Varying without varying: reparameterizations, diffeomorphisms, general covariance, Lie derivatives, and all that

The standard way of deriving Euler-Lagrange (EL) equations given a point particle action is to vary the trajectory and set first variation zero. However, if (i) reparameterisation invariant, (ii) generally covariant, I show that one may derive EL by suitably "nullifying" through judicious coordinate transformation. net result this curve remains fixed, while all other geometrical objects in undergo change, precisely Lie derivatives along vector field. This, then, most direct transparent elucidate connection between general covariance, diffeomorphism invariance, derivatives, without referring covariant derivative. highlight geometric underpinnings generality above ideas applying them simplest field theories, keeping discussion at level easily accessible advanced undergraduates. As non-trivial applications these ideas, Geodesic Deviation Equation using order diffeomorphisms, demonstrate how they can canonical metric stress-energy tensors theories.