Conservative mechanical systems admit a symplectic structure.
However, since real systems typically exhibit energy dissipation, this symplectic structure is often too restrictive for engineering purposes.
Also in economic systems, dissipative phenomena are ubiquitous in the form of consumption and depreciation.

In this thesis, we develop an extension of the symplectic structure that does incorporate dissipation in an intrinsic manner.
This geometric structure is presented in a way that makes it usable for engineering applications, which is done in two steps.

We first construct a contact Hamiltonian system for the damped harmonic oscillator by combining the symplectic structure of conservative mechanics and the contact-geometric description of thermodynamics.
This system is then modified for the harmonic oscillator with both a parallel and serial damper.
We show how the widely adopted Caldirola-Kanai Hamiltonian for the damped harmonic oscillator emerges from the symplectification of the contact Hamiltonian system.

In order to deal with general, multi-degree of freedom systems, the contact structure is then extended to a Jacobi structure.
In contrast to the contact structure, the Jacobi structure encodes the pairing of conjugate variables and the dissipation as two separate entities. We argue that this makes it possible to construct a Hamiltonian system for any mechanical system and illustrate the practicality of this formalism by applying it to a multi-degree of freedom system.

Second, we propose split-quaternions as an alternative to the traditional matrix representation of two-dimensional linear mechanical systems.
We demonstrate how the properties of the dynamical system are directly reflected in its split-quaternion representation.
As a result, the split-quaternion representation offers several advantages for practical applications, e.g., for the classification of fixed points or when computing the system solution.
We use models of the hyperbolic plane to find a relation between the solution geometry of underdamped systems and their split-quaternion representation.