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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, |
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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. |
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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). |
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We have kinetic energy (T) at any
q
= ½mv2.
v = drq/dt
= r dq/dt
= r
q¢.
So, T = ½mv2q¢2.
Potential energy at
q
is U = mgl(1-cosq).
L = T – U = ½ml2q¢
- mgl(1-cosq)
Therefore,
dL/dq¢
= ml2q¢
(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(dq¢/dt)
+ mglsinq
= 0
or, d2q/dt2
+ (g/l) sinq = 0. |
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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/q¢
? 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. |
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