All Questions
Tagged with complex-systems classical-mechanics
73 questions
-2
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1
answer
113
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Advanced math courses for theoretical physics students [closed]
Theoretical physics studies concerning statistical mechanics, dynamical systems and analytical classical mechanics all require working knowledge of mathematical concepts and theories (e.g. manifolds, ...
0
votes
1
answer
58
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Strogatz's condition on definition of energy
In, Nonlinear Dynamics And Chaos, 2nd edition page 160, by Steven H. Strogatz, he writes
Let’s be a bit more general and precise. Given a system $$\dot x =f(x),$$ a conserved quantity is a real-...
6
votes
6
answers
394
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How can I formalize better this proof that angular momentum is conserved for a small impulse?
The book I am studying is discussing Lagrange stability of circular orbits, which assumes fixed angular momentum ${\bf L}$, hence in an introductory paragraph explains why, when studying stability of ...
- 114
1
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0
answers
92
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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
0
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1
answer
35
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Question regarding dynamic modeling of cable driven robot
I am studying rigid body dynamics and am currently exploring the paper "Analytic determination of wrench closure workspace of spatial cable driven parallel mechanisms". In the paper, the ...
- 13
3
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0
answers
101
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Classical mechanics: Hamiltonian perturbation theory. What if the perturbing parameter is < 0?
In Hamiltonian Perturbation theory, we have a Hamiltonian of the form $$H(q,p) = H_0(q,p) + \lambda H_1(q,p).$$
One proceeds by expanding the equations of motion in powers of $\lambda$, assuming $\...
13
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6
answers
1k
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(A modification to) Jon Pérez Laraudogoita’s "Beautiful Supertask" — What assumptions of Noether's theorem fail?
I am curious about the following (physically unrealizable) scenario involving a supertask described here: https://plato.stanford.edu/entries/spacetime-supertasks/#ClasMechSupe. The original paper is ...
- 8,776
1
vote
0
answers
110
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Curvature and stability
In Topological methods in hydrodynamics 1 mentioned that "The Riemannian curvature of a manifold has a profound impact on the behavior of geodesics on it. If the Riemannian curvature of a ...
- 111
0
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1
answer
85
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Double Pendulum Intuition
I have arrived at the equations of motion for a double pendulum, with gravity $g$, masses $m_i$, link lengths $l_i$, angles $\theta_i$, and applied torques $\tau_i$.
Please see the diagram and ...
- 559
1
vote
1
answer
118
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Need help finding Hamiltonian for equations of motion
I have the following equation of motion:
$$\ddot \theta+\dot\theta^2\theta+k^2\theta=0.\tag 1$$
This equation is from this question. I wanted to see if I could find a Hamiltonian for this equation but ...
1
vote
1
answer
127
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Distinguishing between chaos and multiperiodic oscillations from the Fourier spectrum
Consider a system which exhibits multiperiodicity, say with oscillations of the form $x(t) = \sum_{n=0} c_n \cos(n \omega_0 t)$, $\lim_{n \to \infty} c_n = 0$. The Fourier transform $\tilde{x}(\omega)$...
- 39
1
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1
answer
488
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What is difference between a monogenic system and a dynamical system?
What is difference between a monogenic system and a dynamical system?
I am confused in reading about the Hamiltonian principle because some book write system as monogenic and other dynamical.
...
- 21
1
vote
3
answers
748
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What is a phase space?
What is a phase space? And can the phase space be specified with x and y instead of with theta and omega?
I am currently working on a problem where I am graphing the trajectories of three masses (the ...
- 151
1
vote
0
answers
122
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Kasner-Arnold theorem energy role in Needham's "Visual Complex Analysis" (VCA)
In "Visual Complex Analysis" (chapter 5.X.6) Tristan Needham writes
In general, positive energy orbits in either the attractive or repulsive field $F \propto r^A$ map to attractive orbits ...
- 111
0
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2
answers
214
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Dynamics: why do physicists include derivatives like $\dot{\theta}$ in the state space for a system like a pendulum?
I come from statistics, so my experience with physics is spotty, especially on some simple stuff. I have been working on some applications related to control theory lately, and was looking at some ...
- 181
1
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0
answers
119
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Integrability of one-dimensional system of motion?
How can I prove that every one-dimensional system is integrable (meaning that there is a constant of motion)?
It is clear that if $H$ does not depend explicitly on time then $H$ is indeed a constant ...
1
vote
0
answers
52
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Potentials that prevent the phase flow of the system [closed]
I am trying to solve a question that my professor gave.
When a particle moves in one dimension $x$ in a potential $U(x)$ , the resulting motion over a very short time interval is specified by Newton’...
- 9
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0
answers
73
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Is the Hamiltonian of interacting systems integrable if the interaction is linear?
Suppose we allow two integrable systems with Hamiltonians $H_1$ and $H_2$ to interact. Then their combined dynamics can be described by a joint Hamiltonian, $$H = H_1(\mathbf{q}_1,\mathbf{p}_1) + H_2(\...
- 35
0
votes
0
answers
57
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Phase flow and potential energy
I have a question that our teacher gave us and this is my very first time I see concepts like phase flows. Prove that a positive potential energy always guarantees a phase flow. I should use ...
- 101
1
vote
1
answer
251
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Correspondence between quantum and classical integrability
I'm looking for connections between quantum and classical integrability. I know quantum integrability is not well-defined, but let us just take one of the popular definitions by promoting the Poisson ...
- 111
3
votes
4
answers
1k
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What is quasi-periodic motion? [closed]
I'm currently 2nd year physics student (undergraduate). I have seminar which theme is double pendulum.
I'm having trouble understanding quasi-periodic motion in general and more importantly in context ...
- 39
1
vote
0
answers
229
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Mathematical method of classical Mechanics Arnold [closed]
I wanted to know that is there any textbook or PDF file containg suloutin of the book Mathematical methods of classical mechanics by Arnolʹd
Some of the questions are really hard.
5
votes
6
answers
1k
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How is energy conservation & Noether's theorem a non-trivial statement?
Noether's theorem says that energy conservation is a result of temporal translation symmetry of the laws of physics. This is implied to be - and I'm not saying it's not - a very non-trivial statement. ...
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2
votes
2
answers
401
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Are there non-orthogonal "normal" modes for non-identical coupled oscillators?
The question is broad, I will specify an example to elaborate what I'm asking.
Suppose I have two different LC circuits inductively coupled (or capacitively, but the question I have will be relevant ...
0
votes
0
answers
33
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Fixed point of non-linear system: infinite eigenvector
I've come across a $2d$ non-linear dynamical system (autonomous) the stability properties of which I would like to understand better. Computing the stability matrix, its eigenvalues and eigenvectors, ...
- 1,363
3
votes
1
answer
81
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Reference for a type of "multi-hamiltonian" system
Let $H_1,H_2\in\mathcal{C}^1(\mathbb{R}^3;\mathbb{R})$ be two scalar fields. Consider a trajectory $\vec{x}(t)\in\mathbb{R}^3$ such that, for all observable $f\in\mathcal{C}^1(\mathbb{R}^3;\mathbb{R})$...
2
votes
0
answers
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
2
votes
0
answers
135
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Why does the separatrix in phase portraits have infinite period and pass through at least one unstable equillibrium point?
In the case of 1D Hamiltonians not explicitely dependent on time, our professor claims that the "period" of the separatrix is necessarily infinite and must pass through an unsable ...
- 33
0
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0
answers
170
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Changing to action-angle variables to reduce the degrees of freedom
I have a Hamiltonian with two degrees of freedom, and when I change to action angle variables, one of the action variables does not appear in the final Hamiltonian. The reason seems to be because the ...
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2
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0
answers
51
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Destruction of integrals of motion in chaotic systems: Fermi-Pasta-Ulam (FPU) paradox
I am trying to understand behavior of system studied by Fermi, Pasta and Ulam i.e. chain of oscillators interacting via nonlinear forces. I am generally not very familiar with chaos theory and ...
- 41
1
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2
answers
103
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Is there a way to construct a Hamiltonian from a set of DE?
Let's say I have a set of first-order differential equations for set of position $x_i$ and conjugate momenta $p_i$, which might be complicated and time-dependent
$$ \dot{x}_i = f_i(x_j,p_j,t)$$
$$ \...
user245141
1
vote
0
answers
39
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Transformation between a dynamical system to a Hamiltonian system [duplicate]
Consider a dynamical system characterized by these equations
$$\dot{x}=x-xy \\
\dot{y}=-y+xy$$
If we transform $\ln(y)=q$ and $\ln(x)=p$, the system can be changed into a Hamiltonian system with $q$ ...
- 636
2
votes
1
answer
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
votes
2
answers
576
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Arnold's Mathematical Methods of Classical Mechanics and Lyapunov stability
In Arnold's Classical Mechanics of Classical Mechanics, he refers to Lyapunov stability in many of the problems in the second chapter.
E.g. on page 20: "Problem: Consider a periodic motion along the ...
- 378
2
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1
answer
693
views
Discontinuities in a Poincare map for a double pendulum
I'm generating poincare sections of a double pendulum, and they mostly look okay, but some of them have weird discontinuities that seem wrong.
The condition for these sections is the standard $\...
- 171
2
votes
1
answer
124
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Why are marginal eigenvalues of Jacobian of a periodic orbit related to the symmetry?
In ChaosBook, at page 61 of the unstable version of the book, it is stated that
$$J_p (x) \mu (x) = \mu (x,)$$ i.e the velocity vector is an eigenvector of the Jacobian along periodic orbit $p$ ...
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1
vote
1
answer
69
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What is the general definition of thickness of a strange attractor?
Disclaimer: This question is cross posted on Math.SE because I don't know which site is more appropriate for this question.
In Chaosbook, at page 56, it is asked to find the thickness of Rössler ...
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2
votes
1
answer
258
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Chirikov standard map derivation
This might be a stupid question, but I am having trouble understanding the derivation of Standard map by integrating Hamilton's equation of motion over one period. I am going through this dissertation ...
- 41
2
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1
answer
665
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Poincaré plane and Logistic Map
How can we draw Poincaré plane and phase portrait for the Logistic Map for different parameter values?
user212422
6
votes
1
answer
1k
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Is *every* autonomous first-order planar/2D system integrable?
Consider a general autonomous first-order planar/2D system:
$$\begin{cases}
\frac{dx}{dt} = A(x,y)\\
\\
\frac{dy}{dt} = B(x,y),
\end{cases}$$
where $A,B$ are two functions. Reading Classical Mechanics ...
- 647
3
votes
1
answer
900
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A question over Liouville’s Theorem
I have some doubts about Liouville theorem, probably its just something conceptual.
So: I know that for a system in which Liouville’s theorem holds, the volume in the phase space is conserved.
But ...
- 3,735
5
votes
3
answers
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
8
votes
2
answers
501
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When is the ergodic hypothesis reasonable?
Consider an Hamiltonian system.
In which circumstances is it possible to assume that all the states belonging to the hypersurface $H=E_0$ are equally visited?
Is it necessary to have a very high ...
- 1,252
2
votes
2
answers
173
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Closed gravitational orbits and gradient systems
I am currently studying non-linear dynamics on my own time. One of the theorems in the material is that systems that can be written as gradient problems cannot have closed orbits i.e. systems like
$$\...
- 21
5
votes
3
answers
1k
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Control systems from a physicist's perspective
I am highly interested in the study of control systems theory. However it seems that almost all books are written by electronics or mechanical engineers.
Due to this they generally omit many things. ...
1
vote
1
answer
213
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Question about a system with all bounded orbits closed and maximal integrable
Given Hamiltonian system with $2n$-dim phase space, if there exist $k\ge n$ independent integrals of motions then we call it integrable Hamiltonian system. The largest number of independent integrals ...
- 2,157
1
vote
0
answers
565
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Do you know dynamical formulas for hydraulic cylinders, motors and pumps?
I'm looking for dynamic differential equations for hydraulic cylinders, motors and pumps.
I have one differential equation for a hydraulic cylinder, but I'm not sure if it's correct.
Assume a ...
- 161
1
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1
answer
321
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Can a first-order autonomous system not at a fixed point, transition to a fixed point?
The following is from Introduction to Dynamics, by Percival and Richards:
At each zero $x_{k}$ of the velocity field $v\left[x\right],$
$$v\left[x_{k}\right]=0,$$
so that a system initially at $x_{k}$...
- 2,053
2
votes
1
answer
187
views
Why some dynamic systems can undergo sudden changes?
Everybody has observed that the weather may change from beautiful sunshine to extremely bad weather (heavy rain, stormy winds, ...) within less than half hour. What is the fundamental reason for this?
...
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45
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5
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4k
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Why are we sure that integrals of motion don't exist in a chaotic system?
The stadium billiard is known to be a chaotic system. This means that the only integral of motion (quantity which is conserved along any trajectory of motion) is the energy $E=(p_x^2+p_y^2)/2m$.
Why ...
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