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Energy Conservation

Though the word ‘energy itself cannot be defined exactly, we know various forms of energy, like kinetic energy, electro-magnetic energy, etc. which can be precisely quantitatively defined. We also know that one form of energy can be transformed into another form and that the total energy in any process remains conserved. The law of conservation of energy becomes a useful tool in solving several problems in mechanics where the exact dynamics is not always known; we shall this method, the energy method.

The study of motion implies solving Newton’s equation of motion for the object (in non-relativistic classical mechanics). The algorithm is simple: establish the equation of motion, F = ma and integrate it to find the velocity and position of the object with time. However, in practice, the above programme meets two major obstacles: (i) the exact nature of forces is not always known, and (ii) the integration is not always analytically possible. In several such cases, conservation laws (of momentum, energy, etc.) help to reveal all the important features of motion without knowing the exact nature of forces and/or actually solving the equation of motion.

In fact, it turns out that conservation laws in mechanics themselves express the first integrals of equation of motion. For example, let us consider an isolated system; an isolated system is one which is (assumed to be) not subjected to any external force. The equation of motion for such a system is,

Integrating the above equation with respect to time, we get p = constant, which is the statement of conservation of momentum for an isolated system. Thus, given an isolated system, we known that its detailed internal dynamics.

Moreover, we have seen that these conservation laws are based on the fundamental properties of space and time. Hence, their significance is much more fundamental than just being mechanics and play their role in all other fields of physics.