Goldstein Classical Mechanics Notes
Michael Good
Classical mechanics incorporates special relativity. ‘Classical’ refers to the contradistinction
to ‘quantum’ mechanics.
Velocity:
v = dr
dt
.
Linear momentum:
p = mv.
Force:
F = dp
dt
.
In most cases, mass is constant and force is simplified:
F = d
dt
(mv) = m
dv
dt
= ma.
Acceleration:
a = d2r
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
T = dL
dt
.
Work:
W12 =
Z 2
1
F · dr.
In most cases, mass is constant and work simplifies to:
W12 = m
Z 2
1
dv
dt
· vdt = m
Z 2
1
v ·
dv
dt
dt = m
Z 2
1
v · dv
W12 = m
2
(v2
2 − v2
1) = T2 − T1
Kinetic Energy:
T = mv2
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 = −rV (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.
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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:
R =
P
Pmiri
mi
=
P
miri
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.
F(e) M
d2R
dt2 =
X
i
F(e)
i .
Motion of center of mass is unaffected. This is how rockets work in space.
Total linear momentum:
P =
X
i
mi
dri
dt
= M
dR
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
dt
= N(e).
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:
L =
X
i
ri × pi = R ×Mv +
X
i
r0
i × p0
i.
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Total angular momentum about a point O is the angular momentum of motion
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
where T is the total kinetic energy of the system: T = 1
2
P
imiv2
i .
Total kinetic energy:
T =
1
2
X
i
miv2
i =
1
2Mv2 +
1
2
X
i
miv02
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:
V =
X
i
Vi +
1
2
X
i,j i6=j
Vij .
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 dependent
on time...example: bead on rigid curved wire fixed in space
Difficulties with constraints:
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1. Equations of motion are not all independent, because coordinates are no
longer all independent
2. Forces are not known beforehand, and must be obtained from solution.
For holonomic constraints introduce generalized coordinates. Degrees of
freedom are reduced. Use independent variables, eliminate dependent coordinates.
This is called a transformation, going from one set of dependent variables
to another set of independent variables. Generalized coordinates are worthwhile
in problems even without constraints.
Examples of generalized coordinates:
1. Two angles expressing position on the sphere that a particle is constrained
to move on.
2. Two angles for a double pendulum moving in a plane.
3. Amplitudes in a Fourier expansion of rj .
4. Quanities with with dimensions of energy or angular momentum.
For nonholonomic constraints equations expressing the constraint cannot be
used to eliminate the dependent coordinates. Nonholonomic constraints are
HARDER TO SOLVE.
1.4 D’Alembert’s Principle and Lagrange’s Equations
Developed by D’Alembert, and thought of first by Bernoulli, the principle that:
X
i
(F(a)
i −
dpi
dt
) · ri = 0
This is valid for systems which virtual work of the forces of constraint vanishes,
like rigid body systems, and no friction systems. This is the only restriction
on the nature of the constraints: workless in a virtual displacement. This
is again D’Alembert’s principle for the motion of a system, and what is good
about it is that the forces of constraint are not there. This is great news, but it
is not yet in a form that is useful for deriving equations of motion. Transform
this equation into an expression involving virtual displacements of the generalized
coordinates. The generalized coordinates are independent of each other
for holonomic constraints. Once we have the expression in terms of generalized
coordinates the coefficients of the qi can be set separately equal to zero. The
result is:
X
{[ d
dt
( @T
@q˙j
) −
@T
@qj
] − Qj} qj = 0
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Lagrange’s Equations come from this principle. If you remember the individual
coefficients vanish, and allow the forces derivable from a scaler potential
function, and forgive me for skipping some steps, the result is:
d
dt
( @L
@q˙j
) −
@L
@qj
= 0
1.5 Velocity-Dependent Potentials and The Dissipation
Function
The velocity dependent potential is important for the electromagnetic forces on
moving charges, the electromagnetic field.
L = T − U
where U is the generalized potential or velocity-dependent potential.
For a charge mvoing in an electric and magnetic field, the Lorentz force
dictates:
F = q[E + (v × B)].
The equation of motion can be dervied for the x-dirction, and notice they
are identical component wise:
m¨x = q[Ex + (v × B)x].
If frictional forces are present(not all the forces acting on the system are
derivable from a potential), Lagrange’s equations can always be written:
d
dt
( @L
@q˙j
) −
@L
@qj
= Qj .
where Qj represents the forces not arising from a potential, and L contains
the potential of the conservative forces as before.
Friction is commonly,
Ffx = −kxvx.
Rayleigh’s dissipation function:
Fdis =
1
2
X
i
(kxv2
ix + kyv2
iy + kzv2
iz).
The total frictional force is:
Ff = −rvFdis
Work done by system against friction:
dWf = −2Fdisdt
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The rate of energy dissipation due to friction is 2Fdis and the component of
the generalized force resulting from the force of friction is:
Qj = −
@Fdis
@q˙j
.
In use, both L and Fdis must be specified to obtain the equations of motion:
d
dt
( @L
@q˙j
) −
@L
@qj
= −
@Fdis
@q˙j
.
1.6 Applications of the Lagrangian Formulation
The Lagrangian method allows us to eliminate the forces of constraint from the
equations of motion. Scalar functions T and V are much easier to deal with
instead of vector forces and accelerations.
Procedure:
1. Write T and V in generalized coordinates.
2. Form L from them.
3. Put L into Lagrange’s Equations
4. Solve for the equations of motion.
Simple examples are:
1. a single particle is space(Cartesian coordinates, Plane polar coordinates)
2. atwood’s machine
3. a bead sliding on a rotating wire(time-dependent constraint).
Forces of contstraint, do not appear in the Lagrangian formulation. They
also cannot be directly derived.
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