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Average-preserving symmetries and energy equipartition in linear Hamiltonian systems

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Abstract

This paper analyzes energy equipartition in linear Hamiltonian systems in a deterministic setting. We consider the group of phase space symmetries of a stable linear Hamiltonian system, and characterize the subgroup of symmetries whose elements preserve the time averages of quadratic functions along the trajectories of the system. As a corollary, we show that if the system has simple eigenvalues, then every symmetry preserves averages of quadratic functions. As an application of our results to linear undamped lumped-parameter systems, we provide a novel proof of the virial theorem, which states that the total energy is equipartitioned on the average between the kinetic energy and the potential energy. We also show that under the assumption of distinct natural frequencies, the time-averaged energies of two identical substructures of a linear undamped structure are equal. Examples are provided to illustrate the results.

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Authors and Affiliations

  1. Department of Aerospace Engineering, Indian Institute of Technology, Powai, Mumbai, 400076, India

    Sanjay P. Bhat

  2. Department of Aerospace Engineering, The University of Michigan, Ann Arbor, MI, 48109-2140, USA

    Dennis S. Bernstein

Authors
  1. Sanjay P. Bhat
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  2. Dennis S. Bernstein
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Corresponding author

Correspondence to Sanjay P. Bhat.

Additional information

A preliminary version of the results of this paper appeared in the proceedings of the IEEE Conference on Decision and Control, 2004.

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Bhat, S.P., Bernstein, D.S. Average-preserving symmetries and energy equipartition in linear Hamiltonian systems. Math. Control Signals Syst. 21, 127–146 (2009). https://doi.org/10.1007/s00498-009-0039-2

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  • DOI: https://doi.org/10.1007/s00498-009-0039-2

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