The Lagrangian and Hamiltonian Formalisms of Mechanics

Varun Santhosh Menon
The Winchester School, Jebel Ali Dubai, The Winchester School
January, 2018
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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.