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Symmetries and conservation laws of the Liouville equation
V. V. Zharinov Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
Аннотация:
Symmetries and conservation laws of the Liouville equation are studied in the frames of the algebra-geometrical approach to partial differential equations.
Bibliography: 9 titles.
Ключевые слова:
Liouville equation, symmetry, conservation law, evolution derivation, evolution Lie bracket, variational derivative, first integral.
Поступило в редакцию: 17.04.2022 Исправленный вариант: 19.09.2022
§ 1. Introduction Symmetries and conservation laws of the Liouville equation are studied in the frames of the algebra-geometrical approach to partial differential equations. Interest in the Liouville equation and its properties has intensified recently due to its important role in functional mechanics proposed by I. V. Volovich (see [1]–[3]). 1.1. Main ingredients The article is quite computational, to make it more readable we first clarify the designations. Namely, below: - • the summation over repeated upper and lower indices in the prescribed limits is assumed;
- • $\mathbb{N}=\{1,2,3,\dots\}\subset\mathbb{Z}_+=\{0,1,2,\dots\} \subset\mathbb{Z}=\{0,\pm1,\pm2,\dots\}$;
- • $\mathbf{m}=\overline{1,m}=\{1,2,\dots,m\}$, $m\in\mathbb{N}$;
- • $\mathbb{I}=\{\mathrm{i}=(i^\mu)\in\mathbb{Z}_+^\mathbf{m}\mid i^\mu\in\mathbb{Z}_+, \ \mu\in\mathbf{m}\}$;
- • $|\mathrm{i}|=i^1+\dots+i^m$, $\mathrm{i}+(\mu)=(i^1,\dots,i^\mu+1,\dots,i^m)$ for $\mathrm{i}=(i^1,\dots,i^m)\in\mathbb{I}$;
- • $\mathrm{T}=\{t\in\mathbb{R}\}$ is the space of the time variable;
- • $\mathrm{X}=\{\mathrm{x}=(x^\mu)\in\mathbb{R}^\mathbf{m}\mid x^\mu\in\mathbb{R}, \ \mu\in\mathbf{m}\}$ is the space of the spatial variables;
- • $\mathrm{U}=\{u\in\mathbb{R}\}$ is the space of the dependent variable, on the first reading it is useful to consider $u=\phi(t,\mathrm{x})$;
- • $\mathbf{U}\,{=}\,\{\mathbf{u}=(u_\mathrm{i})\,{\in}\,\mathbb{R}_\mathbb{I}\mid u_\mathrm{i}\,{\in}\,\mathbb{R}, \ \mathrm{i}\,{\in}\,\mathbb{I}\}$ is the space of the differential variables, $u\,{=}\,u_0$, on the first reading it is useful to consider $u_\mathrm{i}\,{=}\,(\partial_{x^1})^{i^1}{\cdots}{\kern1pt} (\partial_{x^m})^{i^m}\phi(t,\mathrm{x})$.
The main functional algebras are: Remark 1. Note, the number $r{\kern1pt}{\in}{\kern1pt}\mathbb{Z}_+$ is called the order of a function $f(t,\mathrm{x},\mathbf{u}){\kern1pt}{\in}{\kern1pt}\mathcal{A}$, $\operatorname{ord} f=r$, if the partial derivative $\partial_{u_\mathrm{i}}f\ne0$ for some index $\mathrm{i}\in\mathbb{I}$, $|\mathrm{i}|=r$, while the partial derivatives $\partial_{u_\mathrm{i}}f=0$ for all indexes $\mathrm{i}\in\mathbb{I}$, $|\mathrm{i}|>r$. The main linear operators are: - • $\mathrm{v} \colon \mathrm{X}\to\mathrm{X}$, $\mathrm{x}=(x^\mu)\mapsto\mathrm{v}={v^\mu(\mathrm{x})}$, is a given vector field, governing the dynamics of the model under study;
- • $L=\partial_t+\partial_\mathrm{x}\circ\mathrm{v} \colon \Phi\to\Phi$, $\phi(t,\mathrm{x})\mapsto L\phi(t,\mathrm{x})=\partial_t\phi(t,\mathrm{x}) +\partial_{x^\mu}(v^\mu(\mathrm{x})\phi(t,\mathrm{x}))$, is the Liouville operator;
- • $\frac{d}{dt}=\partial_t+\mathrm{v}\cdot\partial_\mathrm{x} \colon \Phi\to\Phi$, $\phi(t,\mathrm{x})\mapsto \frac{d\phi(t,\mathrm{x})}{dt}=\partial_t\phi(t,\mathrm{x}) +v^\mu\partial_{x^\mu}\phi(t,\mathrm{x})$, is the time derivation along the vector field $\mathrm{v}$;
- • $L=\frac{d}{dt}+\operatorname{div}\mathrm{v}$, where $\operatorname{div}\mathrm{v} =\partial_{x^\mu}v^\mu(\mathrm{x})$ is the divergence of the vector field $\mathrm{v}$;
- • $D_\mu=\partial_{x^\mu}+u_{\mathrm{i}+(\mu)}\partial_{u_\mathrm{i}} \colon \mathcal{A}\to\mathcal{A}$, $g\mapsto D_\mu g=\partial_{x^\mu}g+u_{\mathrm{i} +(\mu)}\partial_{u_\mathrm{i}}g$, $\mu\in\mathbf{m}$, are the total partial spatial derivations, $[D_\mu,D_\nu]=0$, $\mu,\nu\in\mathbf{m}$;
- • $D_\mathrm{i}=(D_1)^{i^1}\circ\dots\circ(D_m)^{i^m}$, $\mathrm{i}=(i^1,\dots,i^m)\in\mathbb{I}$;
- • $D_t=\partial_t+\mathrm{f}_\mathrm{i}\,\partial_{u_\mathrm{i}} \colon \mathcal{A}\to\mathcal{A}$, $g\mapsto D_tg=\partial_t g+\mathrm{f}_\mathrm{i}\,\partial_{u_\mathrm{i}}g$, is the total time derivation, $\mathrm{f}=-D_\mu(v^\mu(\mathrm{x})u)$, $\mathrm{f}_\mathrm{i}=D_\mathrm{i}\mathrm{f}$, $\mathrm{i}\in\mathbb{I}$, $[D_\mu,D_t]=0$, $\mu\in\mathbf{m}$;
- • $\mathbf{D}_t=D_t+\mathrm{f}^*=D_t+v^\mu(\mathrm{x})D_\mu \colon \mathcal{A}\to\mathcal{A}$ is the total time derivation along the vector field $\mathrm{v}$;
- • $\mathbf{L}=D_t-\mathrm{f}_*=D_t+D_\mu\circ v^\mu(\mathrm{x}) \colon \mathcal{A}\to\mathcal{A}$ is the total Liouville operator, $\mathbf{L}=\mathbf{D}_t+\operatorname{div}\mathrm{v}$.
1.2. The Liouville equation The Liouville equation has the form
$$
\begin{equation}
L\phi=\partial_t \phi+\partial_{x^\mu}(v^\mu\cdot \phi) =\frac{d\phi}{dt} +\operatorname{div}\mathrm{v}\cdot\phi=0,
\end{equation}
\tag{1}
$$
where $\phi(t,\mathrm{x})\in\Phi$ is an unknown function, $\mathrm{v} \colon \mathrm{X}\to\mathrm{X}$ is a given vector field. Theorem 1. The general solution $\phi(t,\mathrm{x})\in\Phi$ of the Liouville equation (1) is implicitly defined by the equality
$$
\begin{equation*}
K(\varkappa,s^1,\dots,s^m)=0,
\end{equation*}
\notag
$$
where $K$ is an arbitrary function, while $\varkappa(t,\mathrm{x},\phi)$, $\mathrm{s} =(s^1(t,\mathrm{x}),\dots,s^m(t,\mathrm{x}))$ are the first integrals of the system of the ordinary differential equations
$$
\begin{equation*}
\frac{dt}1=\frac{dx^1}{v^1}=\dots=\frac{dx^m}{v^m} =-\frac{d\phi}{\operatorname{div}\mathrm{v}\cdot\phi}.
\end{equation*}
\notag
$$
The proof of this and the analogous theorems below follows from the theory of differential equations in partial derivatives of the first order (see [4], for example). The situation simplifies when the vector field $\mathrm{v}$ is divergence free, the general solution can be found in the explicit form. Indeed, if $\operatorname{div}\mathrm{v}=0$, then the Liouville equation (1) takes the form
$$
\begin{equation}
L\phi=\frac{d\phi}{dt}=\partial_t\phi+v^\mu\partial_{x^\mu}\phi=0,
\end{equation}
\tag{2}
$$
and has the general solution $\phi=\rho(s)$, where $\rho$ is an arbitrary function, while $\mathrm{s}=(s^1(t,\mathrm{x}),\dots,s^m(t,\mathrm{x}))$ are the first integrals of the shorten system
$$
\begin{equation*}
\frac{dt}1=\frac{dx^1}{v^1}=\dots=\frac{dx^m}{v^m}.
\end{equation*}
\notag
$$
§ 2. Symmetries The Lie algebra of symmetries of the Liouville equation (1) is defined as
$$
\begin{equation}
\operatorname{Sym}(L)=\{\zeta=\operatorname{ev}_g \mid g\in\mathcal{A}, \ \mathbf{L} g =D_tg+D_\mu(v^\mu g)=0\}\simeq\operatorname{Ker}\mathbf{L},
\end{equation}
\tag{3}
$$
where the evolution derivation $\operatorname{ev}_g=D_\mathrm{i} g\cdot\partial_{u_\mathrm{i}}$, the evolution Lie bracket
$$
\begin{equation*}
[\operatorname{ev}_g,\operatorname{ev}_h]=\operatorname{ev}_{\{g,h\}}, \qquad \{g,h\}=\operatorname{ev}_gh-\operatorname{ev}_hg, \quad g,h\in\mathcal{A},
\end{equation*}
\notag
$$
(see [5]–[7], for example). In particular, the linear space $\operatorname{Ker}\mathbf{L}$ here possesses the induced structure of the Lie algebra with the Lie bracket $\{\,{\cdot}\,,{\cdot}\,\}$, where
$$
\begin{equation*}
\operatorname{Ker}\mathbf{L}\times\operatorname{Ker}\mathbf{L}\ni(g,h) \mapsto\{g,h\}\in\operatorname{Ker}\mathbf{L}.
\end{equation*}
\notag
$$
In more detail, the equation $\mathbf{L} g=0$ is written as $Lg-W_\mathrm{i}\cdot\partial_{u_\mathrm{i}}g=0$, where
$$
\begin{equation*}
\begin{gathered} \, Lg =\partial_tg+v^\mu\cdot\partial_{x^\mu}g+\operatorname{div}\mathrm{v}\cdot g, \\ W_\mathrm{i}=\sum_{\mathrm{j}+\mathrm{k}=\mathrm{i},\, \mathrm{j}\ne0}\binom{\mathrm{i}}{\mathrm{k}} \partial_{x^\mathrm{j}}v^\mu\cdot u_{\mathrm{k}+(\mu)} +\sum_{\mathrm{j}+\mathrm{k}=\mathrm{i}} \binom{\mathrm{i}}{\mathrm{k}}\partial_{x^\mathrm{j}}\operatorname{div}\mathrm{v} \cdot u_\mathrm{k}, \end{gathered}
\end{equation*}
\notag
$$
$\partial_{\mathrm{x}^\mathrm{j}}=(\partial_{x^1})^{j^1}\circ\dots\circ(\partial_{x^m})^{j^m}$, $\mathrm{j}=(j^1,\dots,j^m)\in\mathbb{I}$. Proposition 1. The linear operator $\mathbf{L}=L-W_\mathrm{i}\cdot\partial_{u^\mathrm{i}} \colon \mathcal{A}\to\mathcal{A}$ is the order- nonincreasing, i.e., $\operatorname{ord}(\mathbf{L} g)\leqslant\operatorname{ord} g$ for any $g\in\mathcal{A}$. Theorem 2. The equation $\mathbf{L} g=0$ has the general solution $g(t,\mathrm{x},\mathbf{u})\in\mathcal{A}$, implicitly defined by the equality
$$
\begin{equation*}
\mathbf{K}(\mathrm{s},\mathbf{w},\varkappa)=0,
\end{equation*}
\notag
$$
where $\mathbf{K}$ is an arbitrary function of a finite order, while $\mathrm{s}(t,x)$, $\mathbf{w}(t,x,\mathbf{u})$, and $\varkappa(t,x,\mathbf{u},g)$, are the first integrals of the system
$$
\begin{equation*}
\frac{dt}{1}=\frac{dx^\mu}{v^\mu}=-\frac{du_\mathrm{i}}{W_\mathrm{i}} =-\frac{dg}{\operatorname{div}\mathrm{v}\cdot g}, \qquad \mu\in\mathbf{m}, \quad \mathrm{i}\in\mathbb{I} \quad (\textit{without summation}!),
\end{equation*}
\notag
$$
here $\mathrm{i}=(i^\mu)\in\mathbb{I}$, $\mathbf{u}=(u_\mathrm{i})$, $\mathbf{w}=(w_\mathrm{i})\in\mathbb{R}_\mathbb{I}$. Remark 2. Note, the functions $\mathrm{s}(t,x)$ here are solutions of the Liouville equation (2). Again, the situation simplifies when the vector field $\mathrm{v}$ is divergence free. Indeed, if $\operatorname{div}\mathrm{v}=0$, the equation $\mathbf{L} g=0$ takes the form
$$
\begin{equation*}
\mathbf{L} g=\partial_tg+v^\mu\cdot\partial_{x^\mu}g -W'_\mathrm{i}\cdot\partial_{u_\mathrm{i}}g=0, \qquad W'_\mathrm{i}=\sum_{\mathrm{j}+\mathrm{k}=\mathrm{i},\, \mathrm{j}\ne0} \binom{\mathrm{i}}{\mathrm{k}} \partial_{x^\mathrm{j}}v^\mu\cdot u_{\mathrm{k}+(\mu)},
\end{equation*}
\notag
$$
and has the general solution $g=\rho(\mathrm{s},\mathbf{w})$, where $\rho$ is an arbitrary function of a finite order, while $\mathrm{s}(t,x)$, $\mathbf{w}(t,x,\mathbf{u})$ are the first integrals of the shorten system
$$
\begin{equation*}
\frac{dt}{1}=\frac{dx^\mu}{v^\mu}=-\frac{du_\mathrm{i}}{W'_\mathrm{i}} =\frac{du_0}0, \qquad \mu\in\mathbf{m}, \quad \mathrm{i}\in\mathbb{I}\setminus\{0\} \quad \text{(without summation!)}.
\end{equation*}
\notag
$$
§ 3. Conservation laws The linear space of conservation laws for the Liouville equation $L\phi=0$ is defined as the factor-space
$$
\begin{equation}
\mathrm{CL}(L)=\{\rho\in\mathcal{A} \mid D_t\rho\in\operatorname{Div}\mathcal{A}^\mathbf{m}\} \bigm/ \{\rho\in\operatorname{Ker}\delta_u\},
\end{equation}
\tag{4}
$$
where $\operatorname{Div}\mathrm{J}=D_\mu J^\mu$, $\mathrm{J}=(J^\mu)\in\mathcal{A}^\mathbf{m}$, the variational derivative
$$
\begin{equation*}
\delta_u \colon \mathcal{A}\to\mathcal{A}, \qquad \rho\mapsto\delta_u\rho =(-D)_\mathrm{i}(\partial_{u_\mathrm{i}}\rho)
\end{equation*}
\notag
$$
(see [5]–[8], for example). Theorem 3. There is defined the isomorphism of the linear spaces
$$
\begin{equation}
\delta_u \colon \mathrm{CL}(L)\simeq\{\chi\in\operatorname{Ker}\mathbf{D}_t \mid \chi_*=\chi^*\}, \qquad [\rho]\mapsto\chi=\delta_u\rho,
\end{equation}
\tag{5}
$$
where $[\rho]=\rho+\operatorname{Ker}\delta_u$, $\chi\in\mathcal{A}$, Note, the linear operator $\mathbf{D}_t \colon \mathcal{A}\to\mathcal{A}$ is the order-nonincreasing. Theorem 4. The equation $\mathbf{D}_t\chi=0$ has the general solution $\chi=R(\mathrm{s},\mathbf{w})$, where $R$ is an arbitrary function of a finite order, while $\mathrm{s}(t,x)$, $\mathbf{w}(t,x,\mathbf{u})$ are the first integrals of the system
$$
\begin{equation*}
\frac{dt}{1}=\frac{dx^\mu}{v^\mu}=-\frac{du_\mathrm{i}}{W'_\mathrm{i}} =\frac{du_0}0, \qquad \mu\in\mathbf{m}, \quad \mathrm{i}\in\mathbb{I}\setminus\{0\} \quad \textit{(without summation!)},
\end{equation*}
\notag
$$
here $\mathrm{i}=(i^\mu)\in\mathbb{I}$, $\mathbf{u}=(u_\mathrm{i})$, $\mathbf{w}=(w_\mathrm{i})\in\mathbb{R}_\mathbb{I}$. Remark 3. Note, $\mathbf{D}_t=\mathbf{L}|_{\operatorname{div}\mathrm{v}=0}$.
§ 4. Examples Example 1. Let $\mathrm{x}=(x^1,x^2)\in\mathrm{X}=\mathbb{R}^2$, $x^1=q$, $x^2=p$, $H=\frac{p^2}{2m}$, then $\mathrm{v}=(v^1,v^2)$, $v^1=\partial_{x^2}H=\frac{p}{m}$, $v^2=-\partial_{x^1}H=0$ (see [9; Example 6.1]), so here
$$
\begin{equation*}
L=\frac d{dt}=\partial_t+\frac pm\, \partial_q, \qquad \mathbf{L}=\mathbf{D}_t=L-W_{j,k}\cdot\partial_{u_{j,k}}, \quad W_{j,k}=\frac{k}{m} u_{j+1,k-1},
\end{equation*}
\notag
$$
where $\mathrm{i}=(j,k)\in\mathbb{I}=\mathbb{Z}_+\times\mathbb{Z}_+$. In particular, the equation $\mathbf{D}_t\chi=0$ has the general solution $\chi=R(r,s,\mathbf{w})$, where $R(r,s,\mathbf{w})$ is an arbitrary function of a finite order, while $r(t,q,p)$, $s(t,q,p)$, and $\mathbf{w}(t,q,p,\mathbf{u})$ are the first integrals of the system
$$
\begin{equation*}
\frac{dt}1=\frac{mdq}p=\frac{dp}0=-\frac{du_{j,k}}{W_{j,k}}, \qquad j,k\in\mathbb{Z}_+ \quad \text{(without summation!)},
\end{equation*}
\notag
$$
here $\mathrm{i}=(j,k)\in\mathbb{I}=\mathbb{Z}_+\times\mathbb{Z}_+$, $\mathbf{u}=(u_{j,k})$, $\mathbf{w}=(w_{j,k})$. Proposition 2. In the above settings, the first integrals $r$, $s$, and $\mathbf{w}$ are: - • $r(t,q,p)=q-\tau p$, $s(t,q,p)=p$;
- • $w_{j,k}(t,q,p,\mathbf{u})=\sum_{0\leqslant n\leqslant k}\binom kn u_{j+n,k-n}\tau^n$,
where $\tau=\frac{t}{m}$. Corollary 1. The Lie algebra $\operatorname{Sym}(L)\simeq\operatorname{Ker}\mathbf{D}_t$. Corollary 2. The linear space $\mathrm{CL}(L)\simeq\{\chi\in\operatorname{Ker}\mathbf{D}_t \mid \chi_*=\chi^*\}$. Note,
$$
\begin{equation*}
r_*=r^*=0, \qquad s_*=s^*=0, \qquad (w_{j,k})_*=(-1)^{j+k}(w_{j,k})^*.
\end{equation*}
\notag
$$
Remark 4. One may say that $\mathrm{CL}(L)\subset\operatorname{Sym}(L)$. Example 2. Let again $\mathrm{x}=(x^1,x^2)\in\mathrm{X}=\mathbb{R}^2$, $x^1=q$, $x^2=p$, but the Hamiltonian $H=\frac{p^2}{2m}+\frac{\varkappa q^2}2$. In this case $\partial_{x^2}H=\frac{p}{m}$, $\partial_{x^1}H=\varkappa q$, so the vector field $\mathrm{v}=\bigl(\frac{p}{m},-\varkappa q\bigr)$ (see [9; Example 6.3]), so here
$$
\begin{equation*}
L=\frac d{dt}=\partial_t+\frac{p}{m}\,\partial_q-\varkappa q\,\partial_p , \qquad \mathbf{L}=\mathbf{D}_t=L-W_{j,k}\,\partial_{u_{j,k}},
\end{equation*}
\notag
$$
where $ \mathrm{i}=(j,k)\in\mathbb{I}=\mathbb{Z}_+\times\mathbb{Z}_+$, $W_{j,k}=-j\varkappa u_{j-1,k+1}+\frac{k}{m} u_{j+1,k-1}$, $j,k\in\mathbb{Z}_+$. The equation $\mathbf{D}_t\chi=0$ has the general solution $\chi=R(r,s,\mathbf{w})$, where $R$ is an arbitrary function of a finite order, while $r(t,q,p)$, $s(t,q,p)$, and $\mathbf{w}(t,q,p,\mathbf{u})$ are the first integrals of the system
$$
\begin{equation*}
\frac{dt}1=\frac{mdq}p=-\frac{dp}{\varkappa q}=-\frac{du_{j,k}}{W_{j,k}}, \qquad j,k\in\mathbb{Z}_+ \quad \text{(without summation!)},
\end{equation*}
\notag
$$
here $\mathrm{i}=(j,k)\in\mathbb{I}=\mathbb{Z}_+\times\mathbb{Z}_+$, $\mathbf{u}=(u_{j,k})$, $\mathbf{w}=(w_{j,k})$. In the above setting, the first integrals $r$, $s$ are (see [9; Example 6.3]) The first integrals $w_{j,k}$ are defined by the equations
$$
\begin{equation*}
-j\varkappa u_{j-1,k+1}+\dot u_{j,k}+\frac{k}{m} u_{j+1,k-1}=0, \qquad j,k\in\mathbb{Z}_+, \quad \dot u=\frac{du(t)}{dt}.
\end{equation*}
\notag
$$
Simple but tedious calculations give: (0) $j+k=0$: (1) $j+k=1$: hence (2) $j+k=2$: - • $u_{0,2}=w_{0,2}+w_{1,1}\cos{2\omega t}+w_{2,0}\sin{2\omega t}$;
- • $u_{1,1}=m\omega(w_{1,1}\sin{2\omega t}-w_{2,0}\cos{2\omega t})$;
- • $u_{2,0}=(m\omega)^2(w_{0,2}-w_{1,1}\cos{2\omega t}-w_{2,0}\sin{2\omega t})$;
hence - • $w_{0,2}=\frac12\bigl(u_{0,2}+\frac1{(m\omega)^2}u_{2,0}\bigr)$;
- • $w_{1,1}=\frac12\bigl(u_{0,2}- \frac1{(m\omega)^2}u_{2,0}\bigr)\cos{2\omega t} +\frac1{m\omega}u_{1,1}\sin{2\omega t}$;
- • $w_{2,0}=\frac12(u_{0,2}- \frac1{(m\omega)^2}u_{2,0}\bigr)\sin{2\omega t} -\frac1{m\omega}u_{1,1}\cos{2\omega t}$;
(3) $j+k=3$: - • $u_{0,3}=w_{0,3}\cos{\omega t}+w_{1,2}\sin{\omega t} +w_{2,1}\cos{3\omega t} +w_{3,0}\sin{3\omega t}$;
- • $u_{1,2}=\frac{m\omega}3(w_{0,3}\sin{\omega t}-w_{1,2}\cos{\omega t} +3w_{2,1}\sin{3\omega t}-3w_{3,0}\cos{3\omega t})$;
- • $u_{2,1}=\frac{(m\omega)^2}3(w_{0,3}\cos{\omega t} +w_{1,2}\sin{\omega t} -3w_{2,1}\cos{3\omega t}-3w_{3,0}\sin{3\omega t})$;
- • $u_{3,0}=(m\omega)^3(w_{0,3}\sin{\omega t}-w_{1,2}\cos{\omega t} -w_{2,1}\sin{3\omega t} +w_{3,0}\cos{3\omega t})$;
hence - • $w_{0,3}=\frac34\bigl(u_{0,3}+\frac1{(m\omega)^2}u_{2,1}\bigr) \cos{\omega t} +\frac3{4m\omega}\bigl(u_{1,2}+\frac1{(m\omega)^2}u_{3,0}\bigr)\sin{\omega t}$;
- • $w_{1,2}=\frac34\bigl(u_{0,3}+\frac1{(m\omega)^2}u_{2,1}\bigr) \sin{\omega t} -\frac3{4m\omega}\bigl(u_{1,2}+\frac1{(m\omega)^2}u_{3,0}\bigr)\cos{\omega t}$;
- • $w_{2,1}=\frac14\bigl(u_{0,3}-\frac3{(m\omega)^2}u_{2,1}\bigr)\cos{3\omega t} +\frac1{4m\omega}\bigl(3u_{1,2}-\frac1{(m\omega)^2}u_{3,0}\bigr)\sin{3\omega t}$;
- • $w_{3,0}=\frac14\bigl(u_{0,3}-\frac3{(m\omega)^2}u_{2,1}\bigr) \sin{3\omega t} -\frac1{4m\omega}\bigl(3u_{1,2}-\frac1{(m\omega)^2}u_{3,0}\bigr)\cos{3\omega t}$;
and so on, for $j+k=4,5,\dots$ . Again, the statements from the previous example are true. Namely:
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Образец цитирования:
V. V. Zharinov, “Symmetries and conservation laws of the Liouville equation”, Изв. РАН. Сер. матем., 87:5 (2023), 92–98; Izv. Math., 87:5 (2023), 941–946
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/im9356https://doi.org/10.4213/im9356 https://www.mathnet.ru/rus/im/v87/i5/p92
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