You Will Never Look At Laplace The Same Way Again

Get a free ebook: https://animatedmath.gumroad.com/l/se... Become a member:    / @animatedmathofficial   Laplace's equation is usually taught as three second derivatives. Here it becomes a rule about averages, gravity, planetary drift, and empty space. The story begins with Jupiter and Saturn drifting away from Newton's predictions, follows Laplace's attempt to explain their roughly 900-year rhythm, and then asks what the symbol nabla squared actually measures. Starting from Taylor's expansion, we build the one-dimensional idea, carry it onto a sphere, and connect the result to harmonic functions, gravitational potential, Poisson's equation, Earnshaw's theorem, heat, soap films, and the limits of long-range prediction. By the end, you will see why a harmonic value equals the average around it, and why a nonconstant harmonic field cannot hide a strict maximum or minimum inside an empty region. CHAPTERS 0:00 What the triangle really means 1:21 The centuries-long Jupiter-Saturn drift 3:17 Calculate the 5:2 near-resonance 5:52 Why three bodies break the ellipse 9:37 Deriving the neighbourhood-average meaning 16:29 Why empty space cannot contain a bowl Next time, we put the source back in, fix the potential on a boundary, and solve the resulting boundary-value problem from start to finish. Subscribe so you catch it. If I got a detail wrong, tell me in the comments. I read them, and I would rather fix it than defend it. REFERENCES ORIGINAL WORKS AND HISTORICAL SOURCES (FREE) Pierre-Simon Laplace, Theorie des attractions des spheroides et de la figure des planetes (1785): https://fr.wikisource.org/wiki/%C5%92... Pierre-Simon Laplace, A Philosophical Essay on Probabilities, translated excerpt: https://mathshistory.st-andrews.ac.uk... Dictionary of Scientific Biography, Laplace (free PDF): https://mathshistory.st-andrews.ac.uk... MATHEMATICS AND ASTRONOMY (FREE) Encyclopedia of Mathematics, Harmonic function: https://encyclopediaofmath.org/wiki/H... NASA/JPL, Planetary Physical Parameters: https://ssd.jpl.nasa.gov/planets/phys... Scott Tremaine, Is the Solar System Stable?, Institute for Advanced Study: https://www.ias.edu/ideas/2011/tremai... ANIMATION AND COPYRIGHT All animation sequences in this video were designed and built by the Animated Maths team. No stock footage or third-party video clips were used. (c) Animated Maths. All rights reserved. Reuse, re-upload, or redistribution without written permission is not allowed. Covered in this video: Laplace's equation explained, the Laplacian, nabla squared, harmonic functions, the mean-value property, the maximum principle, gravitational potential, Poisson's equation, Earnshaw's theorem, Taylor series, multivariable calculus, partial differential equations, Jupiter and Saturn's great inequality, celestial mechanics, the three-body problem, Laplace's demon, and deterministic chaos. #LaplacesEquation #Laplacian #Mathematics #Calculus #AnimatedMaths