Classical Particle Mechanics and Vibration

Differential equations, whose solutions describe the motion of a particle or system of particles, are called equations of motion. In the mechanics of Isaac Newton (England, 1642-1727), the equations of motion include one of Newton s laws The total force acting on an object equals the mass of the object times the time rate of change of the object s velocity, which is the acceleration, that is, a = dv/dt. Letting F be the vector of force and m the mass of the object, Newton s relation 

Newton s formulation is not the only way in which classical equations of motion can be formulated. Lagrange (Joseph Louis Lagrange, France, 1736-1813), Hamilton (William Rowan Hamilton, Ireland, 1805-1865), and others developed different means, and it is the formulation of Hamilton that has proven the most useful framework for developing the mechanics of quantum systems. It is important to realize that Newtonian, Lagrangian, and Hamiltonian mechanics offer equivalent descriptions of classical systems. 

The f t) found with one formulation is the same as that found with another, even if the route to that function appears different. 

At this point, we shall restrict attention to the mechanics of systems of point-mass particles. This means that particles are taken to have no volume. The mass of each particle is treated as if enclosed in an infinitesimally small region of space, a point. And mostly, we will restrict our attention to conservative systems, which are those for which the potential energy has no explicit dependence on time (i.e., there are no potentials changing with time). These restrictions are not an aspect of the particular mechanical formulation they are used here as a convenience that is in keeping with the types of systems of most inune-diate interest in chemistry. 

The classical kinetic energy, T, of any particle is the square of the momentum divided by twice the particle s mass. Momentum is a vector, and so, in Cartesian space, there is an x-component, a y-component, and a z-component. Thus, 

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