The effect of physically realizable wall potentials (soft walls) on the dynamics of two interacting particles in a one-dimensional (1D) billiard is examined numerically. The 1D walls are modeled by the error function and the transition from hard to soft walls can be analyzed continuously by varying the softness parameter sigma . For sigma-->0 the 1D hard wall limit is obtained and the corresponding wall force on the particles is the delta function. In this limit the interacting particle dynamics agrees with previous results obtained for the 1D hard walls. We show that the two interacting particles in the 1D soft walls model is equivalent to one particle inside a soft right triangular billiard. Very small values of sigma substantiously change the dynamics inside the billiard and the mean finite-time Lyapunov exponent decreases significantly as the consequence of regular islands which appear due to the low-energy double collisions (simultaneous particle-particle-1D wall collisions). The rise of regular islands and sticky trajectories induced by the 1D wall softness is quantified by the number of occurrences of the most probable finite-time Lyapunov exponent. On the other hand, chaotic motion in the system appears due to the high-energy double collisions. In general we observe that the mean finite-time Lyapunov exponent decreases when sigma increases, but the number of occurrences of the most probable finite-time Lyapunov exponent increases, meaning that the phase-space dynamics tends to be more ergodiclike. Our results suggest that the transport efficiency of interacting particles and heat conduction in periodic structures modeled by billiards will strongly be affected by the smoothness of physically realizable walls.
O sistema de Hénon-Heiles foi proposto inicialmente para descrever o comportamento dinâmico de galáxias, mas tem sido amplamente aplicado em sistemas dinâmicos por exibir riqueza de detalhes no espaço de fases. O formalismo e a dinâmica do sistema de Hénon Heiles são investigados neste trabalho, visando uma abordagem qualitativa. Através das Seções de Poincaré, observa-se o crescimento da região caótica no espaço de fases do sistema, quando a energia total aumenta. Ilhas de regularidade permanecem em torno dos pontos estáveis e aparecem fenômenos importantes para a dinâmica, como os "sticky". Palavras-chave: sistemas hamiltonianos, sistema de Hénon-Heiles,órbitas periódicas.The Hénon-Heiles system was originally proposed to describe the dynamical behavior of galaxies, but this system has been widely applied in dynamical systems as it displays many details in phase space. This work presents the formalism to describe the Hénon-Heiles system and a qualitative approach to the dynamical behavior. The growth of the chaotic region in phase space is observed by Poincaré Surface Sections, as the total energy increases. Islands of regularity remain around stable points and there are relevant sticky phenomena. Keywords: Hamiltonian systems, Hénon-Heiles system, periodic orbits. IntroduçãoEm 1964, Michel Hénon e Carl Heiles [1] investigaram a existência de integrais de movimento para um sistema Hamiltoniano particular, que hojeé conhecido como sistema de Hénon-Heiles. Este sistema havia sido proposto para descrever o movimento de galáxias interagindo via força gravitacional. Hénon e Heiles mostraram que a energia total e o momento angular do sistema são constantes de movimento, ou seja, que estas grandezas permanecem inalteradas ao longo da evolução temporal do sistema.O sistema de Hénon-Heiles tem sido amplamente estudado e revela importantes características dos sistemas Hamiltonianos, como o comportamento misto do espaço de fases, no qual apresenta simultaneamente ilhas de regularidade e regiões caóticas. Esse tipo de sistema pode favorecer o fenômeno de transporte [2,3] e mostrar pistas importantes para o estudo deórbitas periódicas estáveis e instáveis.Baseados no trabalho de Hénon e Heiles, analisaremos em detalhe a dinâmica desse sistema. Busca-se descrever o sistema analiticamente através do formalismo Hamiltoniano. As características e o comportamento dinâmico do sistema no espaço de fases são estudados aravés das seções de Poincaré. Na seção 2.é feita uma breve introdução ao formalismo Hamiltoniano. Na seção 3. encontra-se a descrição analítica do sistema de Hénon-Heiles, bem como as suas equações de movimento. Na seção 4. faz-se a análise das transições de fase do sistema. Na seção 5. as conclusões do trabalho e os agradecimentos. Sistemas hamiltonianosOs sistemas dinâmicos podem ser divididos em conservativos e dissipativos. Os sistemas conservativos mantém sua energia constante a medida que o tempo evolui e são chamados de Hamiltonianos. Os sistemas dissipativos não conservam sua energia ao longo do tempo.Um sis...
Physical systems such as optical traps and microwave cavities are realistically modeled by billiards with soft walls. In order to investigate the influence of the wall softness on the billiard dynamics, we study numerically a smooth two-dimensional potential well that has the elliptical (hard-wall) billiard as a limiting case. Considering two parameters, the eccentricity of the elliptical equipotential curves and the wall hardness, which defines the steepness of the well, we show that (1) whereas the hard-wall limit is integrable and thus completely regular, the soft wall elliptical billiard exhibits chaos, (2) the chaotic fraction of the phase space depends nonmonotonically on the hardness of the wall, and (3) the effect of the hardness on the dynamics depends strongly on the eccentricity of the billiard. We further show that the limaçon billiard can exhibit enhanced chaos induced by wall softness, which suggests that our findings generalize to quasi-integrable systems.
The dynamics of three soft interacting particles on a ring is shown to correspond to the motion of one particle inside a soft triangular billiard. The dynamics inside the soft billiard depends only on the masses ratio between particles and softness ratio of the particles interaction. The transition from soft to hard interactions can be appropriately explored using potentials for which the corresponding equations of motion are well defined in the hard wall limit. Numerical examples are shown for the soft Toda-like interaction and the error function.
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