Abstract
Conserved quantities can help to understand and solve the equations of motion of various dynamical systems. Lax pairs are a useful tool to find conserved quantities of some dynamical systems. We give a motivated introduction to the idea of a Lax pair using examples such as the linear harmonic oscillator, Toda chain and Eulerian rigid body. A key step is to write the equations in ‘Lax form’, which makes it easy to read off conserved quantities. In Part II, these ideas will be extended from systems of particles to continuum systems of fields and also given a geometric interpretation in terms of curvature.
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Acknowledgements
We thank an anonymous referee for useful comments and references. This work was supported in part by the Infosys Foundation, J N Tata Trust and grants (MTR/2018/000734, CRG/2018/002040) from the Science and Engineering Research Board, Govt. of India.
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Govind Krishnaswami is on the faculty of the Chennai Mathematical Institute. He works on various problems in theoretical and mathematical physics.
T R Vishnu is a PhD student at the Chennai Mathematical Institute. He has been working on integrable systems.
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Krishnaswami, G.S., Vishnu, T.R. The Idea of a Lax Pair—Part I. Reson 25, 1705–1720 (2020). https://doi.org/10.1007/s12045-020-1091-y
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DOI: https://doi.org/10.1007/s12045-020-1091-y