Action principle, Lagrangian mechanics, Hamiltonian mechanics, and conservation laws when assuming Aristotelian mechanics $F=mv$

Define a physical system when Aristotelian mechanics $F=mv$ instead of Newtonian mechanics $F=ma$.

Then we could have action $I=\int L(q,t)dx$ rather than $\int L(q',q,t)dx$.

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  Is there an action principle?

  
  


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  Will the formula $I=\int p d q$ still hold?

  
  



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  What will be the Hamiltonian and conservation laws look like in this case?