Liouville's formula

Liouville's formula states that the determinant of a fundamental matrix for a linear differential system changes at a rate governed by the trace of its coefficient matrix.

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Fundamental matrix: A matrix whose columns form a basis of solutions to a linear system of differential equations. Its determinant is the quantity whose evolution Liouville's formula describes.
Linear system of differential equations: A system of differential equations in which the unknown functions and their derivatives enter linearly. Liouville's formula concerns the fundamental matrices of these systems.
Abel's identity: A formula describing the Wronskian of solutions to a scalar linear differential equation. It is the scalar-equation version of the determinant evolution captured by Liouville's formula.
Joseph Liouville: A French mathematician whose work spanned analysis, number theory, and mathematical physics. The identity bears his name and belongs to the differential-equation tradition associated with his work.
Jacobi's formula: An identity expressing the derivative of a matrix determinant through the matrix and its derivative. Applying it to a fundamental matrix yields the trace in Liouville's formula.
Initial value problem: A differential equation together with specified values of its unknown functions at an initial point. Choosing an identity initial matrix fixes a standard fundamental matrix for the system.
Liouville's theorem (Hamiltonian mechanics): A theorem stating that Hamiltonian flow preserves phase-space volume. Liouville's formula proves volume preservation for linear Hamiltonian systems through zero trace.
Ordinary differential equation: An equation relating an unknown function of one variable to its derivatives. The formula extends a classical Wronskian result for scalar ordinary differential equations.
Matrix trace: The sum of the diagonal entries of a square matrix. The coefficient matrix's trace determines the logarithmic rate of change of the determinant.
Determinant: A scalar associated with a square matrix that encodes invertibility and oriented volume scaling. The formula tracks this scalar for a matrix of solution columns.