Hamiltonian formulism of mechanics (part 2)

-Following four forms of generator are possible : F₁(q,Q,t) , F₂(q,P,t) , F₃(p,Q,t) , F₄(p,P,t)

  Legendre transforms are
    g(u,y)=f(x,y)-ux
    where u=(∂ F/∂ x)
 (i) F₁=F₁(q,Q,t) putting it in equation 1 and finding derivative of F w.r.t. we get
          p_j = ∂F₁(q,Q,t)/∂q_j

          P_j = - ∂F₁(q,Q,t)/∂Q_j

          K=H+∂F₁(q,Q,t)/∂t
(ii) putting u = -P_j , x = Q_j , y = q_j , g = F₂(q,P,t) , f=F₁ and using Legender transforms we get
          F₂(q_j,P_j,t) = F₁(q_j,Q_j,t)+ΣP_jQ_j
         Putting this in 1 and solving we get
          p_j = ∂F₂(q_j,P_j,t)/∂q_j

          Q_j =  ∂F₂(q_j,P_j,t)/∂P_j

          K=H+∂F₂(q_j,P_j,t)/∂t

 (iii) F₃ = F₃(Q,p,t) is the third form
            Connect it to the first form using legender transforms
            Since p_j = ∂F₁(q_j,P_j,t)/∂q_j and u=∂f/∂x
             this implies that u=p_j , x=q_j , y=Q_j , g=F₃ , f=F₁
              Thus,  F₃ = F₁(Q,q,t)-Σp_jq_j
              or, F₁(Q,q,t)=F₃+Σp_jq_j
              putting these in equation 1
            q_j = -∂F₃(p_j,Q_j,t)/∂p_j
            P_j = -∂F₃(p_j,Q_j,t)/∂Q_j

             K=H+∂F₃(p_j,Q_j,t)/∂t
(iv)    Fourth form is F₄(p,P,t) , connecting F₄ by F₁ through legender transforms and solving we get
         q_j = -∂F₄(p_j,P_j,t)/∂p_j

         Q_j = ∂F₄(p_j,P_j,t)/∂P_j
         K=H+∂F₄(p_j,P_j,t)/∂t

Manage yourself for competetions (think about these thoughts)

1. First of all have a positive attitude it pays and whatever you are thinking of doing do it now i mean start doing whatever you are planning to do or have planned to do so far.


2. Try and manage your stress levels , if stressed most easy part of your syllabus would seem hard and difficult to learn. So maintain your cool.

3. Do not panic if you failed once think about your mistakes, identify your weak link and start preparing again with only success in your mind. Remember that failure means delay not the defeat.

4. Never let yourself be stopped by the loss of one opportunity , think , discover another one and start working for it.

5. Organise yourself so that you can make best out of your limited time.

7. Last but not the least try being fair and genuine with your competitors.

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.