Energetic space

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In mathematics, more precisely in functional analysis, an energetic space is, intuitively, a subspace of a given real Hilbert space equipped with a new "energetic" inner product. The motivation for the name comes from physics, as in many physical problems the energy of a system can be expressed in terms of the energetic inner product. An example of this will be given later in the article.

Energetic space

Formally, consider a real Hilbert space with the inner product and the norm . Let be a linear subspace of and be a strongly monotone symmetric linear operator, that is, a linear operator satisfying

The energetic inner product is defined as

for all in

and the energetic norm{{safesubst:#invoke:anchor|main}} is

for all in

The set together with the energetic inner product is a pre-Hilbert space. The energetic space is defined as the completion of in the energetic norm. can be considered a subset of the original Hilbert space since any Cauchy sequence in the energetic norm is also Cauchy in the norm of (this follows from the strong monotonicity property of ).

The energetic inner product is extended from to by

where and are sequences in Y that converge to points in in the energetic norm.

Energetic extension

The operator admits an energetic extension

defined on with values in the dual space that is given by the formula

for all in

Here, denotes the duality bracket between and so actually denotes

If and are elements in the original subspace then

by the definition of the energetic inner product. If one views which is an element in as an element in the dual via the Riesz representation theorem, then will also be in the dual (by the strong monotonicity property of ). Via these identifications, it follows from the above formula that In different words, the original operator can be viewed as an operator and then is simply the function extension of from to

An example from physics

A string with fixed endpoints under the influence of a force pointing down.

Consider a string whose endpoints are fixed at two points on the real line (here viewed as a horizontal line). Let the vertical outer force density at each point on the string be , where is a unit vector pointing vertically and Let be the deflection of the string at the point under the influence of the force. Assuming that the deflection is small, the elastic energy of the string is

and the total potential energy of the string is

The deflection minimizing the potential energy will satisfy the differential equation

with boundary conditions

To study this equation, consider the space that is, the Lp space of all square integrable functions in respect to the Lebesgue measure. This space is Hilbert in respect to the inner product

with the norm being given by

Let be the set of all twice continuously differentiable functions with the boundary conditionss Then is a linear subspace of

Consider the operator given by the formula

so the deflection satisfies the equation Using integration by parts and the boundary conditions, one can see that

for any and in Therefore, is a symmetric linear operator.

is also strongly monotone, since, by the Friedrichs' inequality

for some

The energetic space in respect to the operator is then the Sobolev space We see that the elastic energy of the string which motivated this study is

so it is half of the energetic inner product of with itself.

To calculate the deflection minimizing the total potential energy of the string, one writes this problem in the form

for all in .

Next, one usually approximates by some , a function in a finite-dimensional subspace of the true solution space. For example, one might let be a continuous piecewise-linear function in the energetic space, which gives the finite element method. The approximation can be computed by solving a linear system of equations.

The energetic norm turns out to be the natural norm in which to measure the error between and , see Céa's lemma.

See also


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