2.3 Calculus of variations

Transcription

1 2.3 Calculus of variations Euler-Lagrange equation The action functional S[x(t)] = L(x, ẋ, t) dt (2.3.1) which maps a curve x(t) to a number, can be expanded in a Taylor series { S[x(t) + δx(t)] = L + L x a δxa + L ẋ a δẋa + O ( δx 2) } dt (2.3.2) { [ L = L + x d ( )] L δx a + d ( ) L a dt ẋ a dt ẋ a δxa + O ( δx 2)} dt (2.3.3) For fixed boundary conditions, δx( ) = δx(t f ) = 0, the last term vanishes, leaving { [ L S[x(t) + δx(t)] = L + x d ( )] L δx a + O ( δx 2)} dt (2.3.4) a dt ẋ a Thus the covariant functional derivative of the action is δx = L a x d ( ) L a dt ẋ a (2.3.5) and an extremum of the action is given by the Euler-Lagrange equation L x d ( ) L = 0 (2.3.6) a dt ẋ a We can also take the functional derivative with respect to the coordinate paths x α (t) to get the coordinate form of the Euler-Lagrange equation δx = L α x d ( ) L = 0 (2.3.7) α dt ẋ α The action for a scalar field φ(x) has the form S[φ(x)] = L(φ, φ, x) ɛ (2.3.8) where ɛ is the spacetime volume form. The covariant Euler-Lagrange equation is δφ = L [ ] φ L a = 0 (2.3.9) ( a φ) where we have used the fact that the volume form is covariantly constant. We can also express the action in terms of coordinates S[φ(x)] = L(φ, φ, x) g d 4 x (2.3.10) Ewan Stewart /12/6

2 The coordinate Euler-Lagrange equation is then expressed in terms of the Lagrangian density L = g L since the components of the volume form depend on the coordinates δφ = 1 { [ ]} L L g φ α = 0 (2.3.11) ( α φ) Conservation laws Eq. (2.3.6) gives the momentum conservation equation dp a dt = L x a (2.3.12) with momentum p a = L (2.3.13) ẋ a which shows that the momentum is conserved if the Lagrangian is independent of x. Multiplying Eq. (2.3.6) by ẋ a we get the energy conservation equation with energy de dt = L t (2.3.14) E = ẋ a L ẋ a L (2.3.15) which shows that the energy is conserved if the Lagrangian is independent of t. Similarly, Eq. (2.3.9) gives the continuity equation a j a = L φ (2.3.16) with field-space momentum 1 current j a = L ( a φ) (2.3.17) which shows that the field-space momentum is conserved if the Lagrangian is independent of φ. Multiplying Eq. (2.3.9) by b φ we get the energy-momentum conservation equation a T a b = L x b (2.3.18) with stress(-energy-momentum) tensor T a b = L ( a φ) bφ Lδ a b (2.3.19) 1 Field-space momentum should not be confused with spacetime momentum. From the spacetime point of view, field-space momentum is a charge. Ewan Stewart /12/6

3 2.3.3 Symmetries and the Lie derivative A continuous symmetry is described by the flow generated by a vector field. The Lie derivative, with respect to a vector field u a, acting on a vector field v a, is L u v a = u b b v a v b b u a (2.3.20) Is the derivative relative to the flow generated by u a, see Figure Note that L u x + δ u δ v(x + δ u) δ v (x + δ u) δ 2 u v δ 2 v u δ 2 L u v δ u(x) lim δ 0 δ u (x + δ v) δ u(x + δ v) x δ v(x) x + δ v Figure 2.3.1: The Lie derivative and its relation to the covariant derivative. v is v(x) parallel transported along u, i.e. transported such that u v = 0, and u is u(x) parallel transported along v. depends on u a and its derivative, bus independent of the metric. If a vector field ξ a satisfies Killing s equation L ξ g ab = a ξ b + b ξ a = 0 (2.3.21) then ξ a is a Killing vector field and generates an isometry of the space. Eqs. (2.3.12) and (2.3.13) give d dt (ξa p a ) = ξ a L a L + ξ (2.3.22) xa ẋ ( a = ξ a a L ξ b b ẋ a ξ ) L a (2.3.23) ẋ a = L ξ L (L ξ ẋ a ) L ẋ a (2.3.24) = L ξ ẋ L (2.3.25) where L ξ ẋ is the partial Lie derivative at fixed ẋ a. Thus ξ a p a is conserved if ξ a generates a symmetry of L. If we choose coordinates such that e a α = ξ a then ξ a p a = p α and its conservation can be seen directly from Eq. (2.3.7). For example, if L = 1 2 mg abẋ a ẋ b V (x) (2.3.26) Ewan Stewart /12/6

4 then and p a = mg ab ẋ b (2.3.27) L ξ ẋ L = 1 2 mẋa ẋ b L ξ g ab L ξ V (2.3.28) If L has a translational symmetry generated by e a x then e a xp a = p x = mg xx ẋ = mẋ (2.3.29) is conserved, while if L has a rotational symmetry generated by e a θ then e a θp a = p θ = mg θθ θ = mr 2 θ (2.3.30) is conserved Actions Particles in spacetime A particle is something that exists as a worldline in spacetime. Figure 2.3.2: A particle in spacetime. A worldline C in a spacetime M has action S[C] = (mσ + qa) (2.3.31) where the worldline volume form σ measures the length along the curve C dτ = σ dx (2.3.32) and A is a covector field in the spacetime. Note that the physics given by = 0 is invariant under A A + λ (2.3.33) since C λ = C λ (2.3.34) Ewan Stewart /12/6

5 is a boundary term. In Lagrangian form S = (mσ a + qa a ) dx a (2.3.35) C = (m ) g ab ẋ a ẋ b + qa a ẋ a dt (2.3.36) The Euler-Lagrange equation C ( ) d L = L (2.3.37) dt ẋ a x a gives where the particle s momentum 2 d dt (p a + qa a ) = q ( a A b ) dxb dt p a = mg ab dx b gcd ẋ c ẋ d dt = mg ab dx b dτ (2.3.38) (2.3.39) Therefore the force on the particle is where the electromagnetic field f a = dp a dτ = mg d 2 x b ab dτ 2 = qf dx b ab dτ (2.3.40) F ab = a A b b A a (2.3.41) Eq. (2.3.40) is the relativistic form of the Lorentz force law, see Eq. (1.3.58). Fields in spacetime A covector field A in a spacetime M has action { 1 S[A] = 2 m2 A A + 1 } ( A) ( A) A J 2 M (2.3.42) where the last term is the continuum form of the particle-field coupling in Eq. (2.3.31). Varying the action { } = m 2 δa A + ( δa) ( A) δa J (2.3.43) M { [ ] } = δa m 2 A + ( A) J + [δa ( A)] 2 Note that p a = mσ a. (2.3.44) Ewan Stewart /12/6

6 or using gives the Euler-Lagrange equation δa = L A + L ( A) (2.3.45) m 2 A + ( A) = J (2.3.46) The mass m causes the field to decay exponentially away from a source, limiting the field s action to microscopic distances, as in the weak nuclear force or superconductivity. However, the particle-field coupling in Eq. (2.3.31) only depends on the part of A that is invariant under a gauge transformation A A + λ (2.3.47) so we may assume that only the gauge invariant part of A is physical. Then the mass term in Eq. (2.3.42) is forbidden, leaving F = J (2.3.48) which, together with the trivial F = 0, is the relativistic form of Maxwell s equations, see Section and Homeworks 8 and 9. Ewan Stewart /12/6