Derive lagrange's equation from hamilton's principle
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Answer:
The contributions of Galileo to the development of classical mechanics are underplayed
in many textbooks. In fact they were crucial. Not only did he formulate the law of
inertia (which became Newton’s 1st law), he also recognized that in general the effect of
the external world on an object is to provide the object with acceleration. This was given
precise formulation in Newton’s 2nd law, but it has wider consequences. For example, if
it is acceleration that really counts, then changing from one reference frame to another
moving at constant velocity relative to the first will not change anything important,
because it does not alter the acceleration. This is called Galilean relativity. (The
equations one uses to make such a change of reference frame had to be revised by
Einstein’s relativity, but the basic point remains valid.)
So if the external world only affects accelerations, then what we need to know initially
about a particle is only its position and velocity. These specify the state of the particle.
Our task in mechanics is to describe changes in that state. That is what we do when we
invoke the 2nd law and find the acceleration.
But there are many situations in which use of the 2nd law is clumsy at best. Consider a
particle sliding without friction on a vertically curved track, subject to gravity. At any
point in its motion there are two forces on it, gravity and the normal force exerted by
the track. Of course gravity has a simple formula (at least near the earth’s surface), but
the normal force is complicated. Its direction changes because the track is curved, and
its magnitude depends on the particle’s speed. For these reasons the actual acceleration
of the particle is a quite complicated function, continually changing both magnitude
and direction. We know the path followed by the particle (assuming it doesn’t leave the
track at any point) but we would be hard pressed to say at what time it reaches a
particular location.