The A - Hamiltonian Dynamics of Hamiltonian Dynamics of

Introduction

In a previous article [1], we have shown that ferroelectric liquid crystals is a realization of a two - dimensional non-integrable Hamiltonian system, following the analogy suggested in [2].

The purpose of the present article, is to study the phase transitions in such system, as bifurcations of the corresponding Hamiltonian vector field. In particular, we show that these bifurcations are Hopf bifurcations [3], depending of two parameters: temperature T, and external magnetic field .

As it is well known in the chiral smectic - phase, the molecular configuration of the liquid crystal is helicoidal, being the molecular director a periodic function: , [4, 5].

In the disordered - phase (A), the equilibrium point is and in the ordered - phase (C), the equilibrium point is . The transition from the constant configuration to the periodic one is just the Hopf bifurcation [6].

As was predicted in [7], and observed in [8], there exists a Lifshitz point in the T - phase diagram of chiral smectic liquid crystal if an external magnetic field , is applied parallel to the smectic layers. The Lifshitz point is described by introducing the Lifshitz function, L, in the Landau - Ginzburg expansion of free energy functional.

The Lifshitz function is not invariant in the presence of the magnetic field, and this, gives rise a non-integrable non-linear Hamiltonian system. We use the linear approximation to perform the stability analysis of the system. Due to the presence of the Lifshitz function, the eigenvalues of the linearized Hamiltonian vector field, are not purely imaginary. How it is well known, under this condition, the qualitative behavior of the non-linear system may be studied from its linearized part at the equilibrium point [9].

The stability analysis of the linear system, gives us the lines - boundaries between the phases in the diagram, in agreement with the results found in [7].

In order to study numerically the solutions of the - phase, we have used the KAOS package to implement the Poincaré section of the two -tori. It was possible, for certain values of the temperature T, and external magnetic field , to show the qualitative equivalence between the non-linear and linearized Hamiltonian system, inside the region of convergence of the Taylor series.

The Poincaré sections of the two tori, with a number of different possible solutions, present characteristics of those system with chaotic behavior. The Poincaré sections became more and more complicated as is increased, according the KAM theorem [6].

For = 0, it is composed only with continuous closed curves corresponding to a quasi-periodic solutions of a completely integrable system.

For sufficiently weak, there are, among continuous closed curves, islands, around fixed points. As is increased beyond a certain value , we observe, among continuous lines, islands and archipelagoes, a sea of scattered points that can not be connected by a single curve. These points corresponds to chaotic solutions.

The complexity of the Poincaré section may be explained by the coupling of the helicoidal motion, described by azimuthal angle , with the tilt motion, described by the tilt angle , between and the axis. This coupling, leads to locked, (periodic), and unlocked, (quasi-periodic) modes, between the helicoidal and tilt oscillations, which gives rise a Hamiltonian quasi-periodic route to chaos.

We complete the study of the diagram, by observing that the C - transition may be described by a second order derivative Hamiltonian system.

The stability analysis is more delicate in this case, because the energy of the system is not definite in sign. Far from the transition line the solution of linearized system is just a helix, while near the transition line the non-linear system has a divergent coupling. This divergency may be in the origin of the reentrant - phase found in ([8].

The article is structured as follows. In section 2, we implement the Hamiltonian study of the A - transition. In section 3, we develop the linear stability analysis. In section 4, the higher order Hamiltonian theory is applied to study of the - C transition. In section 5, the corresponding stability analysis is performed. Finally, section 6, is devoted to some remarks.

The A - Hamiltonian Dynamics of Hamiltonian Dynamics of