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1. LAGRANGIAN MECHANICS 1.1. Hamilton's Principle and The

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1. LAGRANGIAN MECHANICS 1.1. Hamilton's Principle and The 1. LAGRANGIAN MECHANICS Beauty, at least in theoretical physics, is perceived in the simplicity and compactness of the equations that describe the phenomena we observe about us. Dirac has emphasized this point and said “It is more important to have beauty in one’s equations than to have them fit experiment…. It seems that if one is working from the point of view of getting beauty in one’s equations, and if one has really a sound insight, one is on a sure line of progress.” In this sense the beauty of classical physics lies in the fact that it can all be derived from the postulates of relativity together with just one hypothesis, which we call Hamilton’s principle. This includes all of classical mechanics and all of electricity and magnetism. In fact, if we postulate other interactions, such as the Yukawa potential, the mathematical form of these interactions is very restricted. The flexibility in the choice of natural laws is very limited. In the future, as so-called “grand unified theories” are developed, it is expected that even this limited flexibility will be removed. One of the remarkable developments of modern physics has been the growing perception that the laws of physics are inevitable. Hawking may have gone beyond the realm of pure physics when he asked the question “Did God have any choice?” in the way She wrote the laws of physics. However, it seems that if the universe consists of three spatial dimensions and time, and we require causality, then there is little choice in the laws of physics. 1.1. Hamilton’s principle and the postulates of relativity Figure 1 Reference frames Newton stated as his first law of motion that unless acted upon by an outside force, a body at rest will remain at rest, and a body in motion will remain in uniform motion. A frame of reference K in which this is true is called an inertial frame. The postulates of special relativity state that the laws of physics observed in an inertial reference frame K are identical to those observed in another inertial reference frame K ' that moves with respect to K as shown in Figure 1. Clearly, if the reference frame K' is also an inertial frame, it moves relative to K with at most a constant velocity, and vice versa. Newton also said that “…absolute, true, and mathematical time, of itself, and from its own nature, flows equably and without relation to anything external.” We now know that this is true only in the nonrelativistic limit, but we assume here that it is true. Although the discussion really deserves to be fully relativistic, we restrict our attention to the nonrelativistic case. If time is absolute, then the coordinates r and r ' and the times t and t ' in the two inertial reference frames are related by rrV' = − t , (1) tt' = , (2) where V is the velocity of K ' in K . These are known as a Galilean transformation. Hamilton’s principle says that as a system moves from state a to state b , it does so along the trajectory that makes the action integral b Sdt= ∫ L (3) a an extremum, generally a minimum, subject to the constraint that the endpoints a and b (including both the coordinates and the times) are fixed. That is, in the notation of the calculus of variations, b δδSdt= ∫ L = 0 (4) a for variations δr of the trajectory that vanish at the endpoints, as shown in Figure 2. The quantity L is called the Lagrangian for the system, and its form depends on the nature of the system under consideration. The task in classical mechanics and classical field theory therefore consists of two parts. First we must determine the Lagrangian L for the system, and second we must find the equations of motion that minimize the action S . As we shall see, the form of the Lagrangian follows from the postulates of relativity. Only the few parameters that appear in the equations must be determined from experiment. Figure 2 Variation of a trajectory. 1.2. Lagrangian for a Free Particle Up to this point we have not said anything about the physical system we are trying to describe, which may consist of matter, or fields, or both. We begin with a simple, structureless, point particle, described by the coordinates r and time t . We hypothesize that the Lagrangian depends only on the coordinates, the time, and the velocity, but no higher derivatives of the position, so that LL= (rv,,t) (5) where vr= ddt/ is the velocity of the particle. In fact, since space and time are homogeneous, the Lagrangian of a free particle cannot depend explicitly on the coordinates or the time, but only on the velocity. Otherwise, the behavior of the particle would be different at different places and different times. Thus, the Lagrangian must be simply LL= (v) (6) But the Lagrangian cannot depend on the direction of v , since space is isotropic, so it can depend only on the magnitude v2 = vv⋅ and have the form LT= (vv⋅ ) (7) for some function T that we must determine. Hamilton’s principle for a free particle may now be stated in the form b δδSvdt= ∫T ()2 = 0 (8) a Using the methods of the calculus of variations, we compute bb b dddTTδr δδSdtdtdt=vv ⋅ =⋅22 v δ v =⋅ v = 0 (9) ∫∫T () 22 ∫ aadv a dt dv and integrating once by parts, we get b b dddTT⎛⎞ δδSdt=⋅22rv − δ r ⋅ v = 0 (10) 22∫ ⎜⎟ dva a dt⎝⎠ dv The first term vanishes because δr = 0 at the endpoints. Since the second term vanishes for all variations δr , the rest of the integrand must vanish identically: dd⎛⎞T ⎜⎟v 2 = 0 (11) dt⎝⎠ dv Thus, v is a constant. The trajectory of a free particle is a straight line. The form of the function T (vv⋅ ) is determined by the requirements of Galilean relativity, which state that Hamilton’s principle must be equally valid in both the reference frames K ' and K . That is, bb δδSdtdt''''0'''=⋅==+⋅+∫∫T'()vv δ T ⎣⎡ ()() v V v V⎦⎤ (12) aa For this to be true, it is necessary that T ⎣⎡(vV''+⋅+) ( vV)⎦⎤ and T (vv''⋅ ) differ by at most the time derivative of a function of the coordinates and the time, dtΛ(r ', ') TT⎡⎤()()()vV''-''+⋅+ vV vv ⋅= (13) ⎣⎦dt ' for in this case bb dtΛ (r ', ') b δδδTT⎡⎤vV''-'''+⋅+ vV vv ⋅dt = dt '','0 =Λ r t = (14) ∫∫{}⎣⎦()()() ()a aadt ' since the variation of the coordinates vanishes at the endpoints. But dtΛ()r ', ' ∂Λ =∇'' Λ⋅v + (15) dt''∂ t so ∂Λ TT⎡⎤()()()vV''-''''+⋅+ vV vv ⋅=∇Λ⋅+ v (16) ⎣⎦ ∂t ' But T is independent of the coordinates and time, so ∇ 'Λ and ∂Λ∂/'t , which depend only on the coordinates and time, must be constants, and we get TT[vv''2⋅+ Vv ⋅+⋅ ' VV] -( vv '' ⋅) = K12 ⋅+ v 'K (17) It is easily shown (by expanding in a power series, for example), that this can be true only if the Lagrangian is 1 LT()vmvK=⋅=+ (vv ) 2 (18) 2 for some constants m and K . Since it disappears from the equations of motion when the variation is taken, we set K = 0 . We must determine the constant m by comparison with experiment. 1.3. Lagrangian for a Particle Interacting with a Field To describe the interaction of a particle with a field, we postulate a Lagrangian of the form 1 LU=−mv2 ()r, t . (19) 2 where the first term is just the Lagrangian of a free particle. The variation of the action is therefore bbdδr δδS=⋅ m∫ v dt −∫ U dt , (20) aadt But δ U =∇⋅δr , so upon integrating once by parts we get b b ⎛⎞dv δδSm=⋅vr − m +∇⋅ δ r dt =0 (21) a ∫⎜⎟U a ⎝⎠dt According to Hamilton’s principle the first term vanishes because δr = 0 at the endpoints. Then, since the integral must vanish for arbitrary variations δr , the rest of the integrand must vanish identically. We therefore obtain the equations of motion dv m = −∇U (22) dt Comparing this with experiment, we identify the constant m as the mass of the particle and U as the potential energy. 1.4. Invariance and Momentum Figure 3 Translation of a trajectory. We return to (21) for a moment, which we may now write in the form b b ⎛⎞dv δδSm=⋅vr − m +∇⋅ δ r dt =0 (23) a ∫⎜⎟U a ⎝⎠dt and consider the case of a translation of the entire trajectory by the constant amount δr = ε = constant (24) as illustrated in Figure 3. Provided that the potential U is invariant under the translation δr = ε , the Lagrangian is unchanged. Therefore, the action is unchanged by the translation and δ S = 0 for this variation of the trajectory. But the result of the translation remains a valid trajectory, so the integral in (23) still vanishes identically. However, the variation δr is no longer zero at the endpoints. Therefore, we see from the first term in (23) that mmbmavvb =−= v0 (25) a () () That is, the quantity mv is conserved along the trajectory. We call those quantities that are conserved in a translationally invariant system the momenta, so the momentum must be p = mv (26) For a system of particles that attract and repel one another through central potentials, the Lagrangian has the form 112 LU=−∑∑mvii ij()rr i − j (27) iij22, where the factor of ½ appears in the second term because we have counted the interaction between each pair of particles twice. If the positions of all the particles are translated or rotated together, the Lagrangian is unchanged.
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