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The earliest results concerning energy equipartition seem to be presented by Brodsky [2] and Lax and Phillips [12]. In the study of the abstract wave equation, Goldstein [9, 10] has applied the semigroup theory in order to obtain an equipartition theorem stating that the difference between the kinetic energy and the potential energy vanishes as time tends to infinity. By means of the Paley–Wiener theorem, Duffin [8] has shown that if a solution of the classical wave equation has compact support, then after a finite time the kinetic energy of the wave is constant and equals the potential energy.
The result established by Levine [13], using the Lagrange identity, represents a simplified proof that asymptotic equipartition occurs between the Cesàro means of the kinetic and potential energies. The asymptotic equipartition between the mean kinetic and strain energies in the context of linear elastodynamics was studied by Day [7...
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References
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Ghiba, ID. (2014). Partition of Energy. In: Hetnarski, R.B. (eds) Encyclopedia of Thermal Stresses. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2739-7_255
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