In Taylor, read sections 13.1 for today, and review for the second exam on Friday. On Wednesday we will have a review for the exam covering Chapters 7, 10, and 11.
Newton published the Principia in 1687. Lagrange published the Mecanique Analytique in 1788. Hamilton published a third formulation of mechanics in 1834. Though many people worked on parts of this formulation, it is not known as Hamiltonian mechanics. All three formulations of mechanics are equivalent.
Why do we need 3 different formulations? Each has strengths in solving certain types of problems. Newtonian mechanics is easiest for beginners and easily accomodate dissipative, frictional forces. Lagrangian mechanics is useful for solving problems with constraints where friction is not a major issue. Hamiltonian mechanics is useful in advanced topics about stability of systems (such as categorizing systems with no analytic solution) and make a connection to quantum mechanics.
Recall the definition of the Hamiltonian from Chapter 7
While the Lagrangian is expressed in terms of generalized coordinates and generalized velocities (the time derivatives of the generalized coordinates), the Hamiltonian is expressed in terms of generalized coordinates and generalized momenta
We call the space of the n generalized coordinates and n generalized velocities configuration space. The space of the n generalized coordinates and n generalized momenta is called phase space. In both instance, the space has 2n dimensions. The difference in switching from working in configuration space to phase space is subtle but important.
We can imagine the Lagrangian approach as yielding the evolution of a point in a configuration space of 2n dimensions. The Hamiltonian approach yields the evolution of a point in a phase space of 2n dimensions. There are important results that are derivable in phase space, such as Liouville's theorem of Sec. 13.7 of Taylor. You may see other results in a graduate level classical mechanics course.
When the generalized momenta are obvious, the Hamiltonian can be written down directly using H = T + U. When the generalized momenta are not obvious, it is better to use the definition given above for the Hamiltonian in terms of the Lagrangian. We will discuss some examples after deriving Hamilton's equations.
Hamilton's equations are the rules for obtaining differential equations for the motion of a system from Hamiltonian.
Lagrange's equation yields a second order differential equation for each coordinate. In contrast, Hamilton's equations yields two first order differential equations for each coordinate.