Proof of Liouville's Theorem in Dynamics 📂Dynamics

Proof of Liouville's Theorem in Dynamics

Theorem

Let’s consider a vector field for Euclidean space [eq01] and a function [eq02], given as a differential equation. The flow of this vector field [eq03] and the volume of a region [eq04] over time [eq06], having moved according to the flow, can be represented as [eq07]. If [eq08], then for all [eq04] and [eq10], the following holds. eq02

Explanation of the Formulas

[eq11] represents the divergence of the vector field, showing how the vector field spreads or converges.

[eq12] signifies the volume of a given region in the vector field. Liouville’s theorem is more straightforward in formulas; simply put, if the divergence is everywhere [eq13], the volume of the region moved by the flow does not change.

Proof ¹

Strategy: A bit of vector calculus can help here. By representing the vector function and each axis of vectors, it directly follows from basic tools of analysis and linear algebra. eq03

Part 1. When [eq14] is [eq15]

Definition of Volume: When [eq16] is transformed by the vector function [eq17] as [eq18], the volume of [eq19] is as follows. eq04

Following the definition of volume, eq05 Expanding the flow [eq20] in Taylor series around [eq21], eq06 Differentiating [eq22] gives for the identity matrix [eq24], eq07 Consequently, the determinant is eq08 Here [eq25] denotes the Trace, which is the sum of the diagonal elements of a matrix. By taking [eq26] on both ends, eq09 Moving [eq27] to the left side gives, eq10 Dividing both sides by [eq28], eq11 Since Taylor’s approximation is used around [eq29], when [eq30] is [eq31], eq12

Part 2. Extending to [eq32]

Assuming [eq33] allows us to follow the same discussion from Part 1. to arrive at eq13 For some constant [eq34], if [eq35], eq14 [eq36] being transformed by the flow of [eq22] over time [eq38], so [eq39] becomes the volume at time [eq38]. This is true for any [eq41] as well, eq15 The above differential equation has the trivial solution [eq42].

Part 3.

From the last equation of Part 2., if [eq43] then [eq44] is true.

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