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Feynman, Orbits, and the Power of Doing It Step by Step

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​ By the time we reach Chapter 9–7 of  The Feynman Lectures on Physics , the tone of the book has subtly shifted. Feynman is no longer primarily concerned with finding elegant analytical solutions. Instead, he is showing us how physicists actually wrestle complicated systems into submission. The subject is planetary motion, but the deeper lesson is about method, approximation, and the quiet power of Newton’s laws when they are allowed to operate incrementally. Feynman begins by pointing out that techniques which work beautifully for simple systems, such as oscillating springs, do not transfer neatly to planets moving under gravity. The force law is different, and that difference matters. A planet does not experience a restoring force proportional to its displacement. Instead, the force depends on its distance from the Sun in a way that makes the mathematics stubbornly resistant to tidy solutions. Rather than fighting this, Feynman changes strategy. The core idea is to stop thinking...
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Feynman’s second-order accurate method

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One of the quiet pleasures of the  Feynman Lectures on Physics  is discovering just how modern many of Feynman’s instincts were. Long before “computational physics” became a standard course title, Feynman was already teaching students how to think like numerical analysts - cautiously, pragmatically, and with a healthy suspicion of algebraic perfection. Chapter 9-6 is a beautiful example of this mindset in action. Here, Feynman sets aside exact solutions and instead rolls up his sleeves to  actually solve  a differential equation step by step, using numbers. The problem itself is simple: a particle subject to a restoring force proportional to displacement, a(t) = -x(t), the equation of motion for a simple harmonic oscillator. But the point of the chapter is not the oscillator - it is the  method . Feynman begins with the most straightforward numerical idea imaginable: break time into small chunks and march forward. He chooses a time step ε  = 0.10 sec, and s...
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Forces, Free Fall, and a Beautiful Oscillation

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​ In Chapter 9-5 of  The Feynman Lectures on Physics , Richard Feynman turns his attention to a crucial practical question in dynamics:  if Newton’s laws tell us to look for forces, how do we actually find them?  To apply Newton’s sweeping principles of motion, one must know the specific agencies that cause objects to accelerate. This chapter is Feynman’s elegant introduction to the art of identifying those forces and understanding how they give rise to different kinds of motion. Feynman begins by emphasising Newton’s insight that whenever an object accelerates,  some cause must be at work . It is not enough to describe the motion; one must point to the force that produces it. This becomes the guiding programme for all future dynamics:  discover the laws governing the forces themselves . Newton, he reminds us, had already made remarkable progress in this direction. For gravity, Newton provided a precise rule describing how two masses influence each other. For ot...
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