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The total mechanical energy of a system of particles, defined in this short section, is a scalar physical quantity. Note that the condition that total mechanical energy be conserved is much more restrictive than that for the conservation of momentum. The latter conservation law simply requires that there be no net external force on the system while energy conservation requires that all the forces involved are conservative. Indeed, energy conservation is very difficult to achieve in macroscopic many-body systems. On the other hand, most microscopic systems such as collisions between elementary particles are conservative.

The Hamiltonian function introduced here will not be invoked very often in the next few chapters. As you will see, however, it comes to play an important role in quantum mechanics (Chapter 13). The function was introduced to the study of dynamics by (1805 - 1865).