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velocity potential for a rate formulation

I have a question making me no sleep. In the elastic theory we have a separable Lagrangian L = T(v) + W(u) since we can write the internal elastic energy as a function of displacements. What happens if we use a rate form for the strain and the stress? Can we write the potential energy in terms of just velocities? If that is the case the stationary path of the Lagrangian reduces to:

 d/dt (dT/dv - dW/dv)  = 0

that is the trajectories of the system conserves the quantity L = T + W and at each time we look for the solution of the time-indipendent optimization problem of finding the velocities minimizing the Lagrangian. In this way we are just calculating a velocity field and then the body is just advected and represent the domain of integration. The solution will thus be a free velocity, which can be modified by means of Lagrange multipliers prescribing some velocity constraints (boundary conditions). What is wrong in my thinking?

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