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 THE EASIEST PATHS

THE EASIEST PATHS

 A mathematician named Joseph Langrage formed a very economical form of Newtonian dynamics. Suppose, you are given a set of coordinates xi and kinetic and potential energies of a particle in terms of these coordinates. Then, you can find the equation of the path of the particle without much effort. There is one more elegance: the law works for any coordinate, eg. polar, cylindrical or Cartesian, provided you have energies in terms of the coordinates. We won’t derive the equation here, which is quite complicated. But, you can refer to any standard text on mechanics. We just give the equation. Let L = T - U; T is kinetic energy and U is potential energy. If xi¢ represents dxi/dt, time derivative xi, then, for that particle,

if the potential is conservative; else, some more complicative calculations are required. L is the called Langragian.

For example, consider the pendulum motion. We use polar coordinates.

Let us consider the problem of length measurement on a curved surface. Suppose a surface is curved in a particular coordinate system, say, xi (= x1, x2, x3, … xn). They are not powers; they are just superscripts. We apologize for using such notation; but, they are necessary when we use Einstein summation convention (see below).

We have kinetic energy (T) at any q = ½mv2.

v = drq/dt = r dq/dt = r . So, T = ½mv22.

Potential energy at q is U = mgl(1-cosq).

L = T – U = ½ml2 - mgl(1-cosq)

Therefore, dL/dq¢ = ml2 (d/dq¢ considers q to be constant, q has nothing to do with dq/dt as we will see shortly).

dL/dq = -mglsinq

So, the equation is              ml2(d/dt) + mglsinq = 0

or,           d2q/dt2 + (g/l) sinq = 0.

You can easily derive the equations of motion of planet using this.

Thus, this is a very handy calculus and leads to deeper insight into the mechanics. Now, why did we not differentiate cosq with respect to d/ ? Here comes the notion of phase space.

A particle, for example, moving in a plane (say x-y plane), needs four quantities to specify its state at a particular instant. Two are x and y and the rest two are m(dx/dt) and m(dy/dt). So, we need four orthogonal coordinates x, y, dx/dt and dy/dt to completely describe the path. A particle may have the same momentum at various coordinates. So, momentum or velocity is independent of position. Phase space is whole space of 2n quantities, n position coordinates and n momentum coordinates, in which a particle follows a trajectory.

 THE MOST ELEGANT EQUATION THE HYPERCOMPLEXITY THE GALILEAN NON-INVARIANTS THE EMPIRE OF EUCLID THE DISTANCE MEASURER

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