From Force to Energy: Evolution of Classical Mechanics

      Newtonian mechanics is mathematically fairly straightforward, and can be applied to a wide variety of problems. It took the Apollo astronauts to the moon and the voyager spacecraft to the far reaches of the solar system. It is not a unique formulation of mechanics, however; other formulations are possible. Here we will look at two common alternative formulations of classical mechanics: Lagrangian mechanics and Hamiltonian mechanics.

        Its original prescription rested on two principles. First, we should try to express the state of the mechanical system using the minimum representation possible and which reflects the fact that the physics of the problem is coordinate-invariant. Second, a mechanical system tries to optimize its action from one split second to the next; often this corresponds to minimizing its total energy as it evolves from one state to the next. These notes are intended as an elementary introduction into these ideas and the basic prescription of Lagrangian and Hamiltonian mechanics.

         It is important to understand that all of these formulations of mechanics equivalent. In principle, any of them could be used to solve any problem in classical mechanics. The reason they’re important is that in some problems one of the alternative formulations of mechanics may lead to equations that are much easier to solve than the equations that arise from Newtonian mechanics. Unlike Newtonian mechanics, neither Lagrangian nor Hamiltonian mechanics requires the concept of force; instead, these systems are expressed in terms of energy. The equations of Lagrangian and Hamiltonian mechanics are also expressed in the language of partial differential equations.

Lagrangian Mechanics

       The first alternative to Newtonian mechanics we will look at is Lagrangian mechanics. Using Lagrangian mechanics instead of Newtonian mechanics is sometimes advantageous in certain problems, where the equations of Newtonian mechanics would be quite difficult to solve. In Lagrangian mechanics, we begin by defining a quantity called the Lagrangian (L), which is defined as the difference between the kinetic energy K and the potential energy U:

       Since the kinetic energy is a function of velocity v and potential energy will typically be a function of position x, the Lagrangian will (in one dimension) be a function of both x and v: L(x,v). The motion of a particle is then found by solving Lagrange’s equation; in one dimension it is;

Hamiltonian Mechanics

        The second formulation we will look at is Hamiltonian mechanics. In this system, in place of the Lagrangian we define a quantity called the Hamiltonian, to which Hamilton’s equations of motion are applied. While Lagrange’s equation describes the motion of a particle as a single second-order differential equation, Hamilton’s equations describe the motion as a coupled system of two first-order differential equations.

        One of the advantages of Hamiltonian mechanics is that it is similar in form to quantum mechanics, the theory that describes the motion of particles at very tiny (subatomic) distance scales. An understanding of Hamiltonian mechanics provides a good introduction to the mathematics of quantum mechanics. The Hamiltonian H is defined to be the sum of the kinetic and potential energies:

        Here the Hamiltonian should be expressed as a function of position x and momentum p (rather than x and v, as in the Lagrangian), so that H = H(x, p). This means that the kinetic energy should be written as K = p^2/2m, rather than K = ½mv^2. Hamilton’s equations in one dimension have the elegant nearly-symmetrical form;