Ivancevic_Applied-Diff-Geom

24 Applied Differential Geometry: A Modern Introductionand (q 1 , t 1 ) fixed. 29The total energy function called Hamiltonian, denoted by H, is obtainedby performing a Legendre transformation on the Lagrangian. 30 TheHamiltonian is the basis for an alternative formulation of classical mechanicsknown as Hamiltonian mechanics (see below).In 1948, R.P. Feynman invented the path–integral formulation extendingthe principle of least action to quantum mechanics for electrons and pho-29 More generally, a Lagrangian L[ϕ i ] of a dynamical system is a function of the dynamicalvariables ϕ i (x) and concisely describes the equations of motion of the systemin coordinates x i , (i = 1, ..., n). The equations of motion are obtained by means of anaction principle, written asδSδϕ i= 0,where the action is a functionalZS[ϕ i ] = L[ϕ i (s)] d n x,(d n x = dx 1 ...dx n ).The equations of motion obtained by means of the functional derivative are identical tothe usual Euler–Lagrange equations. Dynamical system whose equations of motion areobtainable by means of an action principle on a suitably chosen Lagrangian are knownas Lagrangian dynamical systems. Examples of Lagrangian dynamical systems rangefrom the (classical version of the) Standard Model, to Newton’s equations, to purelymathematical problems such as geodesic equations and the Plateau’s problem.The Lagrangian mechanics is important not just for its broad applications, but alsofor its role in advancing deep understanding of physics. Although Lagrange sought todescribe classical mechanics, the action principle that is used to derive the Lagrange’sequation is now recognized to be deeply tied to quantum mechanics: physical action andquantum–mechanical phase (waves) are related via Planck’s constant, and the Principleof stationary action can be understood in terms of constructive interference of wavefunctions. The same principle, and the Lagrangian formalism, are tied closely to NoetherTheorem, which relates physical conserved quantities to continuous symmetries of aphysical system; and Lagrangian mechanics and Noether’s Theorem together yield anatural formalism for first quantization by including commutators between certain termsof the Lagrange’s equations of motion for a physical system.More specifically, in field theory, occasionally a distinction is made between the LagrangianL, of which the action is the time integral S = R Ldt and the Lagrangiandensity L, which one integrates over all space–time to get the 4D action:ZS[ϕ i ] = L[ϕ i (x)] d 4 x.The Lagrangian is then the spatial integral of the Lagrangian density.30 The Hamiltonian is the Legendre transform of the Lagrangian:H (q, p, t) = X i˙q i p i − L(q, ˙q, t).