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The following sentences in blue can be completed as indicated:
[1] A quantity $F$ of canonical variables is a constant of motion if its Poisson bracket with the Hamiltonian is zero and if
$F$ does not depend explicitly on time.
[2] The Hamiltonian will always have zero Poisson bracket with itself. However there is a situation when the Hamiltonian is not a conserved quantity. This will happen if
the Hamiltonian $H$ depends on time explicitly. For example a charged
[3] Noether's theorem does not always imply that a symmetry transformation of equations of motion, may have a conservation law associated with it. An extra conditions to ensure conservation is are
(a) symmetry transformation must be continuous, and (b) invariance of Lagrangian is required.
[4] For a continuous symmetry transformation of equations of motion to imply a conservation law, an extra sufficient requirement is
existence of an action principle giving rise to the equations of motion.
[5] Examples of systems for which a transformation of coordinates which is symmetry of equation of motion, but does not imply conservation of a physical quantity are
(1) Motion in viscous medium; (2) Charged particle in uniform electric field.
[6] Noether's theorem states that if under a symmetry transformation of equations of motion such that the numerical value of action remains unchanged, then there is a conservation law. However for this statement to hold an important condition on the transformation must be true. This requirement is that
the transformation must be a continuous transformation
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