Numerical Verification and Computational Illustration of the Principle of Least Action in Classical Mechanics
Lagrangian formalism, Newtonian mechanics, Computational techniques, Motion
The principle of least action has served as a fundamental pillar of classical mechanics since its inception in the mid-1700s. Based on the Lagrangian formalism, the principle dictates that the physical trajectory of an object is the one that minimizes (or extremizes) its action functional. The action is a mathematical entity defined as the integration of the system’s Lagrangian over time, where the Lagrangian encapsulates the dynamics of the system as a function of the position and the velocity of the object in question. Newton's second law of motion can be derived from this principle, which is why the Lagrangian formalism and the principle of least action are considered more fundamental. While mathematical proofs and derivations of the principle can be found in the literature, we opted to validate it through a numerical approach as this approach will aid in clarifying its physical significance. This method involves creating a Python program that generates random trajectories for an object moving freely between an initial and final point in a given potential. Using numerical calculus techniques, the program calculates the action integral of the system. For specific examples, we demonstrate that among the generated trajectories, the one that follows the principle of least action indeed matches the solution obtained from solving Newton's equation of motion, with some numerical analysis errors. Ultimately, our project attempts to describe the importance of Lagrangian formalism in mechanics over Newtonian formalism by rudimentary scrutiny of the principle of least action involving computational techniques.
"Numerical Verification and Computational Illustration of the Principle of Least Action in Classical Mechanics", IJSDR - International Journal of Scientific Development and Research (www.IJSDR.org), ISSN:2455-2631, Vol.8, Issue 10, page no.499 - 509, October-2023, Available :https://ijsdr.org/papers/IJSDR2310085.pdf
Volume 8
Issue 10,
October-2023
Pages : 499 - 509
Paper Reg. ID: IJSDR_208768
Published Paper Id: IJSDR2310085
Downloads: 000347197
Research Area: Physics
Country: Guwahati, Assam, India
ISSN: 2455-2631 | IMPACT FACTOR: 9.15 Calculated By Google Scholar | ESTD YEAR: 2016
An International Scholarly Open Access Journal, Peer-Reviewed, Refereed Journal Impact Factor 9.15 Calculate by Google Scholar and Semantic Scholar | AI-Powered Research Tool, Multidisciplinary, Monthly, Multilanguage Journal Indexing in All Major Database & Metadata, Citation Generator
Publisher: IJSDR(IJ Publication) Janvi Wave
