Question

Hamilton's variational principle

Answer 1

Hamilton's variational principle states that the integral J = ∫ t 2 t 1 L d t J=∫t1t2Ldt taken along a path of the possible motion of a physical system is an extremum when evaluated along the path of motion that is actually taken. L = T - V is the Lagrangian of the system, or the difference between its kinetic and potential energy. In other words, out of the myriad ways in which a system could change its configuration during a time interval t 2 − t 1 t2−t1, the actual motion that does occur is the one that either maximizes or minimizes the preceding integral. This statement can be expressed mathematically as δ J = δ ∫ t 2 t 1 L d t = 0 ( 10.1.1 ) δJ=δ∫t1t2Ldt=0(10.1.1) in which [itex]\delta[/tex] is an operation that represents a varitation of any particular system parameter by an infinitesimal amount away from the value taken by the parameter when the integral of equation 10.1.1. is an extremum.