1
Hamiltonian Function: Conservation Laws
Closed System: No interaction with anything outside
the system.
Any closed system has: seven constants or integrals of
motion:
• three components of the linear momentum
• three components of angular momentum
• the total energy
" Conservation of linear momentum:
• Property of inertial frame:
o Space is homogeneous in an inertial system
or frame (a closed system is unaffected by a
translation of the system in space).
• The Lagrangian of a closed system in an inertial
frame is invariant.
.const
q
L
qmp
k
kk =
∂
∂
==
2
In general:
• If the Lagrangian of a system, closed or
otherwise, is invariant with respect to a
translation in a certain direction, then the
linear momentum of the system in that
direction is constant in time.
" Conservation of angular momentum:
• Property of inertial frame:
o Space is isotropic in an inertial frame (a
closed system is unaffected by orientation or
rotation of the entire system).
• The Lagrangian of a closed system remains
invariant if the system is rotated through an
infinitesimal angle.
rq
pp
constprJ
k
kG
G
G
G
G
≡
≡
=×= .
