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On the analytic representation of Newtonian systems

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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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Authors and Affiliations

  1. Department of Physics, Visva-Bharati University, Santiniketan,  731 235, India

    Benoy Talukdar

  2. Department of Physics, Bidhannagar College, EB-2, Sector-1, Salt Lake, Kolkata,  700 064, India

    Supriya Chatterjee

  3. Department of Physics, Kazi Nazrul University,  Asansol,  713 303, India

    Sekh Golam Ali

Authors
  1. Benoy Talukdar
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  2. Supriya Chatterjee
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  3. Sekh Golam Ali
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Correspondence to Sekh Golam Ali.

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Talukdar, B., Chatterjee, S. & Ali, S.G. On the analytic representation of Newtonian systems. Pramana - J Phys 94, 141 (2020). https://doi.org/10.1007/s12043-020-02010-y

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  • DOI: https://doi.org/10.1007/s12043-020-02010-y

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