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

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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

  •   for all   in  
  •   for some constant   and all   in  

The energetic inner product is defined as

  for all   in  

and the energetic norm 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

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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

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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 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

  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 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.

See also

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References

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  • Zeidler, Eberhard (1995). Applied functional analysis: applications to mathematical physics. New York: Springer-Verlag. ISBN 0-387-94442-7.
  • Johnson, Claes (1987). Numerical solution of partial differential equations by the finite element method. Cambridge University Press. ISBN 0-521-34514-6.