Analysis of the classical phase space and energy transfer for two rotating dipoles with and without external electric field

We explore the classical dynamics of two interacting rotating dipoles that are fixed in the space and exposed to an external homogeneous electric field. Kinetic energy transfer mechanisms between the dipoles are investigated by varying both the amount of initial excess kinetic energy of one of them and the strength of the electric field. In the field-free case, and depending on the initial excess energy, an abrupt transition between equipartition and nonequipartition regimes is encountered. The study of the phase space structure of the system as well as the formulation of the Hamiltonian in an appropriate coordinate frame provide a thorough understanding of this sharp transition. When the electric field is turned on, the kinetic energy transfer mechanism is significantly more complex and the system goes through different regimes of equipartition and nonequipartition of the energy including chaotic behavior.

Figure 1

Evolution of the (a) position and (b) energy of the critical points of $V_{\mathcal{M}}({\theta }_1,{\theta }_2,t\ge t_1)$ as a function of the ratio between the electric field parameter $\beta$ and the dipole-dipole interaction parameter $\chi$.

Figure 2

Evolution of the landscape of the potential energy surface $V_{\mathcal{M}}({\theta }_1,{\theta }_2,t)$ for $t>t_1$ and different values of the ratio between the electric field parameter and the dipole-dipole interaction $\beta /\chi$.

Figure 3

Landscape of the rotated potential energy surface $V^{\prime }({\theta }_1^{\prime },{\theta }_2^{\prime },t)$ for $\beta =0$. The period of the rotated potential for $\beta =0$ is $\pi /\sqrt{2}$, but for the sake of clarity, we plot it in the interval $[-\pi ,\pi ]$.

Figure 4

For the field-free system, the normalized time-averaged kinetic energies of the dipoles $\widehat{P_1^2}$ (solid blue line) and $\widehat{P_2^2}$ (dotted red line), see (18), as a function of the initial excess energy of the first dipole $\delta \mathcal{K}$. The dipole-dipole interaction strength is $\chi =1\times {10}^{-5}$.

Figure 5

For the field-free system, time evolution of the kinetic energies $P_1^2\left(t\right)$ (blue lines) and $P_2^2\left(t\right)$ (red lines) and the potential energy $V_{\mathcal{M}}({\theta }_1,{\theta }_2,t)$ (green lines). The initial excess energies of the first dipole are (a) $\delta \mathcal{K}=4\chi$ and (b) $\delta \mathcal{K}=7\chi$. The dipole-dipole interaction strength is $\chi =1\times {10}^{-5}$.

Figure 6

For the field-free system with dipole-dipole interaction $\chi =1\times {10}^{-5}$, Poincaré surfaces of section in the plane $(P_1,{\theta }_1)$ with ${\theta }_2=\pi /2$ for three different initial excess energies in the neighborhood of the critical value $\delta {\mathcal{K}}_c=6\chi$. The red thick points correspond to the trajectory $\tau$ with initial conditions (17).

Figure 7

For the field-free system, time evolution of the momenta $P_1^{\prime }\left(t\right)$ (blue lines) and $P_2^{\prime }\left(t\right)$ (red lines) of the uncoupled pendula (left column) and the product $P_1^{\prime }\left(t\right)P_2^{\prime }\left(t\right)$ (green lines, right column) for the excess energies (a, b) $\delta \mathcal{K}=1.8\chi$, (c, d) $\delta \mathcal{K}=5.9\chi$, and (e, f) $\delta \mathcal{K}=6.1\chi$, for $P_1^{\prime }\left(t\right)$ and $P_2^{\prime }\left(t\right)$, and $P_1^{\prime }\left(t\right)P_2^{\prime }\left(t\right)$, respectively. The dipole-dipole interaction strength is $\chi =1\times {10}^{-5}$.

Figure 8

The normalized time-averaged kinetic energies of the dipoles $\widehat{P_1^2}$ (blue solid line) and $\widehat{P_2^2}$ (red dashed line) as a function of the ratio between the electric field parameter $\beta$ and the dipole-dipole parameter $\chi$ for two initial excess energies of the first rotor (a,c) $\delta \mathcal{K}=4\chi$ and (b,d) $\delta \mathcal{K}=7\chi$. The dipole-dipole interaction parameter is set to $\chi =1\times {10}^{-5}$. The ramp-up time for the electric field is $t_1=1200$ in panels (a) and (b) and $t_1=600$ in panels (c) and (d).

Figure 9

Poincaré surface of section in the plane $(P_1,{\theta }_1)$ with $P_2=0$ for different values of the electric field parameter $\beta$. The dipole-dipole interaction is $\chi =1\times {10}^{-5}$ and the excess kinetic energy of the first dipole is $\delta \mathcal{K}=4\chi$. The thick (red) points correspond to the trajectory $\tau$ with initial conditions (17).

Figure 10

The same as in Fig. 9, but for an excess energy of the first dipole of $\delta \mathcal{K}=7\chi$.

Figure 11

Time evolution of the kinetic energies $P_1^2\left(t\right)$ (blue lines) and $P_2^2\left(t\right)$ (red lines) of the $\tau$ orbit for $\delta \mathcal{K}=4\chi$ and (a) $\beta =4\chi$, (b) $\beta =20\chi$, and (c) $\beta =50\chi$. The dipole-dipole interaction parameter is $\chi =1\times {10}^{-5}$.