All Questions
7 questions
1
vote
0
answers
92
views
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
votes
0
answers
46
views
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
2
votes
1
answer
151
views
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
5
votes
3
answers
698
views
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
answer
624
views
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
votes
1
answer
1k
views
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
votes
1
answer
726
views
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