In the study of ordinary differential equations and their associated boundary value problems, Lagrange's identity, named after Joseph Louis Lagrange, gives the boundary terms arising from integration by parts of a self-adjoint linear differential operator. Lagrange's identity is fundamental in Sturm–Liouville theory. In more than one independent variable, Lagrange's identity is generalized by Green's second identity.
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Statement
In general terms, Lagrange's identity for any pair of functions u and v in function space C2 (that is, twice differentiable) in n dimensions is:
where:
and
The operator L and its adjoint operator L* are given by:
and
If Lagrange's identity is integrated over a bounded region, then the divergence theorem can be used to form Green's second identity in the form:
where S is the surface bounding the volume Ω and n is the unit outward normal to the surface S.
Ordinary differential equations
Any second order ordinary differential equation of the form:
can be put in the form:
This general form motivates introduction of the Sturm–Liouville operator L, defined as an operation upon a function f such that:
It can be shown that for any u and v for which the various derivatives exist, Lagrange's identity for ordinary differential equations holds:
For ordinary differential equations defined in the interval [0, 1], Lagrange's identity can be integrated to obtain an integral form (also known as Green's formula):
where
Proof of form for ordinary differential equations
We have:
and
Subtracting:
The leading multiplied u and v can be moved inside the differentiation, because the extra differentiated terms in u and v are the same in the two subtracted terms and simply cancel each other. Thus,
which is Lagrange's identity. Integrating from zero to one:
as was to be shown.