All Questions
3 questions
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
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
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