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
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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.
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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!
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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
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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,
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