Geometry of second order degenerate lagrangians

The goal of this thesis is to present the Hamiltonian formulations of the dynamical systems generated by the second order Pais-Uhlenbeck, Sarıoğlu-Tekin and Clèment Lagrangians. Pais-Uhlenbeck Lagrangian is non-degenerate in the sense of Ostrogradsky whereas Sarıoğlu- Tekin and Clèment Lagrangians are degenerate. For the degenerate or/and constraint systems, the Legendre transformation is not possible in a straight forward way. For the degenerate systems, one additionally needs to employ, for example, the Dirac-Bergmann algorithm in order to arrive at the Hamiltonian picture. We shall follow several alternative methods while arriving at the Hamiltonian representations of Pais-Uhlenbeck, Sarıoğlu-Tekin and Clèment dynamics. At first, we first shall identify the configuration spaces, the tangent and the cotangent bundles. We shall first use Jacobi-Ostragradskii momenta to define the primary sets of constraints. Accordingly, the total Hamiltonian will be written. The Dirac-Bergmann algorithm will be run in order to identify the final constraint submanifold. In each step of the algorithm, we shall revise the total Hamiltonian by adding the secondary constraints. Once the final constraint set is determined, it is immediate to write the Hamilton's equations governing the dynamics. This is the first and most common way. An alternative way arriving at the Hamilton's equations is to construct the Dirac bracket. To do this, we shall first classify the constraints, determining the final constraint submanifold, into two classes, namely the first and the second. Then, using this classification, we shall define the Dirac brackets associated with the physical systems. There is an alternative way to arrive the Hamilton's equations. In this approach, instead of studying directly with the second order Lagrangians, we shall reduce the second order Pais- Uhlenbeck, Sarıoğlu-Tekin and Clèment Lagrangians to first order Lagrangians by introducing new coordinates and Lagrange multipliers. In this case, the reductions will give degenerate first order Lagrangians even though the second order Lagrangian is non-degenerate. We shall apply the Dirac-Bergmann algorithm for these first order formalisms in order to write the Hamilton's equations.