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Summer Research Fellowship Programme of India's Science Academies

Can equations of motion be of order higher than two?

Abhijeet Singh

Department of Physical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Sector 81, S.A.S. Nagar, Manauli, P.O. 140306, Punjab, India

M. Sami (mentor)

Professor and Director, Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi, 110025, India

Abstract

The differential equation governing the time evolution of the position a point particle, known

as Newton's second law, is second order in time. Even after a lot of progress in our

understanding of the physical world, no physical system has ever been observed whose

dynamical variable evolves in time, in accordance with a higher-order differential equation.

This is not merely a coincidence, and there is a deeper, more fundamental reason to this.

Ostrogradsky's theorem is a strong no-go theorem that imposes a strong restriction on the

types of physical systems that can exist. The theorem states that the Hamiltonian of a system

that does depend on more than two initial-value pieces of information is usually not bounded from

below, and this is detrimental to the existence of the system. The aim of this project is to

understand Ostrogradsky's result in its spirit. The theory of classical mechanics is first learnt

in its Lagrangian and Hamiltonian formulations. Thereafter, Ostrogradsky's theorem is studied

and applied to general physical system which can possibly have a higher order equation of

motion.

Keywords: Lagrangian, Hamiltonian, Cannonical Co-ordinates, Equations of Motion

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