Conservation laws are important because constrain how physical systems change. Hidden in Newton's laws are the conservation of energy, and the conservation of angular momentum. Both of these principles are useful in understanding some of our previous results and are fundamental in physics.
Now the gravitational potential energy is the energy that a body has which can subsequently be used to accelerate the body to a larger magnitude of velocity. For example, if I hold a ball at arms length at rest, and let the ball drop to the Earth, the ball will speed up before hitting the Earth. This potential energy, as I was holding the ball at rest, is given by
Now here's the deal: the gravitational potential energy of the ball at rest in my extended arm, is equal to the maximum kinetic energy that the ball can have just before it reaches the ground. As the ball falls, H decreases. Thus the gravitational energy decreases. Where does it go? Well, the speed of the ball increases. Thus the kinetic energy of the ball increases from the equation for kinetic energy above. Gravitational potential energy is being converted into kinetic energy. This is how energy is conserved.
It is also why you slow down and speed up as you travel up and down in a roller coaster.
Is it consistent with planets in elliptical orbits around the sun speeding up near the the perihelion and slowing down near the aphelion? and Kepler's second law?
Energy conservation is fundamental. Physics can describe to us only how energy in the Universe transforms from one form to another.
Objects executing motion around a point possess a quantity called ANGULAR MOMENTUM. This is an important physical quantity because all experimental evidence indicates that angular momentum is rigorously conserved in our Universe. It can be transferred, but it cannot be created or destroyed. For the simple case of a small mass executing uniform circular motion around a much larger mass (so that we can neglect the effect of the center of mass) the amount of angular momentum takes a simple form. As the adjacent figure illustrates the magnitude of the angular momentum in this case is
Notice how this applies to elliptical planetary orbits. For a planet of mass m in an elliptical orbit, conservation of angular momentum implies that as the object moves closer to the sun it speeds up. That is, if r decreases then v must increase to maintain the same L. Thus near perihelion it speeds up and near aphelion it slows down. Both energy conservation and angular momentum conservation are important to planetary orbits.
Note that the reason planets orbit the sun and do not fall into the sun, is because they have angular momentum and have had this angular momentum from the time they were formed. The planets could have gained this angular momentum before or after their formation, but it is believed that they were likely formed from gas material that was already orbiting the Sun. More on this later.