Examples of using physical intuition to solve math problems - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T09:25:35Z http://mathoverflow.net/feeds/question/46883 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46883/examples-of-using-physical-intuition-to-solve-math-problems Examples of using physical intuition to solve math problems Jack Lemon 2010-11-22T00:24:40Z 2011-09-07T22:16:22Z <p>For the purposes of this question let a "physical intuition" be an intuition that is derived from your everyday experience of physical reality. Your intuitions about how the spin of a ball affects it's subsequent bounce would be considered physical intuitions.</p> <p>Using physical intuitions to solve a math problem means that you are able to translate the math problem into a physical situation where you have physical intuitions, and are able to use these intuitions to solve the problem. One possible example of this is using your intuitions about fluid flow to solve problems concerning what happens in certain types of vector fields.</p> <p>Besides being interesting in its own right, I hope that this list will give people an idea of how and when people can solve math problems in this way.</p> <p>(In its essence, the question is about leveraging personal experience for solving math problems. Using physical intuitions to solve math problems is a special case.)</p> <hr> <p><a href="http://mathoverflow.net/questions/30149/examples-where-physical-heuristics-led-to-incorrect-answers" rel="nofollow">These</a> <a href="http://mathoverflow.net/questions/27943/in-what-ways-is-physical-intuition-about-mathematical-objects-non-rigorous" rel="nofollow">two</a> MO questions are relevant. The first is aimed at identifying when using physical intuitions goes wrong, while the second seems to be an epistemological question about how using physical intuition is unsatisfactory.</p> http://mathoverflow.net/questions/46883/examples-of-using-physical-intuition-to-solve-math-problems/46886#46886 Answer by Qiaochu Yuan for Examples of using physical intuition to solve math problems Qiaochu Yuan 2010-11-22T00:53:57Z 2010-11-22T00:53:57Z <p>Mark Levi's book <a href="http://www.amazon.com/Mathematical-Mechanic-Physical-Reasoning-Problems/dp/0691140200" rel="nofollow">The Mathematical Mechanic</a> is full of elementary and beautiful examples of this kind. Some examples are also given in <a href="http://concretenonsense.wordpress.com/2010/06/24/some-gems-from-physical-mathematics/" rel="nofollow">this blog post</a> by Yan Zhang.</p> <p>A classic example is a "proof" that there exist non-constant meromorphic functions on a compact Riemann surface, which I think is due to Klein: see <a href="http://mathoverflow.net/questions/19649/physical-construction-of-nonconstant-meromorphic-functions-on-compact-riemann-s" rel="nofollow">this MO question</a>.</p> http://mathoverflow.net/questions/46883/examples-of-using-physical-intuition-to-solve-math-problems/46890#46890 Answer by Steven Landsburg for Examples of using physical intuition to solve math problems Steven Landsburg 2010-11-22T02:09:08Z 2010-11-22T02:09:08Z <p>The first and second laws of thermodynamics allow you to recover the inequality between the arithmetic and the geometric means: Bring together n identical heat reservoirs with heat capacity C and temperatures T_1,...T_n and allow them to reach a final temperature T. The first law of thermodynamics tells you that T is the arithmetic mean of the T_i. The second law of thermodynamics demands the non-negativity of the change in entropy, which is </p> <p>Cn Log(T/G) </p> <p>where G is the geometric mean. It follows that T > G.</p> <p>I believe this argument was first made by P.T. Landsberg (no relation!). </p> http://mathoverflow.net/questions/46883/examples-of-using-physical-intuition-to-solve-math-problems/46896#46896 Answer by Aaron Meyerowitz for Examples of using physical intuition to solve math problems Aaron Meyerowitz 2010-11-22T02:58:10Z 2010-11-22T02:58:10Z <p>I gave an answer based on surface tension (which I did not invent) to the <a href="http://mathoverflow.net/questions/37295/advanced-view-of-the-napkin-ring-problem" rel="nofollow">napkin ring problem</a></p> http://mathoverflow.net/questions/46883/examples-of-using-physical-intuition-to-solve-math-problems/46905#46905 Answer by Gideon Schechtman for Examples of using physical intuition to solve math problems Gideon Schechtman 2010-11-22T05:06:40Z 2010-11-22T05:06:40Z <p>For electrical network intuition/applications to random walks see the beautiful little book of Doyle and Snell <a href="http://arxiv.org/abs/math/0001057" rel="nofollow">http://arxiv.org/abs/math/0001057</a></p> http://mathoverflow.net/questions/46883/examples-of-using-physical-intuition-to-solve-math-problems/46918#46918 Answer by Darsh Ranjan for Examples of using physical intuition to solve math problems Darsh Ranjan 2010-11-22T07:27:00Z 2010-11-22T07:27:00Z <p>Polya's <em>Induction and Analogy in Mathematics</em> has a chapter on this, along with some great examples. It's not just physical intuition influencing mathematics; it's more a powerful synergy between physical and mathematical intuition. I'll summarize some of it:</p> <ol> <li><p>Suppose we have two points A and B on the same side of some line L in the plane. What's the shortest path from A to L and then to B? The solution is obvious once we reflect one of the points (and its segment of the path) across L. That solution seems tricky in the abstract, but it's very intuitive if we imagine a reflecting ray of light and think about looking at things in a mirror. </p></li> <li><p>Now suppose A and B are on different sides of L, and a particle moves from A to B, and its speed is different on the two sides of L. What's the shortest path (in time)? (This problem is to a <em>refracting</em> ray of light as the previous one is to a <em>reflecting</em> ray.) It turns out this can be solved by reducing it to a physical problem involving a system of weights and pulleys at equilibrium. I won't try to describe it here, but it might be fun to try to reinvent it. </p></li> <li><p>Now let's take a serious math problem: what plane path minimizes the time an object takes to move from point A (at rest) to point B, assuming constant gravity? (This is the famous "brachistochrone" problem.) By conservation of energy, the speed of the object at a point on the curve depends only on its <em>height</em> (defined relative to its starting point and with respect to the direction of gravity). Thus, we're led to consider light moving in a very particular heterogeneous refracting medium, where the index of refraction depends in a specific way on the height. To find the path taken by light, we simply apply the law of refraction to this medium to obtain a differential equation for the path, which we can then solve. </p></li> </ol> <p>The interplay between mathematical and physical intuition is very interesting here. The first problem is mathematical, but in trying to solve it, it's natural to draw an analogy to optics. The second problem is suggested by optics, but we solve it by analogy with mechanics. The third problem is basically mechanical, but we solve it by analogy with optics, and we actually use the solution to the second problem! </p> http://mathoverflow.net/questions/46883/examples-of-using-physical-intuition-to-solve-math-problems/46919#46919 Answer by Kevin O'Bryant for Examples of using physical intuition to solve math problems Kevin O'Bryant 2010-11-22T07:46:22Z 2010-11-22T07:46:22Z <p>Let $X$ be a random variable taking on $n$ distinct values with probabilities $p_1,\dots,p_n$. The entropy of $X$ is defined by $H(X)=\sum p_i \log_2(1/p_i)$. An early theorem is that $H(X) \leq \log_2(n)$, and here's a physical proof. Place a point with mass $p_i$ at $(x_i,y_i)=(1/p_i,\log(1/p_i))$. The center of mass $$(\bar x,\bar y) = \frac{\sum (m_ix_i, m_iy_i)}{\sum m_i} = (n,H(X))$$ of the $n$ points must lie in the convex hull of the points (this is the physical intuition part). But since $y=\log(x)$ is concave, the convex hull is completely below (or on) the curve $y=\log(x)$. That is, $H(X) \leq \log_2(n)$.</p> http://mathoverflow.net/questions/46883/examples-of-using-physical-intuition-to-solve-math-problems/46923#46923 Answer by Franz Lemmermeyer for Examples of using physical intuition to solve math problems Franz Lemmermeyer 2010-11-22T08:46:08Z 2010-11-22T08:46:08Z <p>Archimedes gave exact proofs as well as mechanically motivated explanations for results like the quadrature of the parabola or the volume of spheres. </p> http://mathoverflow.net/questions/46883/examples-of-using-physical-intuition-to-solve-math-problems/46924#46924 Answer by Christian Blatter for Examples of using physical intuition to solve math problems Christian Blatter 2010-11-22T09:22:24Z 2010-11-22T09:22:24Z <p>Here is a proof of Pick's area theorem $\mu(P)=i +{b\over2}-1$ "using physical intuition": Assume that at time 0 a unit of heat is concentrated at each lattice point. This heat will be distributed over the whole plane by heat conduction, and at time $\infty$ it is equally distributed on the plane with density 1. In particular, the amount of heat contained in $P$ will be $\mu(P)$. Where does this amount of heat come from? Consider a segment $e$ between two consecutive boundary lattice points. The midpoint $m$ of $e$ is a symmetry center of the lattice, so at each instant the heat flow is centrally symmetric with respect to $m$. This implies that the total heat flux across $e$ is 0. As a consequence, the final amount of heat within $P$ comes from the $i$ interior lattice points and from the $b$ boundary lattice points. To account for the latter, orient $\partial P$ so that the interior is to the left of $\partial P$. The amount of heat going from a boundary lattice point into the interior of $P$ is a half, minus the turning angle of $\partial P$ at that point, measured in units of $2\pi$. Since the sum of all turning angles for a simple polygon is known to be one full turn, we arrive at the stated formula.</p> http://mathoverflow.net/questions/46883/examples-of-using-physical-intuition-to-solve-math-problems/60268#60268 Answer by Chandan Singh Dalawat for Examples of using physical intuition to solve math problems Chandan Singh Dalawat 2011-04-01T08:55:22Z 2011-04-01T08:55:22Z <p>Read the following paper for some striking examples.</p> <p>MR2587923 Atiyah, Michael; Dijkgraaf, Robbert; Hitchin, Nigel <a href="http://rsta.royalsocietypublishing.org/content/368/1914/913" rel="nofollow">Geometry and physics.</a> Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 368 (2010), no. 1914, 913–926.</p> http://mathoverflow.net/questions/46883/examples-of-using-physical-intuition-to-solve-math-problems/74803#74803 Answer by Pedro Lauridsen Ribeiro for Examples of using physical intuition to solve math problems Pedro Lauridsen Ribeiro 2011-09-07T22:16:22Z 2011-09-07T22:16:22Z <p>A paradigmatic example is Riemann's original "proof" of his <a href="http://en.wikipedia.org/wiki/Riemann_mapping_theorem" rel="nofollow">mapping theorem</a> in complex analysis. He gave an heuristic argument using Dirichlet's principle which was motivated by electrostatics in the plane.</p>