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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. 95 Three-dimensional motion in a one-dimensional potential: 97 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 96 98 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 99 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! 100