting on a particle
are conservative, then the total energy of the particle, T + V, is conserved.
The Conservation Theorem for the Linear Momentum of a Particle states
that linear momentum, p, is conserved if the total force F, is zero.
The Conservation Theorem for the Angular Momentum of a Particle states
that angular momentum, L, is conserved if the total torque T, is zero.
2
,1.2 Mechanics of Many Particles
Newton’s third law of motion, equal and opposite forces, does not hold for all
forces. It is called the weak law of action and reaction.
Center of mass:
P P
mi ri mi ri
R= P = .
mi M
Center of mass moves as if the total external force were acting on the entire
mass of the system concentrated at the center of mass. Internal forces that obey
Newton’s third law, have no effect on the motion of the center of mass.
d2 R X (e)
F(e) ≡ M = Fi .
dt2 i
Motion of center of mass is unaffected. This is how rockets work in space.
Total linear momentum:
X dri dR
P= mi =M .
i
dt dt
Conservation Theorem for the Linear Momentum of a System of Particles:
If the total external force is zero, the total linear momentum is conserved.
The strong law of action and reaction is the condition that the internal forces
between two particles, in addition to being equal and opposite, also lie along
the line joining the particles. Then the time derivative of angular momentum
is the total external torque:
dL
= N(e) .
dt
Torque is also called the moment of the external force about the given point.
Conservation Theorem for Total Angular Momentum: L is constant in time
if the applied torque is zero.
Linear Momentum Conservation requires weak law of action and reaction.
Angular Momentum Conservation requires strong law of action and reaction.
Total Angular Momentum:
X X
L= ri × pi = R × M v + r0i × p0i .
i i
3
, Total angular momentum about a point O is the angular momentum of mo-
tion concentrated at the center of mass, plus the angular momentum of motion
about the center of mass. If the center of mass is at rest wrt the origin then the
angular momentum is independent of the point of reference.
Total Work:
W12 = T2 − T1
1
mi vi2 .
P
where T is the total kinetic energy of the system: T = 2 i
Total kinetic energy:
1X 1 1X
T = mi vi2 = M v 2 + mi vi02 .
2 i 2 2 i
Kinetic energy, like angular momentum, has two parts: the K.E. obtained if
all the mass were concentrated at the center of mass, plus the K.E. of motion
about the center of mass.
Total potential energy:
X 1X
V = Vi + Vij .
i
2 i,j
i6=j
If the external and internal forces are both derivable from potentials it is
possible to define a total potential energy such that the total energy T + V is
conserved.
The term on the right is called the internal potential energy. For rigid bodies
the internal potential energy will be constant. For a rigid body the internal
forces do no work and the internal potential energy remains constant.
1.3 Constraints
• holonomic constraints: think rigid body, think f (r1 , r2 , r3 , ..., t) = 0, think
a particle constrained to move along any curve or on a given surface.
• nonholonomic constraints: think walls of a gas container, think particle
placed on surface of a sphere because it will eventually slide down part of
the way but will fall off, not moving along the curve of the sphere.
1. rheonomous constraints: time is an explicit variable...example: bead on
moving wire
2. scleronomous constraints: equations of contraint are NOT explicitly de-
pendent on time...example: bead on rigid curved wire fixed in space
Difficulties with constraints:
4
Michael Good
May 30, 2004
1 Chapter 1: Elementary Principles
1.1 Mechanics of a Single Particle
Classical mechanics incorporates special relativity. ‘Classical’ refers to the con-
tradistinction to ‘quantum’ mechanics.
Velocity:
dr
v= .
dt
Linear momentum:
p = mv.
Force:
dp
F= .
dt
In most cases, mass is constant and force is simplified:
d dv
F= (mv) = m = ma.
dt dt
Acceleration:
d2 r
a= .
dt2
Newton’s second law of motion holds in a reference frame that is inertial or
Galilean.
Angular Momentum:
L = r × p.
Torque:
T = r × F.
Torque is the time derivative of angular momentum:
1
, dL
T= .
dt
Work:
Z 2
W12 = F · dr.
1
In most cases, mass is constant and work simplifies to:
Z 2 Z 2 Z 2
dv dv
W12 = m · vdt = m v· dt = m v · dv
1 dt 1 dt 1
m
W12 = (v22 − v12 ) = T2 − T1
2
Kinetic Energy:
mv 2
T =
2
The work is the change in kinetic energy.
A force is considered conservative if the work is the same for any physically
possible path. Independence of W12 on the particular path implies that the
work done around a closed ciruit is zero:
I
F · dr = 0
If friction is present, a system is non-conservative.
Potential Energy:
F = −∇V (r).
The capacity to do work that a body or system has by viture of is position
is called its potential energy. V above is the potential energy. To express work
in a way that is independent of the path taken, a change in a quantity that
depends on only the end points is needed. This quantity is potential energy.
Work is now V1 − V2 . The change is -V.
Energy Conservation Theorem for a Particle: If forces acting on a particle
are conservative, then the total energy of the particle, T + V, is conserved.
The Conservation Theorem for the Linear Momentum of a Particle states
that linear momentum, p, is conserved if the total force F, is zero.
The Conservation Theorem for the Angular Momentum of a Particle states
that angular momentum, L, is conserved if the total torque T, is zero.
2
,1.2 Mechanics of Many Particles
Newton’s third law of motion, equal and opposite forces, does not hold for all
forces. It is called the weak law of action and reaction.
Center of mass:
P P
mi ri mi ri
R= P = .
mi M
Center of mass moves as if the total external force were acting on the entire
mass of the system concentrated at the center of mass. Internal forces that obey
Newton’s third law, have no effect on the motion of the center of mass.
d2 R X (e)
F(e) ≡ M = Fi .
dt2 i
Motion of center of mass is unaffected. This is how rockets work in space.
Total linear momentum:
X dri dR
P= mi =M .
i
dt dt
Conservation Theorem for the Linear Momentum of a System of Particles:
If the total external force is zero, the total linear momentum is conserved.
The strong law of action and reaction is the condition that the internal forces
between two particles, in addition to being equal and opposite, also lie along
the line joining the particles. Then the time derivative of angular momentum
is the total external torque:
dL
= N(e) .
dt
Torque is also called the moment of the external force about the given point.
Conservation Theorem for Total Angular Momentum: L is constant in time
if the applied torque is zero.
Linear Momentum Conservation requires weak law of action and reaction.
Angular Momentum Conservation requires strong law of action and reaction.
Total Angular Momentum:
X X
L= ri × pi = R × M v + r0i × p0i .
i i
3
, Total angular momentum about a point O is the angular momentum of mo-
tion concentrated at the center of mass, plus the angular momentum of motion
about the center of mass. If the center of mass is at rest wrt the origin then the
angular momentum is independent of the point of reference.
Total Work:
W12 = T2 − T1
1
mi vi2 .
P
where T is the total kinetic energy of the system: T = 2 i
Total kinetic energy:
1X 1 1X
T = mi vi2 = M v 2 + mi vi02 .
2 i 2 2 i
Kinetic energy, like angular momentum, has two parts: the K.E. obtained if
all the mass were concentrated at the center of mass, plus the K.E. of motion
about the center of mass.
Total potential energy:
X 1X
V = Vi + Vij .
i
2 i,j
i6=j
If the external and internal forces are both derivable from potentials it is
possible to define a total potential energy such that the total energy T + V is
conserved.
The term on the right is called the internal potential energy. For rigid bodies
the internal potential energy will be constant. For a rigid body the internal
forces do no work and the internal potential energy remains constant.
1.3 Constraints
• holonomic constraints: think rigid body, think f (r1 , r2 , r3 , ..., t) = 0, think
a particle constrained to move along any curve or on a given surface.
• nonholonomic constraints: think walls of a gas container, think particle
placed on surface of a sphere because it will eventually slide down part of
the way but will fall off, not moving along the curve of the sphere.
1. rheonomous constraints: time is an explicit variable...example: bead on
moving wire
2. scleronomous constraints: equations of contraint are NOT explicitly de-
pendent on time...example: bead on rigid curved wire fixed in space
Difficulties with constraints:
4
