mechanics
geometric principles, action, and the discovery of structure
It becomes the task of future engineers, physicists, and mathematicians to formulate an approach to mechanics that underlines a gradual discovery of geometrical principles. I standardize the imperative of differential geometry to be utilized in mechanical systems.
I aim to paint an impressionistic and compressed picture of what we know about physical/mathematical mechanics from the late 20st century body of work, of which all of our institutions are unable to adequetly teach during the dark technocratic ages of the early-21st century. I also hope to instill a passion into the reader to play their part in the renaissance of fundamental physics. Mechanics is the first stop.
the program
Everything in mechanics begins with the question of how nature decides to determine motion. Very quickly the Newtonian perspective becomes inefficient when the roles of the fundamental quantities energy and momentum compel Lagrangian and Hamiltonian mechanics. Motion becomes described on the tangent bundle and is governed by the extremization of action. Geometric reasoning is born when trajectories in Euclidean space are replaced by flows and integral curves of ordinary differential equations written in generalized coordinates. It is also worth studying why some differential equations are solvable and why some are not, by methods of continuous transformation groups. In doing so, fluency with symmetry algebras is ascertained correctly before quantum mechanical descriptions.
Indeed, there are some needed mathematical tools when pursuing mechanics. Linear algebra, numerics, and exterior calculus are skills that must be acquired while following through with the subject in a serious manner. Naturally, local existence and global properties of phase portraits become important characterizations concerning the theory of dynamical systems. The geometrical move of mechanics is then the replacement of the tangent bundle with the cotangent bundle to construct the symplectic form and provide a first real explanation for why mechanics possesses so many conserved structures in a way that is satisfying to the mathematical physicist.
Control theory then invokes the following questions: can motion be optimized, or can we steer the system? And as several vector fields or parameters are present, bifurcation theory elucidates the true modernity of geometrical methods in mechanics. It seeks to understand qualitative changes of dynamics and allowed motions, of which it should be obvious that these notions pertain to the immediate goals of engineering.