Energy transport in a free Euler-Bernoulli beam in terms of Schrödinger's wave function

• URL: http://arxiv.org/abs/2411.04033v1
• Date: Wed, 06 Nov 2024 16:32:11 GMT
• Title: Energy transport in a free Euler-Bernoulli beam in terms of Schrödinger's wave function
• Authors: Serge N. Gavrilov, Anton M. Krivtsov, Ekaterina V. Shishkina,
• Abstract summary: The dynamics of a free infinite Euler-Bernoulli beam can be described by the Schr"odinger equation for a free particle and vice versa. For two corresponding solutions $u$ and $psi$ the mechanical energy density calculated for $u$ propagates in the beam exactly in the same way as the probability density calculated for $psi$.
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• Abstract: The Schr\"odinger equation is not frequently used in the framework of the classical mechanics, though historically this equation was derived as a simplified equation, which is equivalent to the classical Germain-Lagrange dynamic plate equation. The question concerning the exact meaning of this equivalence is still discussed in modern literature. In this note, we consider the one-dimensional case, where the Germain-Lagrange equation reduces to the Euler-Bernoulli equation, which is used in the classical theory of a beam. We establish a one-to-one correspondence between the set of all solutions (i.e., wave functions ${\psi}$) of the 1D time-dependent Schr\"odinger equation for a free particle with arbitrary complex initial data and the set of ordered pairs of quantities (the beam strain and the particle velocity), which characterize solutions $u$ of the beam equation with arbitrary real initial data. Thus, the dynamics of a free infinite Euler-Bernoulli beam can be described by the Schr\"odinger equation for a free particle and vice versa. Finally, we show that for two corresponding solutions $u$ and ${\psi}$ the mechanical energy density calculated for $u$ propagates in the beam exactly in the same way as the probability density calculated for ${\psi}$.

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