Inverse Variational Problem for Nonlinear Dynamical Systems |
| B.A. Khana, S. Chatterjeeb, S.G. Alic, B. Talukdard
aDepartment of Physics, Krishnath College, Berhampore, Murshidabad 742101, India bDepartment of Physics, Bidhannagar College, EB-2, Sector-1, Salt Lake, Kolkata 700064, India cDepartment of Physics, Kazi Nazrul University, Asansol 713303, India dDepartment of Physics, Visva-Bharati University, Santiniketan 731235, India |
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| Two different approaches to solving the inverse problem of the calculus of variation for nonlinear equations are introduced. The first approach is based on an integral representation of the Lagrangian function, while the second one relies on the generalization of Lagrangian symmetry. As an application of the first approach, we initially provide some useful remarks on the Lagrangians of the modified Emden-type equation, and then construct Lagrangian functions for (i) a cubic-quintic Duffing oscillator, (ii) Liénard-type oscillator and (iii) Mathews-Lakshmanan oscillator. Using the second approach, we obtain analytic (Lagrangian) representations for the three velocity-dependent equations, namely, (iv) Abraham-Lorentz oscillator, (v) Helmholtz oscillator and (vi) Van der Pol oscillator. For each of the systems in (i)-(vi) we find the Jacobi integral and thereby provide a method for obtaining the Hamiltonian function. |
DOI:10.12693/APhysPolA.141.64 topics: Lagrangians, Jacobi integrals, Hamiltonians, nonlinear differential equations |