We define #~{linear momentum} of a particle, mass ~m, position #~r:
#~M _ #:= _ ~m deriv{#~r} _ = _ ~m #~v
where _ #~v _ = _ deriv{#~r} _ = _ d#~r ./ d~t
If a force #~F is acting on the particle then we have:
#~F _ = _ ~m deriv2{#~r} _ = _ deriv{#~M}
So if _ #~F _ == _ 0 , _ then _ #~M _ is constant - the #~{principal of conservation of momentum}.
Now _ _ _ #~F ( ~t ) _ = _ deriv{#~M} ( ~t )
which, by the fundamental law of calculus, means
int{#~F ( ~t ),~t_0,~t_1,d~t} _ = _ #~M ( ~t_1 ) - #~M ( ~t_1 )
The quantity _ ~#I #:= ~{∫} #~F d~t _ is called the ~#{impulse} delivered by the force in the given time interval. So impulse is change in momentum, and is useful to describe situations where large forces act for short times.
The #~{angular momentum} #~H_O, about the origin O, of a particle with momenum #~M, and position #~r, is defined:
#~H_O ( ~t ) _ = _ #~r # #~M
The #~{moment} #~G_O ( ~t ), about the origin O, of the force #~F at position #~r is defined:
#~G_O ( ~t ) _ = _ #~r # #~F