All Questions
Tagged with complex-systems mathematical-physics
12 questions
1
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Holonomic constraints as a limit of the motion under potential
In Mathematical Methods of Classical Mechanics, Arnold states the following theorem without proof in pages 75-76:
Let $\gamma$ be a smooth plane curve, and let $q_1, q_2$ be local coordinates where
$...
- 133
2
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0
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46
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Resonant and non-resonant tori density in non-degenerate system
I'm following the discussion on the page 290 of Mathematical Methods of Classical Mechanics by V. I. Arnol'd (you can download it here), and I've encountered the fact that in a nondegenerate system, ...
- 647
5
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1
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438
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The physical meaning of maximal non-integrability of the contact structure
So, basically integrability is equivalent to the existence of an integral manifold of the distribution and I guess, the integral manifold is like a plane of motion where state moves in physical sense. ...
- 383
2
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1
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151
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Arnold's holonomic constraints being limits of potential energy
The following quote comes from Arnold's "Mathematical methods in mechanics" book:
"We consider potential energy $U_N = Nq_2^2 + U_0(q_1, q_2) $, depending
on parameter $N$ (which we ...
- 409
2
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0
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465
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Liouville theorem and the ergodic assumption
I am following a course on statistical mechanics. My instructor presented us the following Liouville theorem in two (claimed) equivalent ways:
Differential statement: The probability distribution $\...
5
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3
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698
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Physical intuition behind Poincaré–Bendixson theorem
The Poincaré–Bendixson theorem states that: In continuous systems, chaotic behaviour can only arise in systems that have 3 or more dimensions. What is the best way to understand this criteria ...
- 1,079
7
votes
1
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624
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necessary and sufficient conditions for an isolated dynamical system which can approach thermal equilibrium automatically
Given an isolated $N$-particle system with only two body interaction, that is
$$H=\sum_{i=1}^N\frac{\mathbf{p}_i^2}{2m}+\sum_{i<j}V(\mathbf{r}_i-\mathbf{r}_j)$$
In the thermodynamic limit, that ...
- 6,071
9
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1
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How to properly use Perturbation Theory in classical systems?
Context: If we consider a particle in upwards motion near the Earth's surface and acted by a quadratic drag we get the non-linear eom:
$$\frac{dv}{dt}=-g-\frac{b}{m}v^2.$$
We can solve it ...
- 18k
6
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1
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726
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When can an autonomous system be written using a Hamiltonian?
If I have an autonomous series of differential equations
$$\tag{1} \frac{dx_i}{dt} ~=~ A_i(x_1,...,x_n)$$
with the condition that
$$\tag{2} \sum_{i=1}^n\frac{\partial A_i}{\partial x_i}~=~0$$
in all ...
- 452
1
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1
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505
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Hill's and Mathieu's equation [closed]
I am supposed to apply Hill's and Mathieu's equation to parametric pendulum.
Can you tell me what is the difference between them?
Why are they used?
What do they describe?
- 21
4
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73
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Solutions of nonlinear systems invariant wrt. perturbations (looking for applications)
I want to ask if the following purely mathematical problem (that I'm working on) might have some applications to physics.
The problem in a nutshell: describe properties of solution sets of real ...
- 163
3
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2
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281
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Infinitesimal input, macroscopic output
I must admit that I never got well how physicists handle infinitesimal quantities, mainly because of my education as a mathematician. So the following lines (taken from the preface of Berezin and ...
- 797