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

Abstract

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