Impulse and change in momentum What exactly is Impulse? Is impulse only defined for "significant large force acting for a small interval of time" or can it have more "generalized" meaning?
My perspective about impulse is that
" It's a change (big OR small) in the momentum of a body in a short period of time due to a resultant force.." is this correct or does this definition need some changes ?
Also,
If impulse,J = ∆P (change in momentum) then this would violate the above stated definition. ( I am not sure whether J = ∆P is true for all physical scenarios )
But then again,
Suppose I punch a wall (don't mind me:) now obviously I have applied a force F on the wall for a small time "dt"..
So according to the definition of impulse in my book and wikipedia, Impulse on this wall would be
J = Fdt right ?
But it should also be equal to ∆P ?
And here ∆P is clearly zero as the wall has not acquired any velocity in the time "dt"
So I would need a clear explanation on what impulse is and how exactly is it different from change in momentum for a physical scenario (and whether J = ∆P is always true or not)
 A: The impulse of a force $\mathbf F(t)$ between times $t_1$ and $t_2$ is defined by
$$\mathbf J=\int_{t_1}^{t_2} \mathbf F\ dt$$
The time interval, ($t_2-t_1$), need not be short, though 'impulse' is especially useful for a large force, $\mathbf F$, acting for a short time interval, because we can then neglect much smaller forces acting on the same body at the same time. [For example if I kick a ball lying on the ground, we can reasonably neglect the frictional force acting on the ball during the short time my foot is in contact with the ball.] So the body's change in momentum is
$$\Delta \mathbf p = \int_{t_1}^{t_2} \frac{d\mathbf p}{dt} dt = \int_{t_1}^{t_2}\mathbf F\ dt = \mathbf J$$
In the case you mention of punching the wall it is not permissible to neglect all other forces on the wall except the punch. This is because the wall is (or should be) firmly attached to the ground, so when you punch it a force acts on it from the ground, in the opposite direction to the force of your punch.
You can still use the notion of impulse to determine a body's change in momentum if more than one significant force acts on a body simultaneously. We then have
$$\Delta{\mathbf p}=\int_{t_1}^{t_2} \frac{d\mathbf p}{dt} dt=\int_{t_1}^{t_2}\sum \mathbf F\ dt=\sum\int_{t_1}^{t_2} \mathbf F dt=\sum \mathbf J,$$
in which $\sum \mathbf F$ is the vector sum of the forces acting on the body at time $t$, and $\sum \mathbf J$ is the vector sum of the impulses over the given time interval of all the forces acting on the body.
