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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
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