Legendre transformation

Optimization is hard to understand.

H=H(q,p;t)=T+V

Hamiltonian represents the total energy of the system, which is the sum of kinetic (T) and potential energy (V). q is generalized coordinates, p is generalized momenta, and t is time.

The Hamiltonian is calculated using the usual definition of H as the Legendre transformation of L.

・ is differentiation.

The momenta are calculated by differentiating the Lagrangian with respect to the generalized velocities.

H=0. It is numerical vectors.

One might think this means that imaginary numbers are just a mathematical game having nothing to do with the real world. From the viewpoint of positivist philosophy, however, one cannot determine what is real. All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons. So what is real and what is imaginary? Is the distinction just in our minds?
— Stephen Hawking


This is the Legendre transformation of L.

You see the bottom (A). y=0.