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Sunday, November 28, 2010

Hamiltonian Formulism of mechanics (part 1)

Hamiltonian is H=T+V

Hamilton's Canonical Equations of motion:-

  • Co-ordinates cyclic in Lagrangian will also be cyclic in Hamiltonian.
  • Canonical transformations are characterized by the property that they leave the form of Hamilton's equations of motion invarient.
  • Lagrange's equation of motion are covarient w.r.t. point transformations (Qj=Qj(qj,t) and if we define Pj as,

           the Hamilton's canonical equation will also be covarient.
  • Consider the transformations
           where Qj and Pj are new set of co-ordinates.
  • For Qj and Pj to be canonical they should be able to be expressed in Hamiltonian form of equations of motion i.e.,

     where,  K=K(Q,P,t) and is substitute of Hamiltonian H of old set in new set of co-ordinates.
  • Qj and Pj to be canonical must also satisfy modified Hamilton's principle i.e.,

  • Using same principle for old set qj and pj

     where F is any function of phase space co-ordinates with continous second derivative.
  • Term ∂F/∂t in 1 contributes to the variation of the action integral only at end points and will therefore vanish if F is a function of (q,p,t) or (Q,P,t) or any mixture of phase space co-ordinates since they have zero variation at end points.
  • F is useful for specifying the exact form of anonical transformations only when half of the variables (except time) are from the old set and half from the new set.
  • F acts as bridge between two sets of canonical variables and is known as generating function of transformations.
In next post I'll discuss more about generating functions , lagrange and poisson brackett.

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