The Stack Overflow podcast is back! Listen to an interview with our new CEO.

Questions tagged [integrals-of-motion]

The tag has no usage guidance.

56 questions
Filter by
Sorted by
Tagged with
51 views

287 views

278 views

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 ...
250 views

239 views

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 ...