Transport of the Euler-Lagrange expression under a C^3 map: E of the substituted Lagrangian is the Jacobian-transpose multiple of E of the ambient Lagrangian

Let $n,m\ge 1$, let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ and $Q\subseteq {\mathbb{R}}^m$ be open and nonempty, and let $X:\mathrm{\Omega }\to Q$ be of class $C^3$ ($C^k$ Euclidean maps and diffeomorphisms). Let $J\subseteq \mathbb{R}$ be an open interval and let $L:J\times Q\times {\mathbb{R}}^m\to \mathbb{R}$ be of class $C^2$, with gradients $L_x$ and $L_w$ in the second and third slots (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$). Define

$\tilde{L}:J\times \mathrm{\Omega }\times {\mathbb{R}}^n⟶\mathbb{R},\tilde{L}(t,q,v):=L(t,X(q),DX(q)v)\mathrm{.}$

Then $\tilde{L}$ is of class $C^2$, and for every interval $J^{\mathrm{\prime }}\subseteq J$ with nonempty interior and every $C^2$ curve $q:J^{\mathrm{\prime }}\to \mathrm{\Omega }$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral) with $x:=X\circ q$ one has, writing ${\tilde{L}}_q$ and ${\tilde{L}}_v$ for the gradients of $\tilde{L}$ in the second and third slots,

${\tilde{L}}_q(t,q(t),q^{\mathrm{\prime }}(t))-\frac{d}{dt}\left({\tilde{L}}_v\right(t,q(t),q^{\mathrm{\prime }}(t)\left)\right)=DX(q(t){)}^{\mathsf{T}}(L_x(t,x(t),x^{\mathrm{\prime }}(t))-\frac{d}{dt}(L_w(t,x(t),x^{\mathrm{\prime }}(t))))$

for every $t\in J^{\mathrm{\prime }}$. In words: the Euler-Lagrange expression of the substituted Lagrangian is the transpose-Jacobian multiple of the Euler-Lagrange expression of the ambient Lagrangian. No rank or invertibility condition on $DX$ is assumed; the identity holds for every $C^3$ map $X$, square or not.

• Use with constrained coordinates. If a map $X$ is used to describe constrained configurations, this identity relates the substituted Euler-Lagrange expression to the ambient expression after multiplication by the explicit matrix $DX(q{)}^{\mathsf{T}}$. The identity is purely mathematical: no physical model, reaction law or constraint hypothesis enters it, and no rank or dimension condition is used beyond the $C^3$ regularity of $X$ and the $C^2$ regularity of $L$.

• Comparison with the invertible case. When $DX$ is invertible the vanishing of the two expressions is equivalent. In general, the identity always sends a zero ambient expression to a zero substituted expression. The converse requires that $DX(q{)}^{\mathsf{T}}u=0$ force $u=0$ for $u\in {\mathbb{R}}^m$; this item assumes no such property.