Newton's laws of motion can be recast into more powerful and general mathematical formalisms. These enable Newton's laws to be applied in the most appropriate coordinate system for the problem at hand. The Euler–Lagrange equations are introduced and used to show how the invariance of the equations under translations and rotations results in the laws of conservation of momentum and angular momentum. Time invariance results in the law of conservation of energy. An important application of the Euler–Lagrange equations is the characterisation the normal modes of oscillation of mechanical systems. Hamilton’s equations, Poisson brackets, the Hamilton–Jacobi equations and action-angle variables are introduced in order to solve increasingly complex problems, including the mathematics necessary for the description of quantum phenomena.