Математические методы классической механики В. И. Арнольда [1] // Николай Тюрин

If we imagine outstanding works of scientific literature as mountain routes leading into the sky, then our short course is nothing more than a stroll with a view of distant snow-white peaks. We are going to explore the visible beginnings of one of the most beautiful routes, leading far beyond the clouds, to the high passes and peaks of classical mechanics. Very soon, yesterday's schoolchildren will set out on this route themselves, but for now... let's practice a little. 1. Infinitesimal Geometry of Configuration Space Knowledge of the origins of vector fields and differential forms will be essential for this route. And since we are preparing for the route laid out by V.I. Arnold, we will follow his path in understanding these geometric objects (as well as what a differential equation is). 2. From Newton's Second Law to Hamiltonian Mechanics Aristotle believed that motion is described by a first-order differential equation. This is perhaps why the ancient Greeks, who were well versed in conic sections, described the motion of celestial bodies using epicycles. Newton described motion with a second-order differential equation; by lowering the order and moving from configuration space to phase space, we can represent everything in a simple and elegant form. 3. Poisson Brackets. Integrable Systems As we were taught in physics classes at school, it is convenient to solve problems using conservation laws. In the most general sense, this principle can be formulated as follows: if a physical quantity (= some function on phase space) commutes with a Hamiltonian (= a distinguished function on phase space that determines the motion of the system) with respect to a skew-symmetric operation (called Poisson brackets), then this quantity is an invariant of motion and is called the integral of motion. 4. Classical Mechanical Systems on Compact Phase Spaces The main difference between V.I. Arnold's approach to classical mechanics and "standard" physics courses is that it is applicable to any phase space, including the case of a compact phase space (some even credit Arnold with generalizing classical mechanics to the compact case, although Dirac clearly understood this issue well when introducing his systems with constraints). This path leads to the beginning of another route to a neighboring, yet unexplored peak—symplectic topology. Nikolai Andreevich Tyurin Vitaly Arnold Summer School "Modern Mathematics," Dubna July 20-25, 2018