Скачать или смотреть Lagrangian Simple Pendulum | Atwood Machine | Compound | Spherical Pendulum

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In this detailed and concept-oriented lecture, we cover some of the most fundamental and highly scoring topics from Classical Mechanics and Analytical Mechanics, essential for CSIR NET Physical Sciences, GATE Physics, JEST, TIFR, B.Sc & M.Sc Physics examinations. The concepts explained here form the backbone of advanced physics and appear repeatedly in PYQs (Previous Year Questions) of competitive exams.

This session begins with a clear understanding of the Simple Pendulum, its derivation using small-angle approximation, oscillation frequency, and energy formulation. We then explore the Atwood Machine, one of the most important systems for understanding constraints, acceleration, tension, generalized coordinates, and Lagrangian formulation.

Next, we move to the Compound Pendulum, analyzing its time period, center of oscillation, and real applicability in laboratory experiments. The lecture also includes an intuitive explanation of the Spherical Pendulum, demonstrating how two degrees of freedom influence the Lagrangian, equations of motion, and trajectory of the bob.

The second part of the lecture focuses on one of the most important topics in advanced mechanics and field theory: Lagrangian in Electromagnetic (EM) Field. We derive the Lagrangian for a charged particle moving under electric and magnetic fields using relativistic intuition and understand how the vector potential and scalar potential contribute to the dynamics.

We then move to Cyclic Coordinates, a cornerstone concept for understanding conservation laws. Students preparing for CSIR NET and GATE will gain clarity on how cyclic (ignorable) coordinates lead to conserved momentum via the Euler–Lagrange equations. The connection between symmetry and conservation laws (via Noether’s theorem) is highlighted.

Finally, we discuss the Gauge Invariance of the Lagrangian, an advanced but extremely important concept. This section explains how adding a total time derivative to the Lagrangian does not affect the equations of motion, how electromagnetic potentials transform under gauge transformation, and why physics remains unchanged. This topic is frequently asked in JEST / TIFR interviews, making it essential for conceptual mastery.

This video is structured for maximum clarity, mathematical rigor, and exam relevance. Whether you are preparing for CSIR NET JRF, GATE PH, JEST, TIFR GS, or university-level exams, this lecture will significantly boost your conceptual understanding and problem-solving skills.

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