Inverse problem for Lagrangian mechanics

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

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

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 and a final state x₁ in Rⁿ, the trajectory that the system determined by L will actually follow must be a minimizer of the action functional S satisfying the boundary conditions u(0) = x₀, u(T) = x₁. Furthermore, the critical points (and hence minimizers) of S must satisfy the Euler–Lagrange equations for S:

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {u}}^{i}}}-{\frac {\partial L}{\partial u^{i}}}=0\quad {\text{for }}1\leq i\leq n,}$

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

In the classical case

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

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

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

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

${\displaystyle m{\ddot {u}}^{i}=-{\frac {\partial V(t,u)}{\partial u^{i}}}\quad {\text{for }}1\leq i\leq n,}$

${\displaystyle {\mbox{i.e. }}m{\ddot {u}}=-\nabla _{u}V(t,u).}$

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

${\displaystyle {\ddot {u}}^{i}=f^{i}(u^{j},{\dot {u}}^{j})\quad {\text{for }}1\leq i,j\leq n,\quad {\mbox{(E)}}}$

that holds for times 0 ≤ t ≤ T, does there exist a Lagrangian L : [0, T] × Rⁿ × Rⁿ → R for which these ordinary differential equations (E) are the Euler–Lagrange equations? In general, this problem is posed not on Euclidean space Rⁿ, but on an n-dimensional manifold M, and the Lagrangian is a function L : [0, T] × TM → R, where TM denotes the tangent bundle of M.

To simplify the notation, let

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

and define a collection of n² functions Φ_jⁱ by

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

Theorem. (Douglas 1941) There exists a Lagrangian L : [0, T] × TM → R such that the equations (E) 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:

${\displaystyle g\Phi =(g\Phi )^{\top },\quad {\mbox{(H1)}}}$

${\displaystyle {\frac {\mathrm {d} g_{ij}}{\mathrm {d} t}}+{\frac {1}{2}}{\frac {\partial f^{k}}{\partial v^{i}}}g_{kj}+{\frac {1}{2}}{\frac {\partial f^{k}}{\partial v^{j}}}g_{ki}=0{\mbox{ for }}1\leq i,j\leq n,\quad {\mbox{(H2)}}}$

${\displaystyle {\frac {\partial g_{ij}}{\partial v^{k}}}={\frac {\partial g_{ik}}{\partial v^{j}}}{\mbox{ for }}1\leq i,j,k\leq n.\quad {\mbox{(H3)}}}$

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

• Douglas, Jesse (1941). "Solution of the inverse problem in the calculus of variations". Transactions of the American Mathematical Society. 50 (1): 71–128. doi:10.2307/1989912. ISSN 0002-9947. JSTOR 1989912.
• Rawashdeh, M., & Thompson, G. (2006). "The inverse problem for six-dimensional codimension two nilradical Lie algebras". Journal of Mathematical Physics. 47 (11): 112901. Bibcode:2006JMP....47k2901R. doi:10.1063/1.2378620. ISSN 0022-2488.{{cite journal}}: CS1 maint: multiple names: authors list (link)