Hamiltonian Formulism of mechanics (part 1)

Hamiltonian is H=T+V
or,

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 (Q_j=Q_j(q_j,t) and if we define P_j as,
  

         

     

           the Hamilton's canonical equation will also be covarient.

• Consider the transformations
  
         Q_j=Q_j(p,q,t)
         P_j=P_j(p,q,t)
         where Q_j and P_j are new set of co-ordinates.
• For Q_j and P_j 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.
• Q_j and P_j to be canonical must also satisfy modified Hamilton's principle i.e.,
  
  
  
  
  
  
  
  
  
  
  
  
  
  
• Using same principle for old set q_j and p_j
  
                             (1)
  
  
   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.