Download Generalized momenta and the Hamiltonian

If the lagrangian does not depend explicitly on the time,
then the hamiltonian is a constant of the motion:
Generalized momenta and the Hamiltonian
based on FW-20
Let’s define generalized momentum (canonical momentum):
time shift invariance implies that
the hamiltonian is conserved
for independent generalized coordinates
Lagrange’s equations can be written as:
Proof:
if the lagrangian does not depend on some
coordinate,
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!
cyclic coordinate
the corresponding momentum is a constant of the
motion, a conserved quantity.
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Three-dimensional motion in a one-dimensional potential:
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If there are only time-independent potentials and time-independent
constraints, then the hamiltonian represents the total energy.
x and y are cyclic coordinates - shift symmetry
Proof:
corresponding generalized momenta:
are conserved:
{
conservation of linear momentum
Three-dimensional motion in a one-dimensional potential:
! is a cyclic coordinates - rotational symmetry
corresponding generalized momentum:
is conserved:
conservation of angular momentum
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Bead on a Rotating Wire Hoop:
! 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.
V
E
R
W
E
I
pendulum equation
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Bead on a Rotating Wire Hoop:
! 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:
is a constant of the motion,
but it does not represent the total energy!
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