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S10l. Diffeomorphism invariant classical mechanics
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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 at 
        http://www.mat.univie.ac.at/~neum/physik-faq.txt

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