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We study the smooth path spaces of Euclidean spaces $\mathbb{R}^N$, asdiffeological spaces. We show that the tangent spaces of the free path space$\mathscr{P}$ are isomorphic to $\mathscr{P}$ itself, and that the tangentspaces of the space $\mathscr{P}_{\mathbf{p}, \mathbf{q}}$ of paths with fixedendpoints $\mathbf{p}$ and $\mathbf{q}$ are isomorphic to the smooth loop spaceof $\mathbb{R}^N$ based at the origin. We also define cotangents and gradientsof smooth maps from these path spaces, and then show that, in the case of theaction functional which arises in the calculus of variations, the gradient isprecisely the path formed out of the terms of the Euler-Lagrange equations. Weshow that solutions of the Euler-Lagrange equations correspond precisely to thezeros of the gradient, and also provide analogous interpretations for theconstrained Euler-Lagrange equations. This gives an illuminating geometricperspective on these equations. Finally, we illustrate the theory with severalconcrete examples from geometry, mechanics and machine learning.

Interpreting the Euler-Lagrange Equations as the Gradient of the Action Functional