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# Questions tagged [integrals-of-motion]

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### Action variables in canonical transformations

Let's suppose we have a Hamiltonian $H(p_k, q_k)$ and we want to transform it via a canonical transformation to one Hamiltonian which doesn't depend on the new coordinates $w_k$, but only in the ...
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The Henon-Heiles potential is $$U(x,y ) = \frac{1}{2} (x^2 + y^2 + 2 x^2 y - \frac{2}{3} y^3) .$$ This is a two degree-of-freedom system. The full Hamiltonian is $$H = p_x^2 + p_y^2 + U(x,y ) .... 0answers 89 views ### First integrals for a particle in a central-force field Consider an arbitrary dimension n>3. What are the independent first integrals for a particle? The Hamiltonian is$$ H = \frac{p^2}{2m} +V (|r|) . $$0answers 585 views ### What variable is the conjugate momentum for angular momentum? From the definition of conjugate momentum for a generalized coordinate we get that the conjugate for angular momentum should be proportonal to its integral with respect to time. According to my ... 1answer 884 views ### Is there an analog to the Runge-Lenz vector for a harmonic potential? The Runge-Lenz vector is an "extra" conserved quantity for Keplerian \frac{1}{r} potentials, which is in addition to the usual energy and angular momentum conservation present in all central force ... 0answers 174 views ### Motion Integrals of a Particle in a Force Field I am trying to wrap my head around the following problem: A point particle is moving in a field, where its potential energy is U=-α/r. Find first motion integrals. In our university we have no ... 0answers 329 views ### Finding conserved quantities from Hamiltonian when Symmetry is not evident [closed] A particle is moving in 3D space, under a potential$$V = -\frac{\alpha}{r}-\frac{\vec{r} \cdot \vec{\mu}}{r^3 } $$where \vec{\mu} is some constant vector. I need to show there are three ... 2answers 2k views ### Non-integrability of the 2D double pendulum Context: For a system with n degrees of freedom (DOF), one has to deal with 2n independent coordinates (2n dimensional phase space), of position q and \dot{q} in Lagrangian formulation, or ... 1answer 277 views ### Integrals of Motion for s Degrees of Freedom From Landau & Lifshitz, Classical Mechanics, the number of integrals of independent integrals of motion for a system of s degrees of freedom is 2s-1. I am considering a spherical pendulum in ... 3answers 733 views ### What could cause an asymmetric orbit in a symmetric potential? My question can be summarized as: Given a potential with a symmetry (e.g. z\rightarrow-z), should I expect orbits in that potential to exhibit the same symmetry? Below is the full motivation for ... 2answers 974 views ### How to prove that a Hamiltonian system is *not* Liouville integrable? To show that a system is Liouville integrable, we just need to find n independent functions f_j such that \{ f_i, f_j \} = 0. But how to prove that such a set of functions do not exist? For ... 0answers 63 views ### Why and how almost periodic series constitute the algebra of observable of integrable systems? In the introduction of his book Noncommutative Geometry, p. 42, Connes explains that when a classical dynamical system has enough constants of motions, the motion of the system is almost periodic, ... 0answers 187 views ### Single particle trajectory in a quadrupole potential I am wondering if there are any studies of a single (classical) particle trajectory in quadrupole potential:$$ V(x,y,z)=A\sqrt[]{\frac{x^2 + y^2}{a} + \frac{z^2}{b}} 
Since the equation of mechanics are of second order in time, we know that for $N$ degrees of freedom we have to specify $2N$ initial conditions. One of them is the initial time $t_0$ and the rest of ...
Integrable systems are systems which have $2n-1$ time-independent, functionally independent conserved quantities ($n$ being the number of degrees of freedom), or $n$ whose Poisson brackets with each ...