Now we can invoke the same initial conditions that we used in the previous

example and this leads to the equation of motion where the position x

scales at the sinusoid frequency times the time.

Where new is a characteristic frequency associated with the system.

In this case the characteristic frequency

scales along with the constant k and the mass of the system m.

Like the Lagrangian formulation that we learned another formulation called

the Hamiltonian formulation of classical mechanics describes the equation of

motion, albeit using a different quantity, H, called the Hamiltonian.

Now the Hamiltonian is defined as the sum of the kinetic and the potential energy.

Now, for the second example that we studied the Hamiltonian of the particle,

it turns out that it would be independent of time.

Now, from these two examples,

we can infer some key conclusions about classical mechanics.

Classical mechanics predicts that the particle motion

is completely deterministic.

That is, the conditions of a particle at any given instant of time

will chart out its future trajectory.

The Lagrangian formulation teaches us that the particle traverses along a path

such that its action S is an extremum, that is, a minimum.

Now we discussed two examples.

The first example taught us that a free particle one that does not have

any influence of any external potential, will maintain a constant velocity,

as proposed by Newton's First Law of Motion.

Second example told us that the motion of a particle

in a time independent potential field, such as the harmonic potential well,

would be governed by a constraint that the total energy is a constant.