Newton's 2nd Law

F = \frac{d p}{d t}

where p : Momentum
If m is not a function of time:

F = m \frac{d u}{d t} = m a

Note

This equation is only valid in an inertial frame - see definition on Newton's 1st Law

The inertial mass of an object measures its reluctance to deviate from its preferred state of uniform motion in a straight line (in an inertial frame).
The 2nd law indicates that force must be a vector quantity, since it is the product of a scalar (mass) and a vector (acceleration). Acceleration is a vector because it is directly related with displacement , which is the prototype of all vectors.

Conservation of Energy

Conservation of energy (for conservative forces) is derived from this law by taking the scalar product with the velocity:

m \vec{u} \cdot \frac{d \vec{u}}{d t} = \frac{m}{2} \frac{d {\vec{u}}^{2}}{d t} = \vec{f} \cdot \vec{u}

\frac{d K}{d t} = \vec{F} \cdot \vec{u}

where $K = \frac{1}{2} m u^{2}$
K - Kinetic Energy represents the energy the object possesses by virtue of motion. The right side of the equation represents the rate at which the force does work on the object
Under action of force the object moves from P to Q:

Δ K = \int_{t_{1}}^{t_{2}} \vec{F} \cdot \vec{u} d t = \int_{P}^{Q} \vec{F} \cdot d r

If the force is conservative the line integral will be independent of the path and the force will satisfy Stokes theorem:

\vec{F} = - \nabla U

for some scalar field U. Also (independent of path):

\int_{P}^{Q} \nabla U = Δ U = U_{Q} - U_{P}

Therefore

Δ K = - Δ U

E = K + U = c o n s t a n t

Where
K : Kinetic Energy
U : Potential Energy
E : Energy - a constant of motion
(only for conservative forces)