All Questions
Tagged with ds.dynamical-systems mp.mathematical-physics
81 questions
1
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0
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48
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Rigorous analysis of phase transitions and universality in a non-linear model of interacting oscillators
Consider a system of interacting non-linear oscillators governed by the McKean-Vlasov equation:
$$\frac{\partial p(x,t)}{\partial t} = \frac{\partial}{\partial x}\left[\frac{\partial V(x)}{\partial x}...
user530766
1
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0
answers
108
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Stability of rigid bodies spinning around $z$-axis under gravity
Consider the problem of a rigid body rotating in 3D space under gravity with one point fixed. I am particularly curious about the equilibrium state where the body is spinning at a constant angular ...
- 807
3
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0
answers
56
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Perturbation method for time-periodic singular system of ODEs
I am studying a problem arising in physics, and I managed to simplify it to a differential system (initial value problem) of the form:
$$
\begin{cases}
\dot{x} = \epsilon f_1(x,y,t) + \epsilon^2 f_2(...
- 131
2
votes
1
answer
119
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Any theorem shows that flowmap $\phi_{\sum_{i=1}^n a_i f_i(x)}^\tau$ can be approximated by $\phi_{f_{\theta(t)}(x)}^{\tau'}$?
Given a control family $F:=\{f_1,\dotsc,f_n\}$, and $\phi_f^\tau(x)$ is the flowmap of the dynamical system
$$
\begin{cases}
z'(t)=f(z),\\
z(0)=x,
\end{cases}
$$ at end time point $\tau$.
Suppose $a_i&...
- 109
21
votes
0
answers
416
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Can a 4D spacecraft, with just a single rigid thruster, achieve any rotational velocity?
(Copied from MSE. Offering four bounties over time, I got no response, other than twenty-nine upvotes.)
It seems preposterous at first glance. I just want to be sure. Even in 3D the behaviour of ...
- 281
6
votes
1
answer
328
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Solution of an ODE upon singular perturbation
The following question originates from a Physics problem, so I apologize if I am not using a suitable mathematical jargon.
The original system involves $N$ massless electric charges at position $\...
- 255
7
votes
1
answer
142
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Hamiltonian-ization of a dynamic system
On affine space, a sufficiently smooth continuous-time Hamiltonian dynamic system $\dot p = \nabla_q H, \dot q = -\nabla_p H$ conserves $H$, preserves the volume form (e.g. if we are looking for ...
- 200
2
votes
1
answer
371
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Examples of ODEs with complex constant coefficients and applications to physics?
This question is asked on stackexchange: Are there examples for ODEs with complex coefficients with applications in physics?
but received no answers. I am reposting it here on the hope that it catches ...
- 852
6
votes
2
answers
3k
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Poincaré recurrence and its implications for statistical physics and the arrow of time
A very important theorem in mathematical physics is Poincaré’s recurrence theorem.
As you recall, this theorem states that given a dynamical system $(M , \phi , \mu)$ with $ \mu M < +\infty$, for ...
- 1,514
6
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0
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498
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“Cohomological equation” in dynamical systems
Let $$\dot{x}=Ax+v_r(x)+v_{r+1}(x)+ \dots$$ with $x \in \mathbb{C}^n$ and $v_r: \mathbb{C}^n \to \mathbb{C}^n$ a homogenous, polynomial function of order $r.$
Then, being able to find a suitable $h$ ...
- 1,514
3
votes
1
answer
524
views
Has the von Neumann entropy ever been used in classical mechanics?
After going through an application of the von Neumann entropy(from quantum information theory) to certain problems in computational neuroscience [2], it occurred to me that this entropy might have ...
- 3,871
18
votes
2
answers
2k
views
Renormalization in physics vs. dynamical systems
I am studying complex dynamics, so to me renormalization of a dynamical system means something like a rescaled first-return map on (a subset of) the underlying space. I understand that in quantum ...
- 181
0
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0
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72
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Li-Yorke sensitivity Vs Li-Yorke dense chaos
Let $X$ be a compact metric space, $X*X$ its cartesian product, and $A$ a subset of $X*X$.
Are the following two properties the same, or e.g. one is stronger than the other?
$A$ is dense and residual ...
- 141
3
votes
0
answers
73
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What is known about discrete versions of the spatially homogenous Boltzmann equation with finitely many (but arbitrarily many) velocities?
Consider a discrete version of spatially homogenous Boltzmann equation with finitely many (but arbitrarily many) velocities $v_i \in \mathbb R^n$ with $i \in I$. Equivalently, consider a system of ...
- 111
0
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0
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64
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Implications for a simple deterministic chaos definition
Among many others, one definition of deterministic chaos terms "chaotic" a classical dynamical system that satisfies the following three properties:
sensitive dependence to initial ...
- 141
0
votes
1
answer
214
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Hamilton equations-Symplectic scheme [closed]
We know that $\dot{q} = \frac{\partial H}{\partial p}$ and $\dot{p} = -\frac{\partial H}{\partial q}$, and we also know the values $Q$ and $P$ respectively of $q$ and $p$ at a later time step $\Delta ...
- 111
2
votes
0
answers
73
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Nonintegrable classical dynamical systems and deterministic chaos
I'm trying to delineate a minimal (and informal) "taxonomy" for classical continuous dynamical systems that could be interested by the phenomenon of "chaos" - unfortunately the ...
- 141
5
votes
3
answers
643
views
What quantities are conserved under a general gradient-flow $\dot X(t) = -\nabla L(X(t))$?
Let $L:\mathbb R^N \to \mathbb R$ be a continuously differential function with gradient $x \mapsto \nabla L(x)$ and consider induced gradient-flow
$$
\dot X(t) = -\nabla L(X(t)).
$$
Question. Is ...
- 6,853
9
votes
1
answer
800
views
Why the least action principle is always (?) used in this particular form?
The least action principle in (mathematical) physics says the following. Given a system, e.g. collection of particles, whose motion satisfies a known system of differential equations (of second order)...
- 21.8k
3
votes
2
answers
434
views
Classification of Lagrangians with given Euler-Lagrange equations
In (mathematical) physics the equations of motion of a system of particles are often interpreted as Euler-Lagrange equations for appropriate Lagrangian $L=L(x,\dot x,t)$ where $x$ is a collection of ...
- 21.8k
1
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0
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76
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What exactly are the benefits of keeping a Hamiltonian system of equations Hamiltonian during solving or transformation?
When faced with a system of differential equations that happens to be Hamiltonian in form, or a perturbation of a Hamiltonian system, we often see in classical work a clear attempt to pursue solutions ...
- 183
3
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0
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127
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Rigorous stability analysis of infinite dimensional ODEs : How to bound the tails?
My question is about linear stability analysis of dynamical systems obtained by discretizing linear(ized) partial differential equations. Consider,
$\dot{x}=Ax$, where $x$ is the infinite dimensional ...
- 2,281
6
votes
2
answers
237
views
Movement of repelled particles in a ball
EDIT:
Given a system of $N\geq 3$ charged point particles in $\mathbb{R}^3$ of the same charge which interact according to Coulomb law (thus they repell one from each other). Is it possible that ...
- 21.8k
4
votes
1
answer
334
views
Examples of particle systems with higher-order collisions
In kinetic theory, one often comes across interacting particle systems with a collisional flavour. I'll currently prefer to think about them as systems of ODEs (or SDEs, Jump Processes, $\ldots$), ...
- 801
2
votes
0
answers
100
views
About geometric quantisation and application to real system
Quantisation is a important step to properly define a quantum system from a classical one. In a nutshell :
On a symplectic manifold $(M,\omega)$ and an algebra of function $f$ on $M$, one defines an ...
- 4,361
2
votes
1
answer
115
views
stationary measure for linear cocycle(random transformation matrices)
Let $(M,\mathcal B, \mu)$ be a probability space which $M=\{A_{1},A_{2},...,A_{N}\}^{\mathbb{N}}$ ($A_{i} \in GL(d ,\mathbb{R})$) and $\mu=p^{\mathbb{N}}$. Let $F:M\times \mathbb R^d\to M\times \...
- 199
7
votes
1
answer
930
views
(In)stability of a two-dimensional dynamical system
Consider the following system of coupled differential equations
\begin{eqnarray*}
\dot{x}_1(t) & = & -x_1(t) - \cos(\omega t)x_1(t) + \cos(\omega t)x_2(t), \ x_1(0)\in\mathbb{R},\\
\dot{x}_2(t)...
- 2,712
4
votes
1
answer
299
views
Symplectic forms and sign of eigenvalues
This question has come out while reading J. Moser "New Aspects in the Theory of Stability
of Hamiltonian Systems". I'm particularly interested to the Appendix, where one investigates the stability of ...
- 255
6
votes
1
answer
238
views
Possibly new solution to equal-mass three-body problem; refinement required
(This is a repost of this question from 18 months ago on the main Mathematics SE site, as the response there has been underwhelming, and I thought here would be a better authority. As you can probably ...
- 201
3
votes
0
answers
79
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Generalisation of Lyapunov time to stochastic dynamical systems
Might there be useful generalisations of the Lyapunov time to stochastic dynamical systems? In particular, I'm interested in methods for calculating confidence intervals around stochastic analogues of ...
- 3,871
7
votes
1
answer
424
views
Naturally occuring counting process with a 1/log asymptotics?
Besides prime numbers, is there another physically realizable counting process that exhibits a 1/log density ? The reason I am posting this question is that we are measuring the response of a quantum ...
- 79
10
votes
2
answers
349
views
Is this Riccati equation ("Josephson junction") always phase-locked at integer rotation numbers?
Given parameters $(a,k,A) \in \mathbb{R}^3$, we consider on $\mathbb{S}^1$ the $2\pi$-periodic ODE
$$ \dot{\theta} \ = \ - a\sin(\theta) + k + A\cos(t) \hspace{4mm} \mathrm{mod} \ 2\pi. $$
Identifying ...
- 708
0
votes
0
answers
3k
views
What is a self-consistent equation in percolation theory
I was reading papers about percolation theory in which I was confused by the expression "self-consistent equation", for example in Temporal percolation in activity-driven networks. I read some ...
- 211
1
vote
1
answer
288
views
Microlocal proof of Wigner semicircle theorem?
Something I really enjoy about Tao's writing is that he proves the same theorem over and over. While I complain a bit sometimes about clarity, this is a heuristic that I very much believe in.
This ...
- 22.8k
4
votes
1
answer
273
views
Ergodicity of the Form Factor in Random Matrix Theory
This question is motivated by recent works in quantum gravity, particularly in the analysis of the Sachdev-Ye-Kitaev (SYK) model. The SYK model is a one-dimensional quantum mechanical model which, in ...
- 192
4
votes
1
answer
368
views
Long wavelength instability: Linear Vs nonlinear phenomenon
I am looking into stability for certain nonlinear PDE on $\mathbb{R}$ around a specific steady solution, $f_0(x)$. The nonlinear Cauchy PDE is given by:
$\dfrac{\partial f(x,t)}{\partial t}=\mathbf{N}...
- 318
3
votes
0
answers
141
views
Which is the number of independent components of a flat spin connection in a 4 dimension Weitzenböck spacetime?
A spin connection $A_{ab\mu}=-A_{ba\mu}$ has 24 components. The number of independent components for a flat spin connection can be counted by subtracting the constrains set by the condition of null ...
- 51
3
votes
0
answers
271
views
Classical analogue of the theorem of equivalence of the S-matrix
In quantum field theory there is a statement called the equivalence theorem of the S-matrix. S-matrix is invariant under reparametrization of the field. Is there in classical mechanics, the analogous ...
- 41
27
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4
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13k
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Hamiltonian, Lagrangian and Newton formalism of mechanics
If my thinking is wrong please let me know. I have little knowledge on beyond-college physics.
For research purposes, I read a few introductions to these three formalisms of classical mechanics [1,2,...
- 8,071
1
vote
0
answers
61
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Stability of Fokker plank solutions with drift not coming from potential: Lyapunov analysis
Consider the FP equation on two dimensional space:
$\dfrac{\partial{\rho(x,y,t)}}{{\partial t}}+u(x,y)\dfrac{\partial\rho}{\partial x}+v(x,y)\dfrac{\partial\rho}{\partial y}=D\Delta\rho(x,y,t)$.
It ...
- 11
4
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0
answers
142
views
KAM stable orbits are smooth
I'm in my final year of my undergraduate studies doing work on modelling the n-body problem numerically and I also have some interest in theoretical guarantees. Now, I've been looking for a theorem ...
- 3,871
9
votes
1
answer
726
views
When does a Lagrangian dynamical system have an equivalent Hamiltonian description?
Let a Lagrangian dynamical system with $n$ degrees of freedom and configuration space $\mathbb{R}^n$
(i.e. phase space $\mathbb{R}^{2n}$), which is described by $L=L(q_{i},\dot{q}_{i},t)$, $i=1,2,......
- 7,746
14
votes
2
answers
390
views
Is there a singularity theorem in higher-dimensional Newtonian gravity?
In classical Newtonian gravity with 3 spatial dimensions, it's hard to get two particles to exactly collide, since at short distance the centrifugal force (~1/$r^3$) beats the gravitational attraction ...
- 273
4
votes
0
answers
116
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Dynamics of pairwise distances in the $n$-body problem
Disclaimer: I have asked this question on Physics SE a week ago, but got no answers. I know that some MO users are interested in the $n$-body problem, so I decided to cross post here as well.
...
2
votes
0
answers
491
views
Is there a Bayesian theory of deterministic signal? Prequel and motivation for my previous question
This is a prequel to my question:
What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)
Clearly my ...
- 1,026
0
votes
1
answer
88
views
underdamped oscillation with quadratic decay
I know that for a 2nd order linear differential equation system, there are 3 possible scenarios: over-damped, critically damped and underdamped. For the underdamped case the solutions are of the form:
...
- 169
6
votes
2
answers
3k
views
What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)
Because I still have no idea how it is possible for me to write down seemingly important equations ... that don't make any sense (at least for me) and because I haven't got any helpful comment so far, ...
- 1,026
5
votes
0
answers
145
views
cohomology ring of stable configuration spaces
Let $M$ be a compact Riemannian manifold without boundary. Distinct $k$-points $x_1,\cdots,x_k\in M$ are called stable if the potential energy given by coulomb forces among $k$ electrons reaches ...
- 543
1
vote
1
answer
213
views
Some quantities which definitions are (somehow) similar to the classical Divergence
Motivated by classical formulas $L_{X}=d\circ i_{X}+i_{X}\circ d$ and $L_{X} \Omega=Div(X) \Omega$ and the essential role of the diff operator $d$ in definition of divergence, we define some ...
- 356
5
votes
0
answers
127
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First return time in an interval for N particles rotating on the circle at constant random speeds
Here is my problem: draw N velocities $v_1,v_2,\dots,v_n$ in $[-\pi,\pi]^N$ from some measure (Haar measure of uniform independent for simplicity) and make $N$ particles rotate around the circle with ...
