Energetic space

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.

Formally, consider a real Hilbert space

be a strongly monotone symmetric linear operator, that is, a linear operator satisfying 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.

can be considered a subset of the original Hilbert space

(this follows from the strong monotonicity property of

are sequences in Y that converge to points in

denotes the duality bracket between

is simply the function extension of

Consider a string whose endpoints are fixed at two points

Let the vertical outer force density at each point

is a unit vector pointing vertically and

be the deflection of the string at the point

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

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

given by the formula so the deflection satisfies the equation

Using integration by parts and the boundary conditions, one can see that for any

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

We see that the elastic energy of the string which motivated this study is so it is half of the energetic inner product of

minimizing the total potential energy

of the string, one writes this problem in the form Next, one usually approximates

, a function in a finite-dimensional subspace of the true solution space.

be a continuous piecewise linear function in the energetic space, which gives the finite element method.

can be computed by solving a system of linear equations.