Dynamically induced conformation depending on excited normal modes of fast oscillation

The Euler-Lagrange equations derived from Eq. (2) are

where $\alpha,\beta,\gamma\in\{1,2,3\}$. These equations are the starting point of our theory.

The assumption (A2) induces the two timescales of $t_{0}=t$ and $t_{1}=\epsilon t$. The fast timescale $t_{0}$ describes the fast spring motion, and the slow timescale $t_{1}$ corresponds to the slow bending motion. The two timescales transform the time derivative into

From the assumptions (A1) and (A2) the variables $l_{j}$ and $\phi$ are expanded as

where $l_{\ast}$ is the natural length of the two springs. As we will observe later, the fast motion of $\phi^{(1)}(t_{0},t_{1})$ is induced by the fast motion of the springs and is of the same order $O(\epsilon)$ as the spring amplitudes. We are interested in $\phi^{(0)}(t_{1})$, which represents large and slow bending motion. We denote the above expansions for simplicity as

We further expand the potential energy function $V$. The spring potential $V_{\rm spring}$ is assumed to be expanded into the Taylor series around the natural length as

That is, the two springs have the same spring constant

The bending potential is assumed to be expanded into the series of $\epsilon$ as

The two assumptions (A1) and (A2) induce

as shown in Appendix C, and hence the leading term of $V_{\rm bend}$ is of $O(\epsilon^{2})$. This ording results from the assumption (A2): The force from the bending potential $V_{\rm bend}$ should be weaker than that of the spring potential $V_{\rm spring}$.

We construct the equations of motion order by order, by substituting Eqs. (8), (10), (11), and (13) into Eq. (7). In $O(\epsilon^{0})$, we have no terms, because $\partial y_{\beta}^{(0)}/\partial t_{0}=0$, $\dot{y}_{\beta},\ddot{y}_{\beta}=O(\epsilon)$, and $\partial V/\partial y_{\alpha}=O(\epsilon)$.

In $O(\epsilon)$ we have

The size-$3$ matrix $\boldsymbol{\rm X}$ is defined by

where

See Appendix D for the explicit form of $\boldsymbol{\rm X}(\boldsymbol{y}^{(0)})$. The third column vector of $\boldsymbol{\rm X}$ is the zero vector, and the right-hand side of Eq. (15) has no contribution from the third element of $\boldsymbol{y}^{(1)}$, namely $\phi^{(1)}$. The fast motion of $\phi^{(1)}$ is hence induced by $l_{1}^{(1)}$ and $l_{2}^{(1)}$, as mentioned after Eq. (9).

The slow motion of $\phi^{(0)}(t_{1})$ is described in $O(\epsilon^{2})$, and the equation of motion for $\phi^{(0)}(t_{1})$ is

Here $(\dot{\boldsymbol{y}})^{(1)}$ is the first order part of $\dot{\boldsymbol{y}}$ and $(\dot{\boldsymbol{y}})^{(1)}\neq{\rm d}\boldsymbol{y}^{(1)}/{\rm d}t$. The explicit forms are

Equation (18) contains the fast oscillation of $\boldsymbol{y}^{(1)}$, and we eliminate it by taking the average over the fast timescale $t_{0}$. The average is defined by

for an arbitrary function $\varphi(t_{0},t_{1})$. After taking the average and recalling $\boldsymbol{y}^{(0)}=(l_{\ast},l_{\ast},\phi^{(0)})$, Eq. (18) is simplified to

The right-hand side represents the force, and the averaged term

represents the effective force yielded by the fast spring motion. Here Tr represents the matrix trace. The right-hand side of Eq. (22) depends on $\boldsymbol{y}^{(0)}$ and $\boldsymbol{y}^{(1)}$. The $\boldsymbol{y}^{(0)}$ dependence can be regarded as the $\phi^{(0)}$ dependence, since $\boldsymbol{y}^{(0)}=(l_{\ast},l_{\ast},\phi^{(0)})$ and $l_{\ast}$ is constant. $\boldsymbol{y}^{(1)}$ depends on $t_{0}$ and $t_{1}$, and the $t_{0}$ dependence is averaged out. We have to eliminate the $t_{1}$ dependence to obtain the effective potential as a function of $\phi^{(0)}$.

We compute the explicit form of the averaged term $\mathcal{A}$ by performing the diagonalization of Eq. (15), and observe the $t_{1}$ dependence, which has to be eliminated. Let $\boldsymbol{\rm P}$ diagonalize $\boldsymbol{\rm X}$ as

The diagonal matrix $\boldsymbol{\rm\Lambda}(\boldsymbol{y}^{(0)})$ consists of the eigenvalues of $\boldsymbol{\rm X}(\boldsymbol{y}^{(0)})$ and is denoted by

where

and

A diagonalizing matrix is

with

The three column vectors $\boldsymbol{p}_{\rm in},\boldsymbol{p}_{\rm anti}$, and $\boldsymbol{p}_{\phi}$ are eigenvectors of $\boldsymbol{\rm X}(\boldsymbol{y}^{(0)})$, and we call the three modes as the in-phase mode, the anti-phase mode, and the zero-eigenvalue mode, respectively.

To solve Eq. (15), we perform the change of variables as

and $\boldsymbol{\eta}$ solves the diagonalized equations

Denoting the amplitudes of the in-phase and the anti-phase modes by $w_{1}(t_{1})$ and $w_{2}(t_{1})$ respectively, which evolve in the slow timescale $t_{1}$ through the coupling with $\phi^{(0)}(t_{1})$, and setting the amplitude of the zero-eigenvalue mode as zero, we introduce the diagonal matrix

Putting all together and remembering that the average of the square of a sinusoidal function is $1/2$, we have

The averaged term $\mathcal{A}$ contains the two evolving amplitudes $w_{1}(t_{1})$ and $w_{2}(t_{1})$. The untrivial evolution of the amplitudes differs from the Kapitza pendulum, which also contains the amplitude of the external oscillation but it is explicitly given. We have to eliminate the two unknown amplitudes from the averaged term $\mathcal{A}$ to obtain a closed equation for $\phi^{(0)}(t_{1})$.

The strategy to eliminate the two unknown amplitudes $w_{1}(t_{1})$ and $w_{2}(t_{1})$ is as follows. First, we introduce a hypothesis, which is inspired from the adiabatic invariant. The hypothesis reduces the number of unknown variables from two to one. Second, we eliminate the remaining unknown variable by using the energy conservation law.

The first step is the introduction of the hypothesis expressed by

where $\nu_{1}$ and $\nu_{2}$ are constants satisfying

A physical interpretation of the hypothesis (H) is that the slow bending motion exchanges energy with the fast spring motion in proportion to its normal mode energy. Validity of the hypothesis (H) is examined in Appendix E. We note that the hypothesis should be valid if the modification of $\phi^{(0)}$ is sufficiently small. The unique unknown variable is now $w(t_{1})$, while the constants $\nu_{1}$ and $\nu_{2}$ have been included in the equations of motion.

The second step is the energy conservation. The leading order of the total energy is of $O(\epsilon^{2})$ and we expand it as $E=\epsilon^{2}E^{(2)}+O(\epsilon^{3})$. The leading term is

Taking the average over $t_{0}$, we have

Substituting Eq. (33), the unique unknown variable $w$ is obtained as

where we denoted $\left\langle E^{(2)}\right\rangle$ by $E^{(2)}$ for simplicity. Finally, we eliminate the unknown amplitudes from the averaged term $\mathcal{A}$ represented in Eq. (32) by substituting Eqs. (33) and (37):

where

We underline that the averaged term $\mathcal{A}$ depends on the constants $\nu_{1},\nu_{2}$ and $E^{(2)}$.

Substituting Eq. (38) into Eq. (21), we have

where

and

Equation (40) is the closed equation for the slow bending motion. It is reproduced as the Euler-Lagrange equation of the effective Lagrangian

Here, the effective (dimensionless) mass $M_{{\rm eff}}(\phi^{(0)})$ is

and the effective potential $V_{{\rm eff}}(\phi^{(0)})$ is

Note that the physical dimension of $V_{{\rm eff}}$ differs from $V_{\rm bend}^{(2)}$ due to the factor $1/C^{33}$.

The effective potential $V_{{\rm eff}}$ of Eq. (46) is the main product of the theory. A remarkable observation is that $V_{{\rm eff}}$ depends on energy $E^{(2)}$ and the normal mode energy distribution $(\nu_{1},\nu_{2})$ through $M_{{\rm eff}}$ [Eq. (45)] and $G_{\nu}$ [Eq. (42)]. Examples of the effective potential are exhibited in Sec. IV.

First of all, we observe the effective potential $V_{{\rm eff}}$ without bending potential, $V_{\rm bend}\equiv 0$, to observe the simplest case. We exhibit effective potentials for some values of $\nu_{1}$ ($\nu_{2}=1-\nu_{1}$) in Fig. 2. The dynamical contribution is completely opposite between the in-phase mode and the anti-phase mode. The in-phase mode makes a valley at $\phi=0$, while the anti-phase mode makes a valley at $\phi=\pi$. A precise analysis reveals that there are the two local minima at $\phi=0$ and $\pi$ in the interval of $\nu_{1}\in(1/4,3/4)$. The coexistence interval is generalized to

for any value of $m_{2}$. See Appendix F for details.

Next, we introduce an example of the bending potential as

This potential has the two minima at $\phi=\pm\pi/2$. We set the equal mass condition, $m_{2}=m$. Since $V_{{\rm eff}}$ depends on the normal mode energy distribution $(\nu_{1},\nu_{2})$ and the total energy $E^{(2)}$, we show graphs of the effective potential for $(\nu_{1},\nu_{2})=(1,0)$ (in-phase), $(1/2,1/2)$ (mixed), and $(0,1)$ (anti-phase) with varying the value of $E^{(2)}$ in Fig. 3. The effective potential $V_{{\rm eff}}$ is similar to the bending potential $V_{\rm bend}^{(2)}$ when the total energy $E^{(2)}$ is small. As the total energy increases, the local minimum points move from $\phi=\pm\pi/2$ towards $\phi=0$ and/or $\phi=\pi$. The local minimum points of $V_{{\rm eff}}$ are the dynamically induced conformations (DICs).

The effective potential is determined at each point on the $(E^{(2)},\nu_{1})$ plane, and yields the set of the local minimum points. We categorize the local minimum points into the three classes: $\phi=0,\pi$, and $\phi_{\sharp}(\neq 0,\pi)$. The three classes induce the seven types of sets as arranged in Table 1. By using the seven types, the $(E^{(2)},\nu_{1})$ plane is divided into regions each of which is assigned by a type of the set as reported in Fig. 4. We stress that the seven types are realized by changing the total energy $E^{(2)}$ and the mode energy distribution $\nu_{1}$.

Finally, we present a phase diagram by varying the center mass $m_{2}$ with fixing $E^{(2)}=4$ in Fig. 5. We give two remarks. First, the seven types are also realized by changing the center mass $m_{2}$. The regions assigned by the types III₀, III_π, IV, and V are enhanced comparing with Fig. 4. This fact suggests that the mass distribution is useful to control the conformation. Second, the dynamical contribution dominates the effective potential when $m_{2}$ is small. This domination can be explained from Eq. (39), which is a part of the averaged term $\mathcal{A}$, and Eq. (26), which is the definitions of $M_{2}$ and $M_{1}$. We have $M_{2}\to M_{1}$ as $m_{2}\to 0$ from Eq. (26). Thus, the denominators of the function $T_{\nu}$ can be close to $0$ near $\phi^{(0)}=0,\pi$, and hence contribution from the averaged term becomes large.

Numerical simulations are performed by using the fourth order symplectic integrator yoshida-90 for the Hamiltonian

which is the Legendre transform of Eq. (1). The time step is set as $\Delta t=10^{-3}$. The relative energy error is suppressed in the reported simulations as $|(E_{\rm num}-E_{0})/E_{0}|<10^{-10}$, where $E_{0}$ and $E_{\rm num}$ are respectively the initial energy and the numerically obtained energy.

The two springs are assumed to be linear and $V_{\rm spring}$ is

because the theory includes only the linear part of the springs. The bending potential is $V_{\rm bend}=\epsilon^{2}V_{\rm bend}^{(2)}$, and we use Eq. (48) as $V_{\rm bend}^{(2)}$. The small parameter $\epsilon$ is fixed as $\epsilon=0.1$. The masses are equal and $m_{1}=m_{2}=m_{3}=m=1$. The spring constant is $k=10$, and the natural length is $l_{\ast}=1$.

We set the initial condition as follows. All the beads have zero initial velocities in the $x$- and $y$-directions. The beads are once placed at the natural lengths of the springs with the bending angle $\phi_{\ast}$. Then, under the hypothesis (H), we give small displacements of $l_{1},l_{2}$, and $\phi$ so as to excite the normal modes of the springs for a given pair of $(\nu_{1},\nu_{2})$. The initial condition, denoted by the subscript $0$, is summarized as

See Eq. (27) for the definitions of $\boldsymbol{p}_{\rm in}$ and $\boldsymbol{p}_{\rm anti}$. We note that the amplitude $w$ is of $O(\epsilon^{0})$. In the Hamiltonian system of Eq. (49), a corresponding initial condition is

where $\boldsymbol{0}$ is the two-dimensional zero vector. The above initial condition gives the second order total energy as

In the next section we set $\phi_{\ast}=\pi/2$, which is a local minimum point of the bending potential $V_{\rm bend}$. The initial condition Eq. (51) hence has two free parameters of the amplitude $w$, which is equivalent with $E^{(2)}$ through Eq. (53), and the normal mode energy ratio $\nu_{1}$ (remember $\nu_{2}=1-\nu_{1}$).

We concentrate on the in-phase mode, $(\nu_{1},\nu_{2})=(1,0)$. Temporal evolution of $\phi(t)$ is exhibited in Figs. 6(d), (e), and (f) for three values of $w$, corresponding to three values of $E^{(2)}$ [see Eq. (53)]. Small and fast oscillation of $\phi(t)$ comes from $\phi^{(1)}$ and is induced from $l_{j}^{(1)}$, which are governed by Eq. (15). When the amplitude $w$ is small, the bending angle $\phi$ almost stays around the initial value $\phi_{0}$ [Fig. 6(d)], as it is predicted from the bending potential $V_{\rm bend}$. However, the amplitude of oscillation of $\phi$ becomes large as $w$ gets large, and the center of oscillation approaches to the zero [Figs. 6(e) and (f)]. We estimate the center of oscillation by the time average

The estimated center is plotted as a function of $E^{(2)}$ in Fig. 6(g) with the minimum $\phi_{\rm min}$ and the maximum $\phi_{\rm max}$ of $\phi(t)$ in $t\in[0,1000]$.

Two remarks are in order. First, the minimum and the maximum of $\phi$ are well predicted by $V_{{\rm eff}}$. There is a gap between $\phi_{\rm ave}$ and the bottom of $V_{{\rm eff}}$ in the energy interval approximated by $E^{(2)}\in[6.8,10]$. The gap is not a counterevidence but a supporting evidence of the theory. This gap comes from the inequality $V_{{\rm eff}}(0)<V_{{\rm eff}}(\phi_{0})$, which implies that the bending angle climbs over the saddle point at $\phi=0$. This passing is confirmed by the jump of the theoretical minimum value of $\phi$. Second, the center of oscillation is continuously modified as the spring energy increases. The conformation is determined not by a local minimum of the bending potential $V_{\rm bend}$ but by a local minimum of the effective potential $V_{{\rm eff}}$ derived from dynamics. We conclude that the effective potential successfully predicts the slow bending motion.

We provide movies in Supplemental Material SM to show dynamics of the system. See Appendix G for explanation on the movies.