Notes on the history of Liouville’s theorem

Jordan Bell
April 10, 2015

1 Introduction

We denote by (n) the set of all linear maps nn. We take it as known that with the operator norm


(n) is a Banach space.

2 Autonomous differential equations

Lemma 1.

If A(n), then


as ϵ0.


Let λ1,,λn be the eigenvalues of A, repeated according to algebraic multiplicity. For ϵ>0, the eigenvalues of I+ϵA+o(ϵ) repeated according to algebraic multiplicity are


as ϵ0. The determinant of a linear map nn is the product of its eigenvalues according to algebraic multiplicity, so


as ϵ0. But


as ϵ0. ∎

Theorem 2.

If A(n), then


We have


As det:(n) is continuous, we have


Then, using Lemma 1,

deteA = limm(det(I+Am))m
= limm(1+1mtrA+o(1m))m
= etrA.

If A(n), then the flow of the vector field A is


For each t we have etA(n), and by Theorem 2 we have


Let λ be Lebesgue measure on n. If U is an open subset of n, then


Therefore, λ is an invariant measure for the flow if and only if trA=0, namely, if and only if A is skew-symmetric.

3 Nonautonomous differential equations

Suppose that I is an open interval and AC(I,(n)). The set X of all functions x:In that satisfy the differential equation


is a vector space. For each tI we define Bt:Xn by Bt(x)=x(t). It is apparent that for each tI the map Bt is linear. For each x0n, by the existence and uniqueness theorem for ordinary differential equations there is a unique xX for which B0(x)=x0, hence for each tI we get that Bt is a bijection, and hence a linear isomorphism.

Suppose that ϕ1,,ϕnX, and for each tI let Φ(t)(n) be defined by Φ(t)ei=ϕi(t). Then


and hence

Φ˙(t)=A(t)Φ(t),tI. (1)

The Wronskian W=W(ϕ1,ϕn) of the ordered set ϕ1,,ϕn is the function that assigns to each tI the oriented volume of the parallelepiped spanned by ϕ1(t),,ϕn(t). That is,

Theorem 3.

Suppose that I is an open interval and that AC(I,(n)). If ϕ1,,ϕnX, then the Wronskian W=W(ϕ1,,ϕn) satisfies


By (1), for each tI we have

Φ(t+Δ) = Φ(t)+Φ˙(t)Δ+o(Δ)
= Φ(t)+A(t)Φ(t)Δ+o(Δ)
= Φ(t)+A(t)Φ(t)Δ+o(Φ(t)Δ)
= Φ(t)(I+A(t)Δ+o(Δ))
= Φ(t)(I+A(t)Δ+o(Δ))

as Δ0. Using Lemma 1 we get

W(t+Δ) = detΦ(t+Δ)
= detΦ(t)det(I+A(t)Δ+o(Δ))
= detΦ(t)(1+trA(t)Δ+o(Δ))
= detΦ(t)+detΦ(t)trA(t)Δ+o(detΦ(t)Δ)
= detΦ(t)+detΦ(t)trA(t)Δ+o(Δ)

as Δ0, i.e.,


which gives us


One checks that for any t0I,


is a solution of the differential equation in Theorem 3. For each tI we have that v(trA(t))v is linear, and in particular is locally Lipschitz, so by the existence and uniqueness theorem it follows that


4 Jacobi’s formula

Let Ω be the volume form on n, and let A(n). One checks that

Ω(x1,,xn)detA=Ω(Ax1,,Axn),x1,,xnn. (2)

If ω is an (n-1)-form on n, then there is a unique xωn such that for all x1,,xn-1n,


Thus, if x0n and we define an (n-1)-form ω by


then there is some xω with which


We define adjA(n) by (adjA)(x0)=xω. Thus, for x1,,xnn, we have

Ω(x1,Ax2,,Axn)=Ω((adjA)x1,x2,,xn). (3)





and because this holds for all x1,,xnn, it follows that


Furthermore, if A(n), then one checks that for all x1,,xnn,

Ω(x1,,xn)trA=i=1nΩ(x1,,Axi,,xn). (4)

If I is an open interval and AC1(I,(n)), we have, using (2) for the first equality, (3) for the third equality, and (4) for the fourth equality,

ddt(Ω(e1,,en)detA(t)) = ddtΩ(A(t)e1,,A(t)en)
= i=1nΩ(A(t)e1,,A˙(t)ei,,A(t)en)
= i=1nΩ(e1,,(adjA(t))A˙(t)ei,,en)
= Ω(e1,,en)tr((adjA(t))A˙(t)),

that is,


Kline p. 798, Jacobi [40], [41, §17], Felix Klein, 19th century, chapter V

5 Reynolds transport theorem

If V is a vector field with flow ϕ and U is a bounded open subset of n with piecewise smooth boundary and f:n×n is smooth, then with Ut=ϕt(U),


this presumes that det(Dϕt)(x)>0. Write DDt=t+VD. We then have

ddtUtf(y,t)𝑑y = ddtUf(ϕt(x),t)det(Dϕt)(x)dx
= U(Df)(ϕt(x),t)ϕ˙t(x)det(Dϕt)(x)
= UDfDt(ϕt(x),t)det(Dϕt)(x)+f(ϕt(x),t)ddtdet(Dϕt)(x)dx.

Writing Jt(x)=det(Dϕt)(x), we have

ddtUtf(y,t)𝑑y = UDfDt(ϕt(x),t)Jt(x)+f(ϕt(x),t)ddtJt(x)dx
= UD(fJ)Dt

Reynolds [73, pp. 12–13, art. 14]

Amann and Escher [4, p. 425, Theorem 2.11]

6 Symplectic geometry

7 Geodesic flow

Invariance of a kinematic measure on the unit tangent bundle.

8 References

Jacobi [42, p. 93]

Truesdell [87, pp. 101, 105, 351]

Whittaker [90, p. 323, §148]

Hartman [35, p. 91]

Barrow-Green [6, p. 83]

Goroff [70, p. I79]

Gray [32, p. 380]

Ostrogradskii [51, pp. 122–123]

Cajori [13, vol. II, p. 101, §464]

Gibbs [30, Chapter XII]

Sklar [77, p. 130] on phase space

Boltzmann [10, pp. 274–290, 443]

Jeans [43, p. 258, §206]

Kac [45, p. 63]

Lenzen [55, p. 129]


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