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Questions tagged [integrals-of-motion]

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14 questions with no upvoted or accepted answers
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Why and how almost periodic series constitute the algebra of observable of integrable systems?

In the introduction of his book Noncommutative Geometry, p. 42, Connes explains that when a classical dynamical system has enough constants of motions, the motion of the system is almost periodic, ...
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81 views

How are action variables linked to first integrals of a Hamiltonian?

Suppose I have an integrable Hamiltonian system $H(q_{1}, p_{1},..., q_{n}, p_{n})$, with first integrals $F_{1} = H, F_{2},..., F_{n}$. Excluding certain singular level sets (i.e. separatrices), one ...
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586 views

What variable is the conjugate momentum for angular momentum?

From the definition of conjugate momentum for a generalized coordinate we get that the conjugate for angular momentum should be proportonal to its integral with respect to time. According to my ...
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45 views

How is BRST symmetry related to local integrals of motion?

I'm hoping someone can confirm or check my reasoning below: In this wiki, they describe caos in a classical system as the spontaneous symmetry breaking of a BRST. In this stackexchange, they clarify ...
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114 views

Any model integrable but not separable?

In textbooks on classical mechanics, the exactly solvable models are all separable. Is there any model integrable but not separable?
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187 views

Single particle trajectory in a quadrupole potential

I am wondering if there are any studies of a single (classical) particle trajectory in quadrupole potential: $$ V(x,y,z)=A\sqrt[]{\frac{x^2 + y^2}{a} + \frac{z^2}{b}} $$
0
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1answer
69 views

How to know the number of constants of a free particle?

Landau-Lifshitz Mechanics says that there are $2s-1$ constants of a system with $s$ degrees of freedom (beginning of the second chapter on Conservation Laws). If this is true, for a single free ...
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18 views

Constants of motion of an electron in a harmonic electromagnetic field in free space

I have encountered a question in Classical Electrodynamics, as below: In free space, an electron, initially at rest at $z=0$, is subjected to an intense laser field $\vec E=\hat x A \cos(\omega t-...
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149 views

What do I need an integral of motion for and why should I care?

I'm fine with Lagrangian mechanics without an integral of motion, but for some reason I keep seeing these integrals mentioned within the context of the Lagrangian with no real explanation as to their ...
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99 views

Existence of time-dependent invariants for a classical Hamiltonian system

Suppose the Hamiltonian is time-dependent, $H(t)$. The condition on a time-dependent invariant is $$ 0 = \frac{d I}{d t } = \frac{\partial I }{\partial t} +\{ I, H\} . $$ We can rewrite it as $$ ...
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1answer
43 views

Regarding $f$ degrees of freedom & $f\!-\!1$ constants & inclusion of these constants

In the classic & famous book "Electromagnetic fields & Interactions" by Richard Becker (Dover publishing), on page 55 (of volume 2) , author says: If the system possesses f degrees of ...
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1answer
88 views

Procedure for finding integrals of motion

In some examples I have read that if you want to find the integral of motion for some equation of motion, say on the form $\ddot{x}+ax=0$ for some constant $a$, you multiply the EOM by $$\dot{x}=q(x) \...
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89 views

First integrals for a particle in a central-force field

Consider an arbitrary dimension $n>3$. What are the independent first integrals for a particle? The Hamiltonian is $$ H = \frac{p^2}{2m} +V (|r|) . $$
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174 views

Motion Integrals of a Particle in a Force Field

I am trying to wrap my head around the following problem: A point particle is moving in a field, where its potential energy is U=-α/r. Find first motion integrals. In our university we have no ...