Scheme representation of the motion of an object (the falling apple) following various variational approaches.

Alternatives to the Newtonian approach [New87] for describing the motion of objects or systems exist. The Lagrangian approach is one of those, based on energy to describe the motion of systems instead of forces.

[New87] Newton I., Philosophiae Naturalis Principia Mathematica, 1687.

Lagrange approach [Lag88] may appear as an overcomplicating mathematical tool compared to Newton's one for simple systems. However, Lagrange approach allows the use of generalized coordinates compared to Newton's approach which can greatly simplify the resolution of a problem depending on its symmetries or on the geometry.

[Lag88] Lagrange J.-L., Méchanique Analitique, 1788.

Hamilton's functional approach [Ham35] is the formulation of the principle of stationary action based on the Lagrangian approach.
From the Physics point of view, its advantage compared to Lagrange approach is the fact that it allows to identify the "natural" boundary conditions during calculation in elastodynamics.

[Ham35] Hamilton W. R., Second essay on a general method in dynamics, Philosophical Transaction of the Royal Society Part I, 1835, pp. 95–144.

Hamilton's functional approach is not the only functional formalism used in elastodynamics. While Hamilton’s functional only depends on the displacement field (and its derivatives), The Reissner’s functional [Rei50] depends on two variable fields: the displacement one and the stress one.

[Rei50] Reissner E., On a variational theorem in elasticity, J. Math. Phys. 29, 1950, pp. 90–95■. The Reissner functional is sometimes call the Hellinger-Prange-Reissner functional to refer also to prior works (in German only) by E. Hellinger and G. Prange.