Abstract: "Symmetry plays a prominent role in classical mechanics. Associated with each symmetry is a redundant degree of freedom in the system. Via Routh reduction, a regular Lagrangian system with symmetry in the configuration space descends to the so-called shape space and its dynamics is then governed by the reduced Euler-Lagrange equations. When Routh reduction is applied to a time-independent Lagrangian system in the framework of least action in Jacobi form, a geodesic principle can be established to describe the shape dynamics of the system in a reparametrization-invariant manner. These observations by the pioneers of mechanics the 18th and 19th centuries can be carried over, mutatis mutandis, to the setting of general relativity. Interesting geometry underlying the constraint dynamics of canonical gravity may be investigated from this vintage point.