Quadratic Hamiltonians: The four classes of quadratic invariants, their interrelations and symmetries

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Abstract
The quadratic invariants of the three basic quadratic Hamiltonian systems‐attractive oscillator, repulsive oscillator, and free particle‐ are shown to be the same. These invariants are divided into two categories, useful and nonuseful. The definition of useful is in terms of contributing to the (quadratic invariant based) symmetry group of the appropriate Hamiltonian. Usefulness is not invariant under a time‐dependent linear canonical transformation. Hence different classes of invariants produce the different symmetry groups for the three different types of quadratic Hamiltonian considered here. The paper concludes with a consideration of a useful transformation of arbitrary quadratic Hamiltonians.