This two-way procedure is indeed one of the most elegant and powerful applications of geometry to physics. Hamiltonian Mechanics 5.1 The Hamiltonian Recall that L= L(q,q,t˙ ), and p ... •As an example, consider a particle moving in three dimensions, described by spherical polar coordinates (r,θ,φ). (An example of this situation is discussed in Section 5 of this article.) Then Furthermore, since much of this book is based on problem solving, this chapter probably won’t be the most rewarding one, because there is rarely any beneflt from using a Hamiltonian instead of a Lagrangian to solve a standard mechanics problem. Hamiltonian Mechanics: A Simple Example Consider the Lagrangian that we looked at before: L = 1 2 mx˙2 − 1 2 mω 2x2 (20) The conjugate momentum (18) is: px = ∂L ∂x˙ = mx˙ (21) Note that as usual, we treat x and ˙x as independent of one another. Lagrange equations consist of a set of k second-order differential equations describing the variables (qk) being the "time" derivatives of the other k variables (qk). 2 Review of Newtonian Mechanics Remark 2.1 In Mechanics one examines the laws that govern the motion of bodies of matter. how useful the Hamiltonian formalism is. Hamiltonian Mechanics The Hamiltonian Formulation of Mechanics is equivalent to Newton's Laws and to the Lagrangian Formulation. The context of this article is more about what Hamiltonian mechanics means in classical mechanics, although I will also give some insights about Hamiltonian mechanics and its significance to other areas in physics. Under motion one understands a change of place as a function of time. Lagrangian (L) T − V = L L = 1 m(x˙ 2 c + L2 φ˙2 cos 2 φ)+ 1 Icφ˙2 − mg L sin φ 2 4 2 2 Equations of Motion The motion happens under the influence of forces, that are assumed to be known. The corresponding that the laws of classical mechanics, once formulated in their Hamiltonian form, can be repaired by suitably introducing h into its equations, thereby yielding quantum mechanics correctly. Example: Falling Stick (Continued) 3 Forces: Conservative [gravity] + Nonconservative [normal]. The constraint force (normal force) does no work. Indeed, many of the examples and problems In this way, Dirac was able to show how quantum mechanics naturally supersedes classical mechanics while reproducing the successes of classical mechanics. [2] 2.1 Point Mechanics and Newtons First Law As we change xc and φ, no virtual work (no dis­ placement in direction of force). Like the Lagrangian Formulation, one can use generalized coordinates with the Hamiltonian, however, the Hamiltonian is written in terms of coordinates and their conjugate momenta rather than the coordinates and their time derivatives as with the Lagrangian. The Hamiltonian formulation of mechanics describes a system in terms of generalised co motion of the system.