On the analytic representation of Newtonian systems
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On the analytic representation of Newtonian systems BENOY TALUKDAR1 , SUPRIYA CHATTERJEE2 and SEKH GOLAM ALI3
,∗
1 Department
of Physics, Visva-Bharati University, Santiniketan 731 235, India of Physics, Bidhannagar College, EB-2, Sector-1, Salt Lake, Kolkata 700 064, India 3 Department of Physics, Kazi Nazrul University, Asansol 713 303, India ∗ Corresponding author. E-mail: [email protected] 2 Department
MS received 27 January 2020; revised 13 June 2020; accepted 3 July 2020 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. Keywords. Calculus of variation; inverse problem; Lagrangians; linear and nonlinear systems. PACS Nos 45.05.+x; 02.30.Zz; 02.03.Hq
1. Introduction In point mechanics the term ‘analytic representation’ refers to the description of Newtonian systems by means of Lagrangians [1]. Understandably, to find the analytic representation of a mechanical system one begins with the equation of motion and then constructs a Lagrangian function by using a strict mathematical procedure discovered by Helmholtz [2,3]. In the calculus of variation, this is the so-called inverse variational problem which is more complicated than the usual direct problem where one first assigns a Lagrangian function using phenomenological consideration and then computes the equation of motion using the Euler–Lagrange equation [4]. However, there are two types of analytic representations, namely, the direct and indirect ones. We can introduce the basic concepts of direct and indirect analytic representations by using a system of two uncoupled harmonic oscillators with equations of motion q(t) ¨ + ω2 q(t) = 0
(1)
and y¨ (t) + ω2 y(t) = 0.
(2) 0123456789().: V,-vol
It is straightforward to verify that the system of eqs (1) and (2) can be analytically represented either by the Lagrangian Ld =
1 2 ω2 2 (q˙ (t) + y˙ 2 (t)) − (q (t) + y 2 (t)) 2 2
(3)
or by the Lagrangian L i = q(t) ˙ y˙ (t) − ω2 q(t)y(t).
(4)
Here overdots denote differentiat
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