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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
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     § 1. IntroductionSymmetries 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 ingredientsThe article is quite computational, to make it more readable we first clarify the designations. Namely, below:
The main functional algebras are:
Remark 1. A number $r\in\mathbb{Z}_+$ is called the order of a function $f(t,\mathrm{x},\mathbf{u})\in\mathcal{A}$, $\operatorname{ord} f= r$, 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:
1.2. The Liouville equationThe 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 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 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
§ 2. SymmetriesThe 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, that is, $\operatorname{ord}(\mathbf{L} g)\leqslant\operatorname{ord} g$ 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 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 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 that 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 lawsThe 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 Note that 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 here $\mathrm{i}=(i^\mu)\in\mathbb{I}$, $\mathbf{u}=(u_\mathrm{i})$, $\mathbf{w}=(w_\mathrm{i})\in\mathbb{R}_\mathbb{I}$. § 4. ExamplesExample 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 Example 6.1 in [9]), so here 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 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: where $\tau=\frac{t}{m}$. Corollary 2. The linear space $\mathrm{CL}(L)\simeq\{\chi\in\operatorname{Ker}\mathbf{D}_t \mid \chi_*=\chi^*\}$. In addition, 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 Example 6.3 in [9]), so here 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 as follows (see Example 6.3 in [9]):
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) $j+k=1$:
Again, the statements from the previous example are true. Namely:
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