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Conserved quantities, i.e. constants of motion, are critical forcharacterizing many dynamical systems in science and engineering. Thesequantities are related to underlying symmetries and they provide fundamentalknowledge about physical laws, describe the evolution of the system, and enablesystem reduction. In this work, we formulate a data-driven architecture fordiscovering conserved quantities based on Koopman theory. The Koopman operatorhas emerged as a principled linear embedding of nonlinear dynamics, and itseigenfunctions establish intrinsic coordinates along which the dynamics behavelinearly. Interestingly, eigenfunctions of the Koopman operator associated withvanishing eigenvalues correspond to conserved quantities of the underlyingsystem. In this paper, we show that these invariants may be identified withdata-driven regression and power series expansions, based on the infinitesimalgenerator of the Koopman operator. We further establish a connection betweenthe Koopman framework, conserved quantities, and the Lie-Poisson bracket. Thisdata-driven method for discovering conserved quantities is demonstrated on thethree-dimensional rigid body equations, where we simultaneously discover thetotal energy and angular momentum and use these intrinsic coordinates todevelop a model predictive controller to track a given reference value.

Discovering conservation laws from data for control