Diffeomorphism invariant classical mechanics

In mechanics, time is a point in a 1-dimensional manifold, and diffeomorphisms are just smooth reparameterizations of the time.

For any Lagrangian of the form L(q,qdot,t):=U(q(t))qdot(t), where q is an n-dimensional column vector and U an n-dimensionaler row vector, the action S=integral L(q,qdot,t)dt is diffeomorphism invariant. As a consequence, the Noether energy (the formal Hamiltonian constructed in the transition from a Lagrangian to a Hamiltonian formulation) vanishes identically and has no physical content.
For one can bring an arbitrary Hamiltonian system xdot=H_p(p,x), pdot=-H_x(p,x), where H is the physically relevant energy, into the above form by putting q^T=(x^T,p^T,s), U(q)=(p^T,0^T,-H(p,x)).

For a careful discussion see Section 4.3 of

Those who can read German can find more in the Section on ''Diffeomorphismeninvariante klassische Mechanik'' in my German Theoretische-Physik-FAQ.

For diffeomorphism invariant reformulations of arbitrary field theories, see

Arnold Neumaier (Arnold.Neumaier@univie.ac.at)
A theoretical physics FAQ