5. Lagrangian and Hamiltonian Methods
In a more sophisticated treatment of classical mechanics, you will hear about the Lagrangian and the Hamiltonian. These formulations can be used in place of Newton's laws, and also provided a lot of the foundation for modern physics (e.g. quantum mechanics). Without a proper in-depth treatment, you will probably struggle to see the connection to Newton's laws, and to understand the conceptual and mathematical basis of these formulations.
As these methods are formally beyond the Olympiad syllabus, you might be better off working on the topics that are in the syllabus first. Nonetheless, many of the ideas in the Lagrangian and Hamiltonian formulations are really beautiful, so we'll use this space to sketch out (very roughly!) how they look like.