This is a contributed entry
Let us asssume from the start that the field's force
is irrotational, i.e.
, that is,
.
In another words, the field's force is conservative if and only if it is irrotational. So the conseravation of mechanical energy
is a consequence of that theorem. Once one imposes
, then one is proving that the necessary condition is:
. Another consequence about the theorem is that the “work” of the field's force is independent of the path described by the particle in its motion. That is, if
and
are two different paths, described by the particle, and joininig its initial and end position
on the time interval
, then the line integrals
must be equal and hence the work
of the field's force, as the particle describes a closed path, must be zero, i.e.
.
The relation
between the force,
, acting on a particle, and the potential energy,
of that particle is:
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(1.1) |
Take the total time derivative of
, giving
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(1.2) |
The kinetic energy of a particle is expressed as
, where
is the mass
of the particle, and
is the magnitude of the particle's velocity. Recall that by Newton's second law,
, where
is the velocity vector. Consider, next, the quantity
, where
is the position vector
of the particle. Expanding
in terms of Newton's second law, it is seen that
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|
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It is assumed that the potential energy is a function
of time and space i.e.
. The time derivative of the potential energy can be expanded through the chain rule as
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(1.3) |
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(1.4) |
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(1.5) |
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(1.6) | |
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(1.7) |
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(1.8) |
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(1.9) |
As of this snapshot date, this entry was owned by bci1.