Inverse problem for Lagrangian mechanics

In mathematics, the inverse problem for Lagrangian mechanics is problem of determining, given a system of ordinary differential equations, whether they could arise as the Euler-Lagrange equations for some Lagrangian function. There has been a great deal of activity in the study of this problem since the early 20th century. A notable advance in this field was a 1941 paper by the American mathematician Jesse Douglas, in which he provided necessary and sufficient conditions for the problem to have a solution; these conditions are now known as the Helmholtz conditions, after the German physicist Hermann von Helmholtz.

Background and statement of the problem

The usual set-up of Lagrangian mechanics on n-dimensional Euclidean space Rⁿ is as follows. Consider a differentiable path u [0, T] → Rⁿ. The action of the path u, denoted S(u), is given by

$S(u)=\int _{0}^{T}L(t,u(t),{\dot {u}}(t))\,\mathrm {d} t,$

where L is a function of time, position and velocity known as the Lagrangian. The principle of least action states that, given an initial state x₀ in Rⁿ, the trajectory that the system determined by L will actually follow must be a minimizer of the action functional S satisfying the initial condition u(0) = x₀. Furthermore, the critical points (and hence minimizers) of S must satisfy the Euler-Lagrange equations for S:

${\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {u}}^{i}}}-{\frac {\partial L}{\partial u^{i}}}{\mbox{ for each }}1\leq i\leq n,$

where the upper indices i denote the components of u = (u¹, ..., uⁿ). In the classical case

$T({\dot {u}})={\frac {1}{2}}m|{\dot {u}}|^{2},$

$V:[0,T]\times \mathbb {R} ^{n}\to \mathbb {R} ,$

$L(t,u,{\dot {u}})=T({\dot {u}})-V(t,u),$

the Euler-Lagrange equations are the second-order ordinary differential equations better known as Newton's laws of motion:

${\ddot {u}}=-\nabla _{u}V(t,u).$

The inverse problem is as follows: given a system of second-order ordinary differential equations

${\ddot {u}}^{i}=f^{i}(u^{j},{\dot {u}}^{j}),1\leq i,j\leq n,\quad (1)$

does there exist a Lagrangian L for which these ordinary differential equations (1) are the Euler-Lagrange equations?

Douglas' theorem and the Helmholtz conditions

To simplify the notation, let

$v^{i}={\dot {u}}^{i}$

and define a collection of n² functions Φ_jⁱ by

$\Phi _{j}^{i}={\frac {1}{2}}{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial f^{i}}{\partial v^{j}}}-{\frac {\partial f^{i}}{\partial u^{j}}}-{\frac {1}{4}}{\frac {\partial f^{i}}{\partial v^{k}}}{\frac {\partial f^{k}}{\partial v^{j}}}.$

Theorem. (Douglas 1941) There exists a Lagrangian L such that the equations (1) are its Euler-Lagrange equations if and only if there exists a non-singular symmetric matrix g with entries g_ij depending on both u and v satisfying the following three Helmholtz conditions:

$g\Phi =(g\Phi )^{\top },\quad (2)$

${\frac {\mathrm {d} g_{ij}}{\mathrm {d} t}}+{\frac {1}{2}}{\frac {\partial f^{k}}{v^{i}}}g_{kj}+{\frac {1}{2}}{\frac {\partial f^{k}}{\partial v^{j}}}g_{ki}=0{\mbox{ for }}1\leq i,j\leq n,\quad (3)$

${\frac {\partial g_{ij}}{\partial v^{k}}}-{\frac {\partial g_{ik}}{\partial v^{j}}}{\mbox{ for }}1\leq i\leq n.\quad (4)$

(The Einstein summation convention is in use for the repeated indices.)

References

• Douglas, Jesse (1941). "Solution of the inverse problem in the calculus of variations". Transactions of the American Mathematical Society. 50: 71–128.
• "The inverse problem for six-dimensional codimension two nilradical Lie algebras". Journal of Mathematical Physics. 47 (112901). 2006. {{cite journal}}: Unknown parameter |name= ignored (help)