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Lagrangian Mechanics Chapter 1 Lagrangian Mechanics Our introduction to Quantum Mechanics will be based on its correspondence to Classical Mechanics. For this purpose we will review the relevant concepts of Classical Mechanics. An important concept is that the equations of motion of Classical Mechanics can be based on a variational principle, namely, that along a path describing classical motion the action integral assumes a minimal value (Hamiltonian Principle of Least Action). 1.1 Basics of Variational Calculus The derivation of the Principle of Least Action requires the tools of the calculus of variation which we will provide now. Definition: A functional S[ ] is a map M S[]: R ; = ~q(t); ~q :[t0; t1] R R ; ~q(t) differentiable (1.1) F! F f ⊂ ! g from a space of vector-valued functions ~q(t) onto the real numbers. ~q(t) is called the trajec- tory of a systemF of M degrees of freedom described by the configurational coordinates ~q(t) = (q1(t); q2(t); : : : qM (t)). In case of N classical particles holds M = 3N, i.e., there are 3N configurational coordinates, namely, the position coordinates of the particles in any kind of coordianate system, often in the Cartesian coordinate system. It is important to note at the outset that for the description of a d classical system it will be necessary to provide information ~q(t) as well as dt ~q(t). The latter is the velocity vector of the system. Definition: A functional S[ ] is differentiable, if for any ~q(t) and δ~q(t) where 2 F 2 F d = δ~q(t); δ~q(t) ; δ~q(t) < , δ~q(t) < , t; t [t0; t1] R (1.2) F f 2 F j j jdt j 8 2 ⊂ g a functional δS[ ; ] exists with the properties · · (i) S[~q(t) + δ~q(t)] = S[~q(t)] + δS[~q(t); δ~q(t)] + O(2) (ii) δS[~q(t); δ~q(t)] is linear in δ~q(t). (1.3) δS[ ; ] is called the differential of S[ ]. The linearity property above implies · · δS[~q(t); α1 δ~q1(t) + α2 δ~q2(t)] = α1 δS[~q(t); δ~q1(t)] + α2 δS[~q(t); δ~q2(t)] : (1.4) 1 2 Lagrangian Mechanics Note: δ~q(t) describes small variations around the trajectory ~q(t), i.e. ~q(t) + δ~q(t) is a `slightly' different trajectory than ~q(t). We will later often assume that only variations of a trajectory ~q(t) are permitted for which δ~q(t0) = 0 and δ~q(t1) = 0 holds, i.e., at the ends of the time interval of the trajectories the variations vanish. It is also important to appreciate that δS[ ; ] in conventional differential calculus does not corre- spond to a differentiated function, but rather· · to a differential of the function which is simply the df differentiated function multiplied by the differential increment of the variable, e.g., df = dx dx or, M @f in case of a function of M variables, df = dxj. Pj=1 @xj We will now consider a particular class of functionals S[ ] which are expressed through an integral d over the the interval [t0; t1] where the integrand is a function L(~q(t); dt ~q(t); t) of the configuration d vector ~q(t), the velocity vector dt ~q(t) and time t. We focus on such functionals because they play a central role in the so-called action integrals of Classical Mechanics. d _ In the following we will often use the notation for velocities and other time derivatives dt ~q(t) = ~q(t) dxj and dt =x _ j. Theorem: Let t1 S[~q(t)] = Z dt L(~q(t); ~q_(t); t) (1.5) t0 where L( ; ; ) is a function differentiable in its three arguments. It holds · · · t1 t M M 1 8 @L d @L 9 @L δS[~q(t); δ~q(t)] = Z dt δqj(t) + δqj(t) : (1.6) <X @q − dt @q_ = X @q_ t0 j=1 j j j=1 j t0 : ; For a proof we can use conventional differential calculus since the functional (1.6) is expressed in terms of `normal' functions. We attempt to evaluate t1 S[~q(t) + δ~q(t)] = Z dt L(~q(t) + δ~q(t); ~q_(t) + δ~q_(t); t) (1.7) t0 through Taylor expansion and identification of terms linear in δqj(t), equating these terms with δS[~q(t); δ~q(t)]. For this purpose we consider M _ _ _ @L @L 2 L(~q(t) + δ~q(t); ~q(t) + δ~q(t); t) = L(~q(t); ~q(t); t) + δqj + δq_j + O( ) (1.8) X @q @q_ j=1 j j d _ We note then using dt f(t)g(t) = f(t)g(t) + f(t)_g(t) @L d @L d @L δq_j = δqj δqj : (1.9) @q_j dt @q_j − dt @q_j This yields for S[~q(t) + δ~q(t)] t1 M t1 M @L d @L d @L 2 S[~q(t)] + Z dt δqj + Z dt δqj + O( ) (1.10) X @q − dt @q_ X dt @q_ t0 j=1 j j t0 j=1 j From this follows (1.6) immediately. 1.1: Variational Calculus 3 We now consider the question for which functions the functionals of the type (1.5) assume extreme values. For this purpose we define Definition: An extremal of a differentiable functional S[ ] is a function qe(t) with the property δS[~qe(t); δ~q(t)] = 0 for all δ~q(t) . (1.11) 2 F The extremals ~qe(t) can be identified through a condition which provides a suitable differential equation for this purpose. This condition is stated in the following theorem. Theorem: Euler{Lagrange Condition For the functional defined through (1.5), it holds in case δ~q(t0) = δ~q(t1) = 0 that ~qe(t) is an extremal, if and only if it satisfies the conditions (j = 1; 2;:::;M) d @L @L = 0 (1.12) dt @q_j − @qj The proof of this theorem is based on the property Lemma: If for a continuous function f(t) f :[t0; t1] R R (1.13) ⊂ ! holds t1 Z dt f(t)h(t) = 0 (1.14) t0 for any continuous function h(t) with h(t0) = h(t1) = 0, then 2 F f(t) 0 on [t0; t1]. (1.15) ≡ We will not provide a proof for this Lemma. The proof of the above theorem starts from (1.6) which reads in the present case t M 1 8 @L d @L 9 δS[~q(t); δ~q(t)] = Z dt δqj(t) : (1.16) <X @q − dt @q_ = t0 j=1 j j : ; This property holds for any δqj with δ~q(t) . According to the Lemma above follows then (1.12) 2 F for j = 1; 2;:::M. On the other side, from (1.12) for j = 1; 2;:::M and δqj(t0) = δqj(t1) = 0 follows according to (1.16) the property δS[~qe(t); ] 0 and, hence, the above theorem. · ≡ An Example As an application of the above rules of the variational calculus we like to prove the well-known result 2 that a straight line in R is the shortest connection (geodesics) between two points (x1; y1) and (x2; y2). Let us assume that the two points are connected by the path y(x); y(x1) = y1; y(x2) = y2. The length of such path can be determined starting from the fact that the incremental length ds in going from point (x; y(x)) to (x + dx; y(x + dx)) is dy dy ds = r(dx)2 + ( dx)2 = dxr1 + ( )2 : (1.17) dx dx 4 Lagrangian Mechanics The total path length is then given by the integral x1 dy s = Z dx r1 + ( )2 : (1.18) x0 dx s is a functional of y(x) of the type (1.5) with L(y(x); dy ) = 1 + (dy=dx)2. The shortest path dx p is an extremal of s[y(x)] which must, according to the theorems above, obey the Euler{Lagrange dy condition. Using y0 = dx the condition reads d @L d y = 0 ! = 0 : (1.19) 2 dx @y0 dx 1 + (y ) p 0 From this follows y = 1 + (y )2 = const and, hence, y = const. This in turn yields y(x) = 0 p 0 0 ax + b. The constants a and b are readily identified through the conditons y(x1) = y1 and y(x2) = y2. One obtains y1 y2 y(x) = − (x x2) + y2 : (1.20) x1 x2 − − Exercise 1.1.1: Show that the shortest path between two points on a sphere are great circles, i.e., circles whose centers lie at the center of the sphere. 1.2 Lagrangian Mechanics The results of variational calculus derived above allow us now to formulate the Hamiltonian Prin- ciple of Least Action of Classical Mechanics and study its equivalence to the Newtonian equations of motion. Threorem: Hamiltonian Principle of Least Action The trajectories ~q(t) of systems of particles described through the Newtonian equations of motion d @U (mjq_j) + = 0 ; j = 1; 2;:::M (1.21) dt @qj are extremals of the functional, the so-called action integral, t1 S[~q(t)] = Z dt L(~q(t); ~q_(t); t) (1.22) t0 where L(~q(t); ~q_(t); t) is the so-called Lagrangian M 1 _ 2 L(~q(t); ~q(t); t) = mjq_ U(q1; q2; : : : ; qM ) : (1.23) X 2 j − j=1 Presently we consider only velocity{independent potentials.
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