72
votes
Accepted
Six yolks in a bowl: Why not optimal circle packing?
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(...
- 10.3k
29
votes
Do bubbles between plates approximate Voronoi diagrams?
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 ...
- 188k
26
votes
Six yolks in a bowl: Why not optimal circle packing?
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 ...
- 369
23
votes
Accepted
Example of ODE not equivalent to Euler-Lagrange equation
Note: I'm updating my answer to give a better (i.e., simpler) example plus a little more information about how to derive the example from Douglas' results (which may not be entirely clear upon first ...
- 108k
22
votes
Hamiltonian, Lagrangian and Newton formalism of mechanics
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 ...
- 5,824
22
votes
Accepted
Why is the billiard problem for obtuse triangles so hard?
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, ...
22
votes
Why is the billiard problem for obtuse triangles so hard?
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 ...
- 156k
22
votes
Interpretation of the action in classical mechanics
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 ...
- 6,870
18
votes
Hamiltonian, Lagrangian and Newton formalism of mechanics
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 ...
- 7,746
18
votes
Applications of symplectic geometry to classical mechanics
The list will be long, very long indeed. But to start:
Questions about dynamics of Hamiltonian systems are at the heart of symplectic topology, symplectic capacities are precisely introduced for that ...
- 8,098
17
votes
Accepted
Construction of an optimal electron cage
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 ...
- 54.2k
17
votes
Decidability of 3 body problem
In Church's thesis meets the N-body problem Warren Smith argues that unsimulable physical systems exist in Newton’s laws of gravity and motion for point masses, because an uncountably infinite number ...
- 188k
16
votes
Accepted
Do bubbles between plates approximate Voronoi diagrams?
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 (...
- 13.6k
16
votes
Why is the billiard problem for obtuse triangles so hard?
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 ...
- 5,103
16
votes
Interpretation of the action in classical mechanics
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 ...
- 188k
15
votes
Conjecture: Finitely many points where gravitational field due to N masses vanishes
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).
...
- 91.8k
14
votes
Applications of symplectic geometry to classical mechanics
V.I. Arnold's Mathematical Methods of Classical Mechanics is entirely based on the ideas and methods of symplectic geometry, such as the Birkhoff normal form, the Kolmogorov- Arnold-Moser theorem on ...
- 188k
14
votes
Accepted
Decidability of 3 body problem
The paper Undecidability in $\mathbb{R}^n$: Riddled Basins,
the KAM Tori, and the Stability of the Solar System by Matthew W. Parker (Philosophy of Science 70 (April 2003), 359–382) comes close to ...
- 82.7k
12
votes
Accepted
Why are Lagrangian submanifolds called Lagrangian?
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 ...
- 188k
12
votes
Six yolks in a bowl: Why not optimal circle packing?
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.
...
- 221
10
votes
Accepted
Ping-pong progress through a quincunx
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 ...
- 188k
10
votes
Mathematical physics without partial derivatives
As to question 2, there are certainly plenty of non-trivial discrete models in statistical physics, such as the Ising or Potts models, or lattice gauge theories with discrete gauge groups, that ...
- 3,065
8
votes
Geometric interpretation of the half-derivative?
A simple perspective for all fractional integro-derivative operators (FID) of this type is that they satisfy the group power sum property (law of exponents)
$$D_x^{\alpha}D_x^{\beta} = D_x^{\alpha+\...
- 10.5k
8
votes
Tying knots via gravity-assisted spaceship trajectories
Yes, with some caveats. For an authoritative source, please see the monograph Koon et al. "Dynamical systems, the three-body problem and space mission design"
The short story is that with 2 or more ...
- 2,281
8
votes
Accepted
Movement of repelled particles in a ball
If all the particles remained in a bounded domain, the virial theorem would apply. In the case of a radial inverse square power law, it states that twice the asymptotic time average of kinetic energy ...
- 3,017
8
votes
history of geometric mechanics
Darboux was perhaps the first to argue fully explicitly that “the general problem of mechanics is nothing but the generalisation to an arbitrary number of variables of the problem of the study of ...
- 291
7
votes
Mathematical physics without partial derivatives
Well if you take out partial derivatives, at least quantum field theory and in particular conformal field theory will survive the massacre. The reason is explained in my MO answer:
$p$-adic numbers in ...
7
votes
Interpretation of the action in classical mechanics
While the following may be less “physically” intuitive than the question intends, I liked it when I heard it. If you believe in the importance of Poisson brackets, it is natural to ask if they make ...
- 31.5k
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