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Newtons second law, Schrodingers equation and Maxwells equations are alltheories composed of at most second-time derivatives. Indeed, it is not oftenwe need to take the time derivative of the acceleration. So why are we notseeing more higher-order derivative theories? Although several studies presenthigher derivatives usefulness in quadratic gravity and scalar-field theories,one will eventually encounter a problem. In 1850, the physicist MikhailOstrogradsky presented a theorem that stated that a non-degenerate Lagrangiancomposed of finite higher-order time derivatives results in a Hamiltonianunbounded from below. Explicitly, it was shown that the Hamiltonian of such asystem includes linearity in physical momenta, often referred to as theOstrogradsky ghost. This thesis studies how one can avoid the Ostrogradskyghost by considering degenerate Lagrangians to put constraints on the momenta.The study begins by showing the existence of the ghost and later cover theessential Hamiltonian formalism needed to conduct Hamiltonian constraintanalyses of second-order time derivative systems, both single-variable andsystems coupled to a regular one. Ultimately, the degenerate second-orderLagrangians successfully eliminate the Ostrogradsky ghost by generatingsecondary constraints restricting the physical momenta. Moreover, an outline ofa Hamiltonian analysis of a general higher-order Lagrangian is presented at theend.

Theories with higher-order time derivatives and the Ostrogradsky ghost