Hamilton's equations

Deriving Hamilton's equations

Hamilton's equations have another advantage over Lagrange's equations: if a system has a symmetry, so that some coordinate ${\displaystyle q_{i}}$ does not occur in the Hamiltonian (i.e. a cyclic coordinate), the corresponding momentum coordinate ${\displaystyle p_{i}}$ is conserved along each trajectory, and that coordinate can be reduced to a constant in the other equations of the set. This effectively reduces the problem from n coordinates to (n − 1) coordinates: this is the basis of symplectic reduction in geometry. In the Lagrangian framework, the conservation of momentum also follows immediately, however all the generalized velocities ${\displaystyle {\dot {q}}_{i}}$ still occur in the Lagrangian, and a system of equations in n coordinates still has to be solved.