# Liouville's theorem

1. Oct 4, 2009

### Petar Mali

Phase volume is constant.

$$\int_{G_0}dx^0=\int_{G_t}dx^t$$

$$x=(x_1,...,x_{6N})$$

$$\int_{G_0}dx^0=\int_{G_t}dx^t=\int_{G_0}Jdx^0$$

We must prove that $$J=1$$

$$J=\frac{\partial (x_1^t,...,x_{6N}^t)}{\partial (x_1^0,...,x_{6N}^0)}$$

$$J$$ is determinant with elements

$$a_{ik}=\frac{\partial x_i^t}{\partial x_k^0}$$

Minor for $$a_{ik}$$ is

$$D_{ik}=\frac{\partial J}{\partial a_{ik}}$$

And now $$J$$ is define like

$$J=\sum_{k}D_{ik}a_{ik}$$

Why is define like this? Why not

$$J=\sum_{i,k}D_{ik}a_{ik}$$ ?