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

PJ Olver, Applications of Lie groups to differential equations, Springer, New York 1993.

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

C.G. Torre, Covariant phase space formulation of parameterized field theories, J. Math. Phys. 33 (1992) 3802-3812 hep-th/9204055