Quantum-mechanics equations {Hamilton's equations} {Hamilton equations} relate particle positions and momenta. Potential-energy change plus kinetic-energy change equals zero, by conservation of energy. Energy conserves between kinetic-energy and potential-energy exchanges, so potential-energy change and kinetic-energy change are equal and opposite. Therefore, potential-energy change equals negative of kinetic-energy change. Potential energy depends on field position. Kinetic energy depends on momentum. Potential-energy change is energy gradient. Kinetic-energy change is momentum-change rate. Hamilton equation states that energy gradient, dH / dx, equals negative of momentum-change rate (force), dp / dt. Partial derivative of potential-energy function (Hamiltonian) with position is negative of derivative of momentum with time: DH / Dx = - dp / dt, where D is partial derivative, H is Hamiltonian potential energy, x is position, p is momentum, and t is time. Hamiltonians are wavefunctions that solve Hamilton equation.
Rearranging makes Hamiltonian potential-energy change dH equal negative of momentum change dp times position change dx divided by time change dt: dH = - dp * (dx / dt) = - m * dv * v = - m * v * dv, where v is velocity.
Rearranging makes negative of first derivative of Hamiltonian with momentum equal position derivative with time: - dH / dp = dx / dt = v. Velocity v = dx / dt equals negative of derivative of potential-energy change with momentum change dH / dp.
comparison
Hamilton's method substitutes two first-order differential equations for Lagrange's one second-order differential equation.
time
If particles are stationary, so positions do not depend on time, derivatives with time equal zero, and energy gradient equals zero, so energy is constant over all positions.
If particles move, so positions depend on time, use angle instead of position, and action instead of momentum, to find particle matter-wave frequencies and particle energies. Physical action is energy over time, so momentum is energy gradient over time. Angle indicates phase which indicates frequency, and angle varies directly with position, so position is angle gradient over time.
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Date Modified: 2022.0224