Principle of Balance of Mechanical Energy
Jason Har, Kumar K. Tamma · 2012
Thus far, we have described the fundamental aspects of classical mechanics as evident from much of the open literature, namely, the three branches, Newtonian, Lagrangian, and Hamiltonian mechanics. Summarizing, the principal conclusions are that Newtonian mechanics reflects the statement of the principle of balance of linear momentum, and it required the description of field variables as vectors with the Newtonian dynamical system represented as second-order in time, the presence of the k-number of constraints leading an N-body system to a dynamical system with 3N – k degrees of freedom, and an inertial reference frame. The Newtonian dynamical system involves 3N-number of Cartesian variables. In contrast, although there is no new physics that has been brought about in comparison to Newtonian mechanics, Lagrangian mechanics introduced the concept of generalized co-ordinates in that termed as the configuration space, eliminated the constraints, and provided a dynamical system with the ndof number of generalized coordinates (note that the number of degrees of freedom is ndof = 3N – k). The representation of the equation of motion is also of second-order in time, but involving the scalar quantity, namely, the Lagrangian. On the other hand, Hamiltonian mechanics introduced the concept of phase space with 2ndof-number of canonical variables, also did not have any constraints, but inherits the representation of the equations of motion as a system of first-order in time via a scalar function, namely, the Hamiltonian. It also has not brought forth any new physics in contrast to the two previous branches of mechanics. As described subsequently in later chapters of this textbook, they are all equivalent to each other and each of the three branches of mechanics inherits certain pros and cons.