In classical mechanics, the energy of a system is expressed
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In classical mechanics, the energy of a system is expressed in terms of a function called the Hamiltonian. When the ener
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In classical mechanics, the energy of a system is expressed in terms of a function called the Hamiltonian. When the energy is independent of time, the Hamiltonian depends only on the positions, qi , and the momenta, pi , of the particles in the system, that is, H D H.q1; ; qn; p1; ; pn/. There is also another function, called the Lagrangian, that depends on the positions qi and the velocities qPi , that is, L D L.q1; ; qn; qP1; ; qPn/, such that the Hamiltonian is a Legendre transformation of the Lagrangian with respect to the velocity variables: H.q1; ; qn; p1; ; pn/ D X i pi qPi ? L.q1; ; qn; qP1; ; qPn/:(a) What variables are conjugate in this Legendre transformation? What partial derivatives of L are implicitly determined by it? (b) In the absence of external forces, the principle of least action requires that @L @qi D Ppi . By taking the differential of H and using the result of part (a), show that @H @qi D?Ppi and @H @pi D Pqi . These are known as Hamilton
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