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
where q is an n-dimensional column vector and U an n-dimensionaler
row vector, the action
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
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