Mathesis Classical Mechanics 2: Analytical Mechanics (New)

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Mathesis courses are offered free of charge and will continue to be so. You can attend them until the end and participate in the weekly tests and the final exam. However, in most courses you will be asked for a small fee —around 20€— for the issuance of a certificate of successful attendance, provided that you are entitled to it having secured the required minimum performance of 50%.

In the first lessons of mechanics we learn that, if we know the forces acting on a body, then Newton's second law is sufficient to determine its motion. However, this is not always enough. In many systems, motion is limited by constraints, and then forces appear that are not known in advance, but are determined by the motion itself.

A typical example is the ball moving in the groove of a roulette wheel: the force exerted by the groove is not a given force like weight, but depends on the speed and trajectory of the bead. Thus it becomes clear that the Newtonian formulation, although fundamental, is not always the most natural or efficient language for describing motion.https://www.youtube.com/watch?v=xzxEXU9hU34This is precisely the gap that Analytical Mechanics comes to fill. Instead of starting from individual forces, it describes motion in terms of generalized coordinates and principles that incorporate the constraints of the system from the outset. In this way, it offers a unified, elegant, and powerful framework for the study of classical mechanics, but also the necessary background for the transition to modern theoretical physics.The course is an introduction to Analytical Classical Mechanics for those who already know basic Newtonian Mechanics. The first part presents the Lagrangian formulation, starting from d'Alembert's principle, and then the Euler–Lagrange equations arise from Hamilton's principle of least action. Through this new perspective, a formulation of mechanics emerges that goes beyond the exclusively Newtonian scheme and paves the way for 20th century theoretical physics, such as quantum mechanics and general relativity. Noether's theorem is also presented, at an introductory level, in order to show the relationship between the symmetries of nature and the conservation laws, as well as the way in which conserved quantities can be identified from the form of the Lagrangian. This part concludes with a first introduction to the Hamiltonian formulation and the normal equations of motion.

In the second part of the course, analytical mechanics is applied to the study of solid body motion. Euler angles, the general description of the position and orientation of a solid body, and the general expression of its kinetic energy are introduced. Moments of inertia, the inertia tensor, and principal axes are presented, followed by the study of characteristic examples, such as rotation around a fixed axis, the natural pendulum, the spinning top, and the tippe top, all within the framework of the Lagrangian approach. The aim of the course is to provide not only technical tools for solving problems, but also a deeper understanding of the concepts that connect classical mechanics with modern physics.

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Prerequisites: Differential calculus, vector analysis, differential equations, Newtonian mechanics

https://mathesis.cup.gr/courses/course-v1:Physics+Phys4.2+26B/about

TeacherCostas Tassis Kostas Tassis is Professor of Theoretical Astrophysics at the University of Crete. He received his B.A. in Physics from the Aristotle University of Thessaloniki and his Ph.D. in Astrophysics from the University of Illinois. He worked as a postdoctoral fellow at the Institute of Cosmology at the University of Chicago from 2005 to 2008, at NASA's JPL from 2008 to 2011, and at the Max Planck Institute in Bonn from 2011 to 2012, when he moved to Crete. His research interests focus on the theory of Stellar Genesis, Plasma Physics (Magnetohydrodynamics) and Cosmology. He leads the international PASIPHAE experiment (http://pasiphae.science), which aims to map the interstellar magnetic field and interstellar magnetic dust in our Milky Way, to enable the detection of the imprint of the first moments of the Universe.

Volunteer lesson assistantChrissi Koukouraki