Hamiltonian as total energy for natural systems?

[CrazyNeutrino]

Could someone please prove that the H=T+V for systems where the transformation from Cartesian to Generalized coordinates is time independent. I have read through the proofs in Taylor's classical mechanics and Goldstein's but do not understand them.

 

[Valerie Park]

Thanks in advance.The proof of this statement relies on the fact that the kinetic energy and potential energy can be expressed in terms of the generalized coordinates and their derivatives. In general, the kinetic energy is given by:T = ½m i θi·2 + ½m ij θi θj·2 + ½m ijk θi θj θk·2 + ...where m i , m ij , m ijk , ... are the components of the mass matrix and θi , θj , θk , ... are the generalized coordinates. Similarly, the potential energy is given by:V = V(q1, q2, ..., qn)where q1, q2, ..., qn are the generalized coordinates. Now, since the transformation from Cartesian to Generalized coordinates is time independent, the kinetic energy and potential energy can be expressed as functions of the generalized coordinates only, without any explicit dependence on time. This means that the total energy H of the system is given by = T + V where T and V are the kinetic and potential energies expressed in terms of the generalized coordinates, as above. Thus, we have shown that the total energy of a system where the transformation from Cartesian to Generalized coordinates is time independent is given by H = T + V.