Define the Hamiltonism and derive C anonical Equations of Motion
Hamiltonian approach is a reformulation of Lagrangian mechanics which consists in replacing the velocities with momenta . However, the interest of such reformulation is that Hamiltonian mechanics give a much more general approach to mechanics, as generalized coordinates and generalized momenta play much more symmetric (and thus less constrained) role than the pairs of Lagrangian mechanics. In particular, Hamiltonian approach can be generalized for quantum mechanics, but neither Newtonian nor Lagrangian mechanics can not.
To derive the formalism and the canonical equations let us consider the lagrangian of a free particle :
We see that the momentum is given by
Which motivates the definition for the general case :
We also define the Hamiltonian function as
Strictly speaking, from a mathematical point of view we are performing a Legendre transformation to pass from the coordinates to the coordinates .
Now minimizing the action
, and integrating by parts the term gives us a set of equations, called canonical equations of motion :
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