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Liouville's theorem (Theorem)

Let

$\displaystyle \dot{x}=f(x)$ (1)

be a autonomous ordinary differential equation in $ \mathbb{R}^n$ defined by a smooth vector field $ f\colon \mathbb{R}^n\to \mathbb{R}^n$ and the Jacobian of $ f$ is denoted $ \frac{\partial f}{\partial x}$. Also let $ \Phi_t(x)$ be the flow associated with (1). Let
$\displaystyle V(t) = \int_{\Phi_t(D)} dx$
be the volume of the image of $ D$ under this flow after a time $ t$.
Theorem 1 (Liouville's theorem)   If $ D\subseteq \mathbb{R}^n$ is a bounded measurable domain. Then
$\displaystyle \dot{V}(t)= \int_{\Phi_t(D)} \operatorname{div}\, f(x) dx$
Proof. Let $ V(t)$ be defined as above then
$\displaystyle V(t_0+h)$ $\displaystyle =$ $\displaystyle \int_{\Phi_{t_0+h}(D)}dy$  
  $\displaystyle =$ $\displaystyle \int_{\Phi_h(\Phi_{t_0}(D))}dy$  
  $\displaystyle =$ $\displaystyle \int_{\Phi_{t_0}(D)} \operatorname{det}\left(\frac{\partial\Phi_h}{\partial x}(x)\right) dx.$  

We claim that, for $ x\in \Phi_{t_0}(D)$,

$\displaystyle \frac{\partial\Phi_t}{\partial x}(x) = I + t\frac{\partial f}{\partial x}(x) + o(t)$
as $ t\to 0$.

In fact,

$\displaystyle \Phi_t(x) = x + \int_{0}^t f(\Phi_s(x))ds,$
and by the Leibniz integral rule
$\displaystyle \frac{\partial \Phi_t}{\partial x}(x) = I + \int_{0}^t \frac{\partial}{\partial x}f(\Phi_s(x)) ds,$
so that
$\displaystyle \frac{\partial}{\partial t} \frac{\partial \Phi_t}{\partial x}(x) = \frac{\partial}{\partial x}f(\Phi_t(x))$
and evaluating at $ t=0$ we get
$\displaystyle {\frac{\partial}{\partial t} \frac{\partial \Phi_t}{\partial x}(x... ... = \frac{\partial}{\partial x}f(\Phi_0(x)) = \frac{\partial f}{\partial x} (x).$
Our claim follows from this and from the definition of derivative.

Hence

$\displaystyle \operatorname{det}\left(\frac{\partial\Phi_t}{\partial x}(x)\right)$ $\displaystyle =$ $\displaystyle \operatorname{det}\left(I + t\frac{\partial f}{\partial x}(x)\right) + o(t)$  
  $\displaystyle =$ $\displaystyle \prod_{i=1}^n(1 + \frac{\partial f_i}{\partial x_i}(x)) + o(t)$  
  $\displaystyle =$ $\displaystyle 1+t\sum_{i=1}^n\frac{\partial f_i}{\partial x_i}(x) +o(t)$  
  $\displaystyle =$ $\displaystyle 1 + t\operatorname{div}\, f(x) + o(t)$  

as $ t\to0$. It follows that
$\displaystyle V(t_0+h) = \int_{\Phi_{t_0}(D)} 1 + h\operatorname{div}\, f(x) + o(h) dx$
and
$\displaystyle \dot{V}(t_0)$ $\displaystyle =$ $\displaystyle \lim_{h\to 0}\frac{V(t_0+h)-V(t_0)}{h}$  
  $\displaystyle =$ $\displaystyle \frac{\int_{\Phi_{t_0}(D)} 1 + h\operatorname{div}\, f(x) + o(h) dx -V(t_0)}{h}$  
  $\displaystyle =$ $\displaystyle \frac{V(t_0) + h\int_{\Phi_{t_0}(D)}\operatorname{div}\, f(x)dx + o(h) -V(t_0)}{h}$  
  $\displaystyle =$ $\displaystyle \int_{\Phi_{t_0}(D)}\operatorname{div}\, f(x)dx + \lim_{h\to 0}\frac{o(h)}{h}$  
  $\displaystyle =$ $\displaystyle \int_{\Phi_{t_0}(D)}\operatorname{div}\, f(x)dx.$  

$ \qedsymbol$
Corollary 1   The flow of an Hamiltonian system preserves volume.
Proof. It follows directly since the vector field of an Hamiltonian system has divergence equal to zero. Hence $ \dot{V}=0$ implies that the volume is constant. $ \qedsymbol$

References

TG
Teschl, Gerald: Ordinary Differential Equations and Dynamical Systems. http://www.mat.univie.ac.at/˜gerald/ftp/book-ode/index.html, 2004.



"Liouville's theorem" is owned by Koro. [ full author list (2) | owner history (1) ]
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Cross-references: implies, divergence, Hamiltonian, preserves, derivative, integral, domain, measurable, bounded, flow, image, volume, Jacobian, vector field, smooth, ordinary differential equation, autonomous
There are 2 references to this entry.

This is version 17 of Liouville's theorem, born on 2005-05-08, modified 2006-10-15.
Object id is 7027, canonical name is LiouvillesTheorem3.
Accessed 2383 times total.

Classification:
AMS MSC34A34 (Ordinary differential equations :: General theory :: Nonlinear equations and systems, general)

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