# How is the Routhian of classical mechanics defined?

The Hamiltonian is a function on the cotangent bundle to a configuration manifold $H:T^*M\rightarrow \mathbb R$. The Lagrangian is a function on the tangent bundle to the configuration manifold $\mathscr L:TM\rightarrow \mathbb R$. What is the Routhian function $R$ defined on?

My guess is $TM\bigoplus T^*M$?

• Isn't that a bit big? I would have thought we want a subbundle of $TM$ corresponding to whatever coordinates are being treated in a Lagrangian way, added to the subbundle of $T^*M$ dual to the complement of this.
– user10851
Sep 5 '15 at 0:59
• I found what you're looking for, please see my answer physics.stackexchange.com/questions/265248/…
– user122132
Jul 9 '16 at 17:28