68

The system doesn't try to minimise the radius of the enclosing circle, but its potential energy. We can idealise this as non-overlapping disks in a convex rotationally symmetric potential $V$ with $V(0) = 0$. The configuration that was physically realised then has potential energy $5 V(d)$ (with $d$ the diameter of the yolks) while the configuration from ...


45

A possible mechanical interpretation of the half-derivative can be given in terms of Abel's solution to a classical problem from the calculus of variations (the tautochrone problem). Let there be a heavy particle which is constrained to slide without friction along the curve $y=y(t)$ in uniform gravity to its lowest point. Then, given a function $T(...


32

Due to chaotic behaviour of the Solar System, it is not possible to precisely predict the evolution of the Solar System over 5 Gyr and the question of its long-term stability can only be answered in a statistical sense. For example, in http://www.nature.com/nature/journal/v459/n7248/full/nature08096.html (Existence of collisional trajectories of Mercury, ...


29

A possibly new example: a ball bouncing on a parabola ($=$ graph of a quadratic polynomial) is never chaotic. The associated dynamical system has an extra invariant that makes it "integrable" (if I have the terminology correctly). The general orbit is an elliptic curve $\cal E$ equipped with a point $P$ such that going from one bounce to the next corresponds ...


29

The soap froth has a dynamics that Voronoi diagrams lack. The two-dimensional network of soap bubbles evolves in time according to the area law $$\frac{dA}{dt}=k(n-6),\qquad\qquad(*)$$ where $A$ is the area of a cell, $n$ the number of sides it has, and $k$ a coefficient determined by the surface tension of the bubbles. Supplemented by a mechanism by which ...


26

Those packing rules only apply for rigid circles. Anyone who's ever cracked an egg knows that yolks are not rigid. As a result of that, you can clearly see that the sides of yolks are flattened as they touch another yolk. So those packing rules simply don't apply.


20

Each of the different formalism of classical mechanics has its advantages and disadvantages. However, in the end all three frameworks tend to be equivalent, and thus the following list is very subjective and there may be exceptions to all points. Newton: Easily includes dissipative systems and is the only formalism that can handle non-potential forces. ...


20

It's not the case that the action is always minimized in physics -- the result is purely that it's a stationary point of the action. This has come up several times at Physics Stack Exchange: When is the principle of stationary action not the principle of least action? Is Action Always “Locally” Least? This raises the question of "Why a stationary point?", ...


19

Perhaps the cosmic censorship conjecture (the absence of singularities outside event horizons) is the most compelling, at least that is what is argued by Klainerman in Cosmic censorship and other great mathematical challenges of general relativity, with reference to Hilbert's requirement that a great problem in mathematics "should be clear and easy to ...


19

Theorem 1.1 in Richard Swartz's paper Obtuse Triangular Billiards I: Near the (2,3,6) triangle rules out easy proofs: He shows that, for any $\epsilon>0$ and any $N>0$, there is a triangle whose angles are within $\epsilon$ of $(\pi/2, \pi/3, \pi/6)$ and for which any closed path involves more than $N$ bounces. So we can't write down some finite list ...


17

The three formalisms of classical mechanics, i.e. the Newtonian, the Lagrangian (analytical mechanics) and the Hamiltonian (canonical formalism) are generally not equivalent to each other -at least not in some strict sense of the word "equivalent" which could be considered valid for the totality of physical systems. But in fact they are, for lots of systems ...


16

The answer is yes, and you can play with them at MoMath. Specifically, the Twist and Roll exhibit has several convex shapes, which will roll arbitrarily far sideways when started at an angle that is not directly downhill. Click the image to see a video of the exhibit, which shows the motion of several of the objects.


16

This paper (Batyrin and Laughlin, 2008) seems to indicate that we are doomed.


16

If $B$ depends on a single parameter $t$ then derivating with respect to $t$ the equality $$ B n_i =\lambda_i n_i $$ we deduce $$\dot{B} n_i +B\dot{n}_i=\dot{\lambda}_i n_i +\lambda_i\dot{n}_i. $$ Here we assume that $\Vert n_i\Vert =1$. Hence $\dot{n}_i\perp n_i$, $\forall i$. Taking the inner product of the above equality with $n_i$ and observing ...


16

There are two nice connections to generalizations of Voronoi diagrams that I'm aware of. Moukarzel showed that 2D soap bubbles are sectional multiplicative Voronoi partition (SMVP), i.e. 2D slices (sections) of a 3D generalized (multiplicative) Voronoi diagram where each source $f_i$ has a weight $a_i$ so that a cell $\Omega_i$ in the diagram consists of ...


15

Electrostatic potential is a harmonic function on any region without charges. It has no local minimum, in fact the value at the origin is the average of the potential over a sphere centred at the origin within your "cage". Therefore it is impossible to do what you want: there will always be a path the electron can take to escape to infinity. EDIT: Just ...


15

Start from the simplest path, a triangle with angles $\alpha, \beta, \gamma$, and build the unique triangle for which this path is a billiard path. It's easy to see that the latter triangle has angles $\frac{\alpha+\beta}{2}, \frac{\gamma+\beta}{2}, \frac{\alpha+\gamma}{2}$ and is therefore acute. Any acute triangle can be obtained in such a way. The next ...


15

I asked Rich Schwartz, who is one of the experts in this area (as noted by the OP). Here, with Rich's permission, is his response: I am not sure why it is so hard. All I can really say is that, after a lot of experimentation, I can't really see any pattern to it. It might be hard in the same way that building the fountain of youth is hard: nobody ...


14

I have some partial answers. I. It is not hard to construct a Dirichlet series $$f(z)=\sum_{n=1}^\infty a_ne^{\lambda_n z}$$ which converges to $0$ absolutely and uniformly on the real line but does not converge at some points of the complex plane. It is constructed as a sum of 3 series $f=f_0+f_1+f_2.$ Let $f_1$ be a series with imaginary exponents $\...


14

Is the following physical argument "intuitive" enough? Mud is a mixture of soil and water, and as it dries, the proportion of water decreases. As this occurs, the mud would like to contract relative to its original volume. Thus tensile stresses are generated. A crack that forms opens up a gap in the surface of the mud, which allows the mud to relax the ...


14

Even though the historical order is the other way around, it is helpful to start from wave/quantum mechanics and arrive at classical mechanics in the limit that the wave length of the particle goes to zero. Mathematically, that limit is the stationary phase approximation, meaning that classical trajectories are perpendicular to surfaces of constant phase. ...


13

Consider what path is traced out by the projectile in the 2d velocity space (horizontal velocity x-axis; horizontal is "after rotating up so the ground is flat, gravity no longer vertical"). It starts somewhere on a circular arc, and thereafter follows a path 'down and to the right' at an angle $\phi$ to the vertical, at constant speed (corresponding to ...


12

What do you mean, "the yolks don't follow optimal packing"? Sure they do. The configuration with one yolk in the center has the exact same radius as the one with six yolks distributed along the edge. It also has lower potential energy, thus the 6-circle solution you cited is a non-global optimum at best. In fact it's probably metastable, given egg yolks' ...


11

Well, there is always the trivially enforced solution $$\tag{1} S[x,\lambda]~=~\int\! dt \sum_{i=1}^3\lambda_i(t) \left(\ddot{x}^i(t)+\alpha \dot{x}^i(t) \right),$$ where $\lambda_i(t)$ are three Lagrange multiplier variables. From now on we assume that we are not allowed to use other variables than $x^i(t)$. Whether the 3D ODE $\ddot{\bf x}+\alpha\dot{\bf ...


11

Since questions on the interface between GR and QFT are admissible, here's what I consider the open problem in that direction. Fix a manifold $M$ and consider the set of globally hyperbolic solutions $\mathcal{S}(M)$ of Einstein's equations, possibly with appropriate asymptotic conditions. (a) Give $\mathcal{S}(M)$ the structure or an infinite dimensional ...


11

It seems to me that Newton formalisms is incapable of describing a quantum system(The uncertainty principle is not well addressed) while both Hamiltonian and Lagrangian are capable of describing a quantum system. So the Hamiltonian and the Lagrangian is simply a more general framework including Newton. This part is not correct. The Newton, Hamilton, and ...


11

Since you are talking about gravitation (rather than electrostatics) I assume that all charges are positive. (With charges of different sign it is easy to arrange a whole curve of equilibrium points). It is certainly non-zero if the number of points is at least 2. (From very general consideration it must be at least $N-1$ if you count multiplicitis properly....


10

This echos the 2017 comments, but since the question has now been bumped to the front page it might be helpful to give the actual source in Maslov's book [1]. [1] V.P. Maslov, Perturbation Theory and Asymptotic Methods (in Russian, Moscow, 1965); Théorie des pertubations et méthodes asymptotique (French translation, Paris, 1972). The Lagrangian manifold ...


9

A comprehensive study of the Galton board has been published by Arthur Lue and Howard Brenner, Phase flow and statistical structure of Galton-board systems (1993). Without any friction the influence of gravity vanishes in the long-time limit, because the kinetic energy overwhelms it. In that limit the Galton board reduces to a Lorentz gas, with a chaotic ...


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