On the Inverse Problem of the Calculus of Variations for Systems of Second-Order Ordinary Differential Equations

Abstract

The inverse problem of the calculus of variations is the problem of characterizing those systems of second-order ordinary differential equations of the form

$${\ddot x^i} = {f^i}(t,{x^k},{\dot x^k}), i = 1,2,...,n$$

for which one can find a non-singular symmetric matrix ${g_{ij}}(t,{x^k},{\dot x^k})$ such that the equivalent system ${g_{ij}}({\ddot x^j} - {f^j}) = 0$ (summation convention in force) is the set of Euler-Lagrange equations of some Lagrangian function $L(t,{x^k},{\dot x^k})$. The necessary and sufficient conditions for the existence of such a multiplier matrix g _ij , known as the Helmholtz conditions, may be specified purely analytically as follows. Replace ${\dot x^k}$ by v ^k, for clarity. In terms of the given second-order system, define the differential operator

$$\Gamma = \frac{\partial }{{\partial t}} + {\upsilon ^i}\frac{\partial }{{\partial {x^i}}} + {f^i}\frac{\partial }{{\partial {\upsilon ^i}}}$$

where f ⁱ = f ⁱ(t, x ^k, v ^k), and the matrices Γ ⁱ _j (t, x ^k, v ^k) and Φ ⁱ _j (t, x ^k, v ^k) by

$$\Gamma _j^i = - \tfrac{1}{2}\frac{{\partial {f^i}}}{{\partial {\upsilon ^j}}},\Phi _j^i = - \frac{{\partial {f^i}}}{{\partial {x^j}}} - \Gamma _k^i\Gamma _j^k - \Gamma (\Gamma _j^i)$$

.