Lecture 12: Introduction to Analytical Mechanics of Continuous Systems Lagrangian Density for Continuous Systems The kinetic and potential energies as T = 1 2 i η2 i (1 and V = 1 2 i+1 η i 2, i (2 where η i is the displacement of the i-th particle from its equilibrium position.the Lagrangian L = T V = 1 2 (m i η2 i k(η i+1 η i 2. (3 For going from the discrete to continuous case, this equation may be rewritten in the form L = 1 m 2 i a[ a η2 i ka ( η i+1 η i 2 ]. (4 a η i+1 η i a The Hooke s law F = Y ξ, (6 with a discrete analog L = 1 2 η(x + a η(x = lim a 0 a = dη dx. (5 F = k(η i+1 η i = ka ( η i+1 η i, (7 a (µ η 2 Y ( dη dx 2 dx L(x dx. (8 1
The Lagrangian will be expressed by the three-dimensional integral L = L dx dy dz. (9 L is the Lagrangian density: L = 1 2 (µ η2 Y ( dη dx 2 (10 In general case of the three-dimensional system the Lagrangian density may be written as L = L ( η, η x, η y, η z, η t, x, y, z,. (11 t 2
Three- and Four-Dimensional Lagrangian Formulation Hamiltons principle δi = δ 2 2 1 1 Ldx dy dz dt, (12 t 2 t 1. (13 For convenience, we will change (x, y, z to (x 1, x 2, x 3. Then we obtain δi = 2 δldx 1 dx 2 dx 3 dt = 2 [ L 1 δi = 2 1 1 η δη+ L η δ η+ L ( η x i (14 δη { L η d L dt η d [ L ]} dx i ( η dx1 dx 2 dx 3 dt. x i (15 In accordance with the Hamilton principle, this integral must be equal to zero, but with an arbitrary η it can be the case only if d L dt η + d [ L ] L dx i ( η x i η = 0. (16 δ( η x i ] dx 1 3
L η = µ η t, L ( η x = Y η x, L η = 0, (17 µ 2 η t Y 2 η = 0, (18 2 x2 which is nothing but the equation for the longitudinal wave, propagating in elastic solid. In more general cases, such as oscillations in elastic solids, the displacement in all three directions take place. Accordingly, in such cases we need to introduce three generalized coordinates η j (x 1, x 2, x 3, t, j = 1, 2, 3. d L + d [ L ] L dt η j dx i ( η j = 0 (j = 1, 2,... x i η j (19 This equation can be rewritten in a very compact mathematical form if we use the so-called covariant notation introducing a four-dimentional space with coordinates x 0 = t, x 1 = x, x 2 = y, x 3 = z. (20 η ρ,ν η ρ x ν ; η,j η x j ; η i,µν 2 η i x µ x ν. (21 In this notation d ( L L = 0. (22 dx ν η ρ,ν η ρ Equations represent a set of partial differential equations for the generalized coordinates (or the field 4
quantities η ρ, with as many equations as there are different values of ρ. 5
Stress-Energy Tensor and Conservation Theorems dl dx µ = dl = L η ρ,µ + L η ρ,µν + L. (23 dx µ η ρ η ρ,ν x µ d dx ν ( L d dx ν η ρ,ν ηρ,µ + L η ρ,ν η ρ,µ x ν + L x µ = ( L L η ρ,µ Lδ µν =. (25 η ρ,ν x µ T µν = L η ρ,µ Lδ µν, (26 η ρ,ν dt µν = L. (27 dx ν x µ T µν forms a four-tensor of the second-rank that contains more rich physical information than in the case of an ordinary Jacobi s integral. T 00 = L η ρ,0 Lδ 00 = L η ρ L. (28 η ρ,0 η ρ If the Lagrangian density can be defined as the difference between a kinetic energy density T and a potential energy density V, L = T V, (29 T = 1 2 µ η ρ η ρ, (30 T 00 = T + V, (31 6 d ( L η ρ,µ + dx ν η ρ,ν (24
and thus T 00 can be identified as a total energy density. 7
The case when L does not depend explicitly on x µ dt µν dx ν = 0. (32 dt µ0 dt + dt µj dx j = 0, (33 dt µ0 dt + T µ = 0, (34 Tµ0 dv = T µ dv = T µ ds, (35 where the surface integral defines the total flux of the vector T µ through the surface of the system. T00 dv = T 0 dv = T 0 ds. (36 If T 00 is an energy density then the volume integral in l.h.s. of this equation may be identified as the total energy. The three-dimensional tensor T ij happens to be just the well-known stress tensor which appear if one consider the displacement of an elasic solids. After these identifications the four-dimensional tensor T µν has been called the stress-energy tensor. 8
Hamilton and Poisson Bracket Formulation π ρ (x µ = L η ρ (37 H = π ρ η ρ L. (38 d ( L = 0, dx µ η ρ,µ (39 dπ ρ dt + d ( L = 0. dx i (40 Π ρ = dv π ρ (r, t, (41 dπ ρ = η ρ + π λ π ρ L π ρ = η ρ, (42 dη ρ = π λ η ρ L η ρ L η ρ = L η ρ (43 = d ( L = πρ d ( L dη ρ dx µ η ρ,µ dx i (44 dη ρ,i = π λ L = L (45 d dη ρ dx i ( dη ρ,i δ δψ ψ d dx i 9 = πρ. (46 ψ,i, (47
η ρ = δh δπ ρ, d ( L dt η ρ π ρ = δh. (48 δη ρ δl δη ρ = 0, (49 10
Finally, we consider the Poisson bracket formulation of continuous mechanics. Let us consider some density U that is a function of the phase space coordinates (η ρ, π ρ, their spatial gradients(η ρ,i, π ρ,i and, possibly, x µ : U = U(η ρ, π ρ, η ρ,i, π ρ,i, x µ. (50 The corresponding total (or integral quantity is U(t = U dv, (51 where the volume integral extends over all space enclosed by the bounding surface on which η ρ and π ρ vanish. Differentiating U with respect to time we have du dt = du dt dv = dv { U η ρ η ρ + U η ρ,i + U π ρ π ρ + U π ρ,i π ρ,i + (52 dv U η ρ,i = du dt = du dt = dv U η ρ x i = dv { δu η ρ + δu π ρ + U } δη ρ δπ ρ t dv { δu δh δh δu } + δη ρ δπ ρ δη ρ δπ ρ d ( U dv η ρ dx i (53 (54 dv U t. (55. [U, W ] = dv { δu δw δw δu }, (56 δη ρ δπ ρ δη ρ δπ ρ U t = dv U t. (57 11
du dt dt 00 U = [U, H] + t, (58 dt = H t, (59 + dt 0i dx i = L t. (60 dt dt = dt 0i L dx i t. (61 H t = L t. (62 + H dx i t. (63 dt = dt 0i Finally, we notice that although all the abovediscussed formalisms have been developed for the description of the continuous mechanical systems such as elastic solids, these formalisms can be applied to any field irrespective of its nature or origin, say, for the electromagnetic field, for the Schrodinger wave function field, for a classical field of a Dirac electron etc. Hamilton s principle becomes in effect a convenient and compact description both the mechanical (discrete or continuous systems and fields. 12