Newton's law with Cantor potential

In summary, we have a system where a particle with initial conditions x(0) < 0, \dot{x}(0)>0 follows a potential function U(x) defined as a Cantor steps function. The behavior of the particle's motion depends on the initial velocity, with two cases being analyzed: when the initial velocity is less than \sqrt{2}, the particle will be reflected at x=0 and travel along the negative x-axis with constant velocity, while when the initial velocity is greater than \sqrt{2}, the particle will traverse the Cantor steps region without any change in velocity until reaching x=1, where it will be instantaneously slowed down. This is due to the cumulative effect of the plateaus in the
  • #1
jostpuur
2,116
19
I want that [itex][0,\infty[\to\mathbb{R}[/itex], [itex]t\mapsto x(t)[/itex] satisfies

[tex]
\ddot{x}(t) = -\partial_x U(x)
[/tex]

where [itex]U:\mathbb{R}\to\mathbb{R}[/itex] is some potential function. Then I set the initial conditions [itex]x(0) < 0[/itex], [itex]\dot{x}(0)>0[/itex], and define

[tex]
U(x) = \left\{\begin{array}{ll}
0,&\quad x < 0\\
\textrm{Cantor steps},&\quad 0\leq x\leq 1\\
1,&\quad x > 1\\
\end{array}\right.
[/tex]

What's going to happen? How should the Newton's law be interpreted?

I've got a feeling that this has something to do with weak solutions, but it doesn't seem clear to me. For example

[tex]
0 = \int\Big(\ddot{\rho}(t) x(t) + \rho(t) \partial_xU(x(t))\Big) dt
[/tex]

with some smooth test function [itex]\rho[/itex] doesn't solve the problem, because you cannot get rid of [itex]\partial_x U[/itex] with integration by parts.
 
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  • #2
By energy conservation you see that a step in U makes an instantaneous change in velocity of the particle. You represent the Cantor steps as a limit of step functions of increasing detail. For each such approximating 'pre Cantor' potential the motion is thus well defined. My further proceeding would be to produce the (x,t)-diagram for a few pre Cantor potentials (e.g. with Mathematica) and let me inspire in guessing a limit for the final U. The behavior of x(t) will, of course, depend decisevely on the initial velocity: is the particles kinetic energy large enough to hopp over the first step of U.
Good luck for your research, looks like an interesting problem.
 
  • #3
What I proposed in my previous post as an computer experiment can be done by mere thinking:
I assume that 'Cantor steps' means indicator function of Cantor's ternary set. In particular
[itex] U(x) \in \{0,1\}[/itex].
Two cases:
[itex](a) \quad \dot{x}(0) < \sqrt{2}[/itex] (i.e. [itex] E_\text{kin} < U_\text{max}[/itex])
[itex](b) \quad \dot{x}(0) > \sqrt{2}[/itex]
The case of equality is unphysical i.e. corresponds not to a
real world situation, even after accepting reasonable idealizations.
Solution:
(a) The particle will become reflected at x=0 and will travel along the negative x-axis with constant velocity.
(b) The particle will traverse the Cantor steps region without any change of the original velocity and will at x=1 instantaneously become slowed down to velocity
[itex]\sqrt{{\dot{x}(0)}^2 - 2}[/itex]
and will continue to travel with this constant value of velocity.
The reason that the Cantor steps region is traversed with unmodified velocity is as follows:
Consider any member of the sequence, the limit of which defines Cantor's set. Then the potential is a sum of finitely many plateaus (hight 1) with valleys (hight 0) in between. We see the particle slowed down to the previously mentioned value during traversing a Cantor plateau. At the end of the plateau, the original velocity will suddenly restituted. When following the sequence towards the limit, the cummulative length's of plateaus tends to zero (this is an elementary propery of Cantor's ternary set, explained e.g. in Wikipedia) so that the effect of the occasionally reduced velocity tends to zero.
 
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What is Newton's law with Cantor potential?

Newton's law with Cantor potential is a modified version of Newton's law of universal gravitation proposed by German mathematician Georg Cantor. It takes into account the fractal nature of the universe and suggests that gravity is not a constant force but varies depending on the distance between two objects.

How does Newton's law with Cantor potential differ from traditional Newton's law?

Unlike traditional Newton's law, which assumes a constant gravitational force between two objects, Newton's law with Cantor potential suggests that gravity is not constant and varies with distance. This means that the force of gravity decreases as the distance between two objects increases, instead of remaining constant.

What evidence supports the validity of Newton's law with Cantor potential?

Some evidence for the validity of Newton's law with Cantor potential comes from observations of the orbits of stars and galaxies. These observations show that the force of gravity decreases with distance, which is consistent with the predictions of this modified law. Additionally, Cantor's theory has been used to successfully explain the behavior of black holes and other astrophysical phenomena.

How does Newton's law with Cantor potential affect our understanding of the universe?

Newton's law with Cantor potential challenges our traditional understanding of gravity and opens up new possibilities for understanding the universe. It suggests that gravity is not a constant force, but rather a variable force that can change depending on the distance between objects. This could potentially lead to new discoveries and advancements in our understanding of the fundamental laws of the universe.

Are there any criticisms of Newton's law with Cantor potential?

As with any scientific theory, there are some criticisms of Newton's law with Cantor potential. Some physicists argue that there is not enough evidence to support this modified law and that traditional Newton's law of universal gravitation is sufficient. Others argue that Cantor's theory is too complex and does not offer any significant improvements to our understanding of gravity. However, ongoing research and observations may help to address these criticisms in the future.

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