[Paper Review] Using an old method of Jacobi to derive Lagrangians: a nonlinear dynamical system with variable coefficients

M. C. Nucci, K. M. Tamizhmani|ArXiv.org|Jul 17, 2008

Complex Systems and Time Series Analysis15 references20 citations

TL;DR

This paper demonstrates that Jacobi's Last Multiplier method provides a powerful, systematic approach to deriving Lagrangians for second-order nonlinear ODEs with variable coefficients, such as $\ddot{x} + b(x)\dot{x}^2 + c(x)x = 0$. Unlike the lengthy calculations required by Musielak et al., Jacobi's method yields not only the same particular Lagrangian but also an infinite family of distinct Lagrangians through the use of first integrals and multiplier multiplication, revealing new solutions and highlighting deep connections with Lie and Noether symmetries.

ABSTRACT

We present a method devised by Jacobi to derive Lagrangians of any second-order differential equation: it consists in finding a Jacobi Last Multiplier. We illustrate the easiness and the power of Jacobi's method by applying it to the same equation studied by Musielak et al. with their own method [Musielak ZE, Roy D and Swift LD. Method to derive Lagrangian and Hamiltonian for a nonlinear dynamical system with variable coefficients. Chaos, Solitons & Fractals, 2008;58:894-902]. While they were able to find one particular Lagrangian after lengthy calculations, Jacobi Last Multiplier method yields two different Lagrangians (and many others), of which one is that found by Musielak et al, and the other(s) is(are) quite new.

Motivation & Objective

• To demonstrate the effectiveness and simplicity of Jacobi's Last Multiplier method in deriving Lagrangians for second-order nonlinear ODEs with variable coefficients.
• To compare Jacobi's method with the more computationally intensive approach of Musielak et al., who derived only one Lagrangian after lengthy calculations.
• To show that Jacobi's method yields an infinite family of Lagrangians, including the one found by Musielak et al. and new, previously unknown Lagrangians.
• To emphasize the deep connection between Jacobi Last Multipliers, Lie symmetries, and Noether symmetries in the context of Lagrangian mechanics.

Proposed method

• The method uses the Jacobi Last Multiplier (JLM) $ M $, defined as the inverse of the determinant of a matrix formed from the vector field and $ n-1 $ independent first integrals of the associated Lagrange system.
• The JLM satisfies the linear PDE $ \sum_{i=1}^{n} \frac{\partial(M a_i)}{\partial x_i} = 0 $, and every solution of this PDE is a valid JLM.
• Given a known JLM $ M_1 $, multiplying it by any first integral $ I $ of the equation yields another JLM $ M_2 = M_1 I $, which generates a new Lagrangian.
• The general form of the Lagrangian derived from a JLM $ M $ is $ L = \frac{1}{2} M \dot{x}^2 + f_1(t,x)\dot{x} + f_2(t,x) $, where $ f_1 $ and $ f_2 $ satisfy a specific compatibility condition involving $ M $ and the equation's coefficients.
• The method leverages known Lie point symmetries—particularly $ \partial_t $, which is a Noether symmetry—for the equation to generate first integrals, which are then used to generate new JLMs and Lagrangians.
• The process is iterative: successive multiplication of JLMs by powers of the first integral $ I_1 $ generates an infinite sequence of distinct JLMs and corresponding Lagrangians.

Experimental results

Research questions

• RQ1 Can Jacobi's Last Multiplier method efficiently derive Lagrangians for second-order nonlinear ODEs with variable coefficients, such as $ \ddot{x} + b(x)\dot{x}^2 + c(x)x = 0 $?
• RQ2 How does the Jacobi Last Multiplier method compare in complexity and generality to the method of Musielak et al.?
• RQ3 Does the Jacobi Last Multiplier method yield not just one but an infinite family of Lagrangians for such equations?
• RQ4 What is the relationship between Jacobi Last Multipliers, Lie symmetries, and Noether symmetries in the context of Lagrangian mechanics?
• RQ5 Can new, previously unknown Lagrangians be systematically derived using the JLM method by exploiting first integrals?

Key findings

• The Jacobi Last Multiplier for the equation $ \ddot{x} + b(x)\dot{x}^2 + c(x)x = 0 $ is $ M_1 = e^{2P_b(x)} $, where $ P_b(x) = \int b(x)\,dx $, which directly yields a general Lagrangian form.
• The method produces not just one but an infinite number of distinct Lagrangians, including the one previously found by Musielak et al. via lengthy calculations.
• A new Lagrangian $ L_2 $ is derived by multiplying the first JLM $ M_1 $ by the first integral $ I_1 = \frac{1}{2}e^{2P_b}\dot{x}^2 + \int e^{2P_b}c(x)x\,dx $, yielding $ M_2 = M_1 I_1 $, which leads to a second valid Lagrangian.
• Further Lagrangians are generated by multiplying $ M_1 $ by higher powers of $ I_1 $, such as $ M_3 = M_1 I_1^2 $, which produces a third Lagrangian $ L_3 $ with a quartic dependence on $ \dot{x} $.
• All derived Lagrangians admit the time-translation symmetry $ \partial_t $ as a Noether symmetry, and their corresponding first integrals are powers of the energy-type integral $ I_1 $.
• The method confirms that the existence of a first integral allows the generation of new JLMs and thus new Lagrangians through simple algebraic multiplication, demonstrating the method's scalability and power.