Atwood’s machine (elevator):

constraint:

(reaction forces correspond to variations of generalized coordinates

that violate the constraints)

the forces of constraint are tensions:

3 eqns. for 3 unknown

tension force93

One cylinder rolling on another: (with r and θ and θ as generalized coordinates)1 2(although there is just one degree of

freedom, θ , if cylinder is not slipping and remains in contact with the other one)

1

constraints:

the forces of constraint are the normal force, and friction force:

5 eqns. for 5 unknown

rotational kinetic energy

potential energy

kinetic energy of the center of mass

94

One cylinder rolling on another: (with r and θ and θ as generalized coordinates)1 2

constraints:

5 eqns. for 5 unknown:

(1)

(2)

(3)

(4)

(5)

(1):

(3):(5)

(2):

can be integrated:

eq. of motion for the only independent coordinate

constant corresponds to cylinder starting at rest at the top

cylinders stay in contact as far as .

angle of separation:beyond this point we need all three variables, the motion is

described by eqs. of motion with lagrange multipliers set to 0.

95

Generalized momenta and the Hamiltonianbased on FW-20

Let’s define generalized momentum (canonical momentum):

for independent generalized coordinates

Lagrange’s equations can be written as:

if the lagrangian does not depend on some coordinate,

cyclic coordinatethe corresponding momentum is a constant of the motion, a conserved quantity.

related to the symmetry of the problem - the system is

invariant under some continuous transformation.

For each such symmetry operation there is a

conserved quantity!

96

Three-dimensional motion in a one-dimensional potential:

x and y are cyclic coordinates - shift symmetry

corresponding generalized momenta:

are conserved:

conservation of linear momentum

Three-dimensional motion in a one-dimensional potential:

ϕ is a cyclic coordinates - rotational symmetry

corresponding generalized momentum:

is conserved:

conservation of angular momentum

97

Proof:

If the lagrangian does not depend explicitly on the time, then the hamiltonian is a constant of the motion:

time shift invariance implies that the hamiltonian is conserved

98

If there are only time-independent potentials and time-independent constraints, then the hamiltonian represents the total energy.

Proof:

{99

Bead on a Rotating Wire Hoop:θ is the generalized coordinate

hoop rotates with constant angular velocity about an axis perpendicular to the plane of the hoop and passing through the edge of the hoop. No friction, no gravity.

pendulum equation

REVIEW

100

Bead on a Rotating Wire Hoop:θ is the generalized coordinate

hoop rotates with constant angular velocity about an axis perpendicular to the plane of the hoop and passing through the edge of the hoop. No friction, no gravity.

generalized momentum:

the hamiltonian:

but it does not represent the total energy!is a constant of the motion,

101