Varun Santhosh Menon |

The Winchester School, Jebel Ali Dubai, The Winchester School |

January, 2018 |

Full text (external site) |

## Abstract |

This paper serves to demonstrate and summarize my research into Analytical Mechanics. Advanced mechanics is an area of physics that Physics students are certainly never exposed to due to the mathematical complexity of the subject. However, concepts in analytical mechanics are extended to cutting edge areas of modern physics, particularly quantum physics, statistical physics and relativistic physics. The Lagrangian and Hamiltonian reformulations of classical mechanics are rich in beautiful, elegant mathematics and present a far more general and abstract way of understanding observable phenomena than purely Newtonian approaches. Moreover, these concepts are extended and abstracted to other areas of theoretical physics, so an understanding of these formulations is invaluable to a physics student. Computing physical problems using Lagrangians and Hamiltonians are remarkably easy and reduce the computational complexity of solving for the dynamics of large complex physical systems. The difficult part is following and understanding the logic behind the development of these esoteric concepts. The mathematical proofs are not arduous by any means, yet the challenge lies in understanding how and why some of the greatest physicists and mathematicians of antiquity even thought of these things. This was by far the biggest enigma I faced in my research as there are little to no resources that explain the motivation and more importantly, the thought process behind analytical mechanics. A lot of proofs in textbooks, though logically sound, will work backwards from the premise instead of working towards the premise which leaves one wondering how some concepts were conceived of at all. Thus, this paper will largely focus on logic and intuition with minimal computation and problem solving as there are already a plethora of detailed resources available that provide myriad rigorous examples and practice problems. Furthermore, I will also investigate the Lagrangian and Hamiltonian formalisms through a geometric interpretation. |