Routh Symmetry in the Chaplygin’s Rolling Ball

The Routh integral in the symmetric Chaplygin’s rolling ball has been regarded as a mysterious conservation law due to its interesting form of $\sqrt{I_1I_3 + m ⟨Is, s⟩}\Omega_3$. In this paper, a new form of the Routh integral is proposed as a Noether’s pairing form of a conservation law. An explicit symmetry vector for the Routh integral is proved to associate the conserved quantity with the invariance of the Lagrangian function under the rollingly constrained nonholonomic variation. Then, the form of the Routh symmetry vector is discussed for its origin as the linear combination of the configurational vectors.