# On oscillations of two joined pendulums with cavities partially filled with an incompressible ideal fluid.

Let $G_1$ and $G_2$ be two joined bodies with masses $m_1$ and $m_2$. Each of them has
a cavity partially ﬁlled with homogeneous incompressible ideal ﬂuids situated in domains ${\Omega}_1$
и ${\Omega}_2$ with free boundaries ${\Gamma}_1(t)$, ${\Gamma}_2(t)$ and rigid parts $S_1$, $S_2$. Let ${\rho}_1$, ${\rho}_2$ be densities of ﬂuids.
We suppose that the system oscillates (with friction) near the points $O_1$, $O_2$ which are spherical
hinges.

We use the vectors of small angular displacement

$$\overrightarrow{\delta_k}(t) = \sum\limits_{j=1}^{3} \delta^j_k(t) \overrightarrow{e_k^j},\ k = 1, 2,$$

to determine motions of the removable coordinate systems $O_k x_k^1 x_k^2 x_k^3$ (connected with bodies)
relative to stable coordinate system $O_1 x^1x^2x^3$. Then angular velocities $\overrightarrow{\omega_k}(t)$ of bodies $G_k$ is
equal to $\frac{d\overrightarrow{\delta_k}}{dt}$.

Let $\overrightarrow{u_k}(x, t) = \overrightarrow{w_k}(x, t) + \bigtriangledown \Phi_k(x, t), w_k \in \overrightarrow{J_0}(Ω_k), \bigtriangledown Φ_k \in \overrightarrow{G_{h,S_k}}(\Omega_k)$ and $p_k(x, t) \in H^1(\Omega_k)$
be ﬁelds of ﬂuids velocities and dynamical pressures in $\Omega_k$ (in removable coordinate systems),
$\zeta_k(x, t) \in L_{2,\Gamma_k} := L_2(\Gamma_k) \ominus sp\ 1_{\Gamma_k}$ are functions of normal deviation of $\Gamma_k(t)$ from equilibrium
plane surfaces $\Gamma_k(0) = \Gamma_k$. Then we consider initial boundary value problem (2.1), (2.4)–(2.6)
with conditions (2.7)–(2.11).

We obtain the law of full energy balance (2.12). Using the method of orthogonal projections
with some additional requirements initial problem can be reduced to the Cauchy problem for the
system of diﬀerential equations

$$C_1\frac{dz_1}{dt} + A_1z_1 + gB_{12}z_2 = f_1(t), z_1(0) = z_1^0,$$

$$gC_2 \frac{dz_2}{dt} + gB_{21}z_1 = 0, z_2(0) = z_2^0,$$

$$z_1 = (\overrightarrow{w_1}; \triangledown\Phi_1; \overrightarrow{\omega_1}; \overrightarrow{w_2}; \triangledown\Phi_2; \overrightarrow{ω_2})^{\tau} \in \mathscr{H_1}, z_2 = (\zeta_1; P_2\overrightarrow{\delta_1}; \zeta_2; P2\overrightarrow{\delta_2})^{\tau} \in \mathscr{H_2},$$

in Hilbert spaces

$$\mathscr{H_1} = (\overrightarrow{J꣠_0}(\Omega_1)\oplus\overrightarrow{G_{h,S_1}}(\Omega_1)\oplus\mathbb{C^3})\oplus(\overrightarrow{J_0}(\Omega_2)\oplus\overrightarrow{G_{h,S_2}}(\Omega_2)\oplus\mathbb{C^3}), \mathscr{H_2} = (L_{2,\Gamma_1}\oplus\mathbb{C^2})\oplus(L_{2,\Gamma_2}\oplus\mathbb{C^2}).$$

Here operators of potential energy $C_k$ is bounded, $C_1$ is positive deﬁnite, $A_1$ is bounded and
nonnegative, $B_{ij}$ is skew self-adjoint operators. Using this properties we prove theorem on
existence of unique strong solution for $t \in [0; T]$ if some natural conditions for initial data
and given functions $f_1(t)$ are satisﬁed. As a corollary we obtain theorem on solvability of initial
Cauchy problem.
If friction is absent then operator $A_1 = 0$ and for $z(x, t) = e^{i \lambda t}z(x)$ we obtain spectral
operator problem. For the eigenvalues $\mu = \frac{\lambda^2}{g}$ we ﬁnd new variational principle and prove that
spectrum is discrete. It consists of positive eigenvalues with limit point $+\infty$ in stable case, or
the positive branch and not more then ﬁnite number of negative eigenvalues in unstable case.

Keywords: equation of angular momentum deviation, operator matrix, self-adjoint operator,
strong solution, discrete spectrum.

UDC:
517.98, 517.955, 532.5