Lie Symmetries and Invariants of General Time Dependent Quadratic Hamiltonian System

Eight Lie algebras of point-symmetric groups and corresponding generators are admitted by the equation of motion, which a general time-dependent quadratic Hamiltonian obtains. We show that invariant quantities obtained by eight algebraic generators are the Wronskian constant, three time-dependent conserved quantities, which are quadratic forms in position and momentum, and the trivial, 0. All obtained invariant quantities are represented by auxiliary conditions, which are two linearly independent solutions of a homogeneous differential equation of the equations of motion. Invariant variables associated with an invariant consisting of the linearity of x and p is defined. It shows that if the motion of the system is oscillatory, the Poisson bracket of the two invariant variables 10 is obtained as i, and in the case of monotonic motion, it is obtained as 1.