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Wikipedia / 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.

Energetic space

with the inner product and the norm 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

together with the energetic inner product is a pre-Hilbert space. The energetic space is defined as the completion since any Cauchy sequence).

by

are sequences in Y in the energetic norm.

Energetic extension

admits an energetic extension

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

then

via the Riesz representation theorem

An example from physics

Consider a string on the real line (here viewed as a horizontal line). Let the vertical outer force density is a unit vector be the deflection 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

minimizing the potential energy will satisfy the differential equation

with boundary conditions

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

be the set of all Locally integrable function that are twice continuously differentiable, with the boundary conditions

given by the formula

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

is a symmetric linear operator.

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

is then the Sobolev space We see that the elastic energy of the string which motivated this study is

with itself.

of the string, one writes this problem in the form

  1. .

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.

, see Céa's lemma.

See also

References