In this work we propose two mutually Legendre-dual integral equations as the first principles of physics. The first equation expresses the time integral of energy as the sum of the line integral of momentum over space and the line integral of angular momentum over rotation angles; the second is its dual, expressing the energy integral of time as the sum of the line integral of position over momentum and the line integral of rotation angles over angular momentum. These two equations reveal the three fundamental duality pairs: energy--time, momentum--position, and angular momentum--angle. Starting from these postulates, we systematically derive, via variational principles, Legendre transformations, and field-theoretic generalizations, all major equations of classical mechanics (Newtonian, Lagrangian, Hamiltonian), electromagnetism (Maxwell equations), special relativity, general relativity, thermodynamics, and statistical mechanics. Moreover, by multiplying the two postulates by wave functions and applying canonical quantization, we naturally obtain the Schr\"odinger equation and its dual in the energy representation, where the three duality pairs become the three fundamental commutation relations. The duality principle is enforced throughout: every derived equation has a strict dual counterpart. This work demonstrates that the two integral equations constitute a unified foundation for all of physics, from classical to quantum, revealing a deep intrinsic symmetry of physical laws.
S. B. Liu (Wed,) studied this question.