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Tuesday, September 27, 2011

What are Legendre transformations

The transition from Lagrangian to Hamiltonian formulism corresponds to changing the variables in our mechanical functions from ( q , q . , t ) to ( q , p , t )

where , p is related to q and q . by the equation

p i = L ( q j , q ˙ j , t ) q ˙ i

The procedures for switching variables in this manner is provided by the legendre transformations.

Consider a function of only two variables f(x,y), so that differential of f has the form


where , u = df dx and v = df dy                      (1)

To change the basis of description from x,y to a new set of variables u,y , so that differential quantities are expressed in terms of differential du and dy. Let g be the function of u and y defined by the equation


differential of g is given as




which is exactly in the desired form. The quantities x and v are now functions of variables u and y given by the relations

x = g u , v = g y

which are exactly converse of equation 1

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