On the analytic representation of Newtonian systems

Benoy Talukdar; Supriya Chatterjee; Sekh Golam Ali

doi:10.1007/s12043-020-02010-y

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arXiv · 2006.08597

On the analytic representation of Newtonian systems

Benoy Talukdar·Supriya Chatterjee·Sekh Golam Ali

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Abstract

We show that the theory of self-adjoint differential equations can be used to provide a satisfactory solution of the inverse variational problem in classical mechanics. A Newtonian equation when transformed to the self-adjoint form allows one to find an appropriate Lagrangian representation (direct analytic representation) for it. On the other hand, the same Newtonian equation in conjunction with its adjoint provides a basis to construct a different Lagrangian representation (indirect analytic representation) for the system. We obtain the time-dependent Lagrangian of the damped Harmonic oscillator from the self-adjoint form of the equation of motion and at the same time identify the adjoint of the equation with the so called Bateman image equation with a view to construct a time-independent indirect Lagrangian representation. We provide a number of case studies to demonstrate the usefulness of the approach derived by us. We also present similar results for a number of nonlinear differential equations by using an integral representation of the Lagrangian function and make some useful comments.

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physics.class-ph math-ph math.MP nlin.SI

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Benoy Talukdar, Supriya Chatterjee, Sekh Golam Ali. 2020-06-12. On the analytic representation of Newtonian systems. https://doi.org/10.1007/s12043-020-02010-y

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