We define #~{kinetic energy} of a particle, mass ~m, velocity #~v:
~T _ #:= _ &hlf. ~m #~v &dot. #~v _ = _ &hlf. ~m #~v^2
Note that
deriv{~T} _ = _ &hlf. ~m fract{d ( #~v &dot. #~v ), d ~t} _ = _ &hlf. ~m ( deriv{#~v} &dot. #~v + #~v &dot. deriv{#~v} ) _ = _ ~m deriv{#~v} &dot. #~v _ = _ #~F &dot. #~v
Where #~F is the force acting on the particle. The quantity _ #~F &dot. #~v _ is called the #~{activity} of the force
Now by the fundamental theorem
int{#~F ( ~t ) &dot. #~v ( ~t ),~t_0,~t_1,d~t} _ = _ ~T ( ~t_1 ) - ~T ( ~t_0 )
The quantity _ ~W #:= ~{∫} #~F &dot. #~v d~t _ is called the ~#{work} done by the force in the given time interval.
int{#~F ( ~t ) &dot. #~v ( ~t ),~t_0,~t_1,d~t} _ = _ int{#~F ( ~t ) &dot. fract{d#~r, d~t} ,~t_0,~t_1,d~t} _ = _ int{#~F ( ~t ) &dot. ,~A,~B,d~#r}
Where _ ~A = ~#r ( ~t_0 ) _ and _ ~B = ~#r ( ~t_1 ) .
Note that work, as defined above, is a tangental line integral along the path taken by the particle, and, in general, is dependent on that path. If ~W is only dependent on the end-points, ~A and ~B, and not on the actual path between them, then the force field ~#F is said to be #~{conservative}.
For the rest of these notes we will assume that the force field is conservative.
If #~F is a conservative field, then the work done when we move a particle from any point ~P say, to a fixed reference point, ~O, along ~{any} path is
~V ( ~P ) _ #:= _ int{#~F ( ~t ) &dot. ,~P,~O,d~#r}
This is called the #~{potential energy} of the particle at point ~P in the field #~F.
A particle moves from ~A to ~B along a path &Gamma.. Then since the field is conservative
int{#~F ( ~t ) &dot. ,~A,~B,d~#r} _ _ = _ _ int{#~F ( ~t ) &dot. ,~A,~O,d~#r} _ + _ int{#~F ( ~t ) &dot. ,~O,~B,d~#r}
_
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = _ _ ~V ( ~A ) _ - _ ~V ( ~B )
But
int{#~F ( ~t ) &dot. ,~A,~B,d~#r} _ _ = _ _ ~T ( ~t_1 ) _ - _ ~T ( ~t_0 )
Now _ ~A = ~A ( ~t_0 ) , _ and _ ~B = ~B ( ~t_1 ), so we write
~V ( ~A ) _ = _ ~V ( ~t_0 ) , _ and _ ~V ( ~B ) _ = _ ~V ( ~t_1 )
Which means that
~V ( ~t_0 ) _ - _ ~V ( ~t_1 ) _ _ = _ _ ~T ( ~t_1 ) _ - _ ~T ( ~t_0 )
or~V ( ~t_0 ) _ + _ ~T ( ~t_0 ) _ _ = _ _ ~V ( ~t_1 ) _ + _ ~T ( ~t_1 )
i.e. _ ~V ( ~t ) + ~T ( ~t ) _ is constant. _ This is the #~{principal of conservation of energy}.