Symmetry Principles and Conservation Laws in Atomic and ...

GENERAL ARTICLEacceleration. Newton's second law contains the heartof this stimulus{response relation, expressed as a di®erentialequation. It is interesting that laws of classicalmechanics can be built alternatively on the basis of an`integral principle', namely the `principle of variation',discussed in the next section.3. Principle of VariationThe connection between symmetry and conservation lawsbecomes even more transparent in the alternative formalismof classical mechanics, namely the Lagrangian/Hamiltonian formulation. It is instructive to ¯rst seethat this alternative formalism is based not on the linearresponse relationship embodied in the Newtonian principleof causality, but in a completely di®erent approach,namely the `principle of variation'.It is interesting thatlaws of classicalmechanics can bebuilt alternativelyon the basis of an‘integral principle’,namely the‘principle ofvariation’,Newtonian mechanics o®ers an accurate description ofclassical motion by accounting for the same by the `causeand e®ect' relationship. An alternative and equivalentdescription makes it redundant to invoke such a causaldescription. This alternative description dispenses theNewtonian notion of the 'cause-e®ect' relationship, andinstead of it invokes a variational principle, namely, thatthe `action integral' is an extremum. Those who are usedto thinking in terms of the Newtonian formulation alonewould ¯nd it strange that one gets equivalent descriptionof classical mechanics without invoking the notion offorce at all!Let us ¯rst state the principle of extremum action. Webegin on common ground with the Newtonian formulation,namely that the position q and velocity : q specifythe mechanical state of a system. A well-de¯ned functionof q and : q would also then specify the mechanicalstate of the system. What is known as the Lagrangian ofa system L(q; : q) is just that; it is named after its originatorLagrange (1736{1813). Furthermore, in a homoge-RESONANCE September 2010835