In mathematics, more precisely in functional analysis, an energetic space is, intuitively, a subspace of a given realHilbert 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.
The energetic inner product is defined as and the energetic norm is 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 . 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 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 . 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
Consider a string whose endpoints are fixed at two points on the real 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 conditions 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's 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 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 continuouspiecewise linear function in the energetic space, which gives the finite element method. The approximation can be computed by solving a system of linear equations. The energetic norm turns out to be the natural norm in which to measure the error between and, see Céa's lemma.
