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Complete the following sentences:
[1] A quantity $F$ of canonical variables is a constant of motion if its Poisson bracket with the Hamiltonian is zero and if
[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
[3] Noether's theorem does not always imply that a symmetry transformation may have a conservation law associated with it. An extra condition(s) to ensure a conservation is (are)
[4] For a symmetry transformation of equations of motion to imply a conservation law, an extra sufficient requirement is
[5] An example of a transformation of coordinates which is symmetry of equation of motion, but does not imply conservation of a physical quantity is
[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
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