Classical Mechanics in Hamiltonian Form

Transcription

1 Classical Mechanics in Hamiltonian Form We consider a point particle of mass m, position x moving in a potential V (x). It moves according to Newton s law, mẍ + V (x) = 0 (1) This is the usual and simplest description of classical mechanics. However, for the purposes of quantum mechanics it is useful to cast these classical equations of motion in a different form. This material will be familiar to students who have done the Advanced Classical Mechanics course. The details of what follows are not essential to the FQM course but familiarity with these results is expected. Action and Lagrangian We first consider the Lagrangian form, in which it is asserted that the equations of motion Eq.(1) arise as the condition determining the configuration x(t) which extremizes the action t2 S = dt L(x, ẋ) (2) t 1 subject to x(t) being fixed at initial and final times. The calculus of variations shows that for given Lagrangian funcion L(x, ẋ), the funtion x(t) extremizing S satisfies the Euler-Lagrange equation ( ) d L L dt ẋ x = 0 (3) We take the Lagrangian to be L = 1 2 mẋ2 V (x) (4) and Eq.(1) follows. Alternatively, Eq.(1) may be derived directly by requiring that the first order variation δs of the action S is zero, under a variation δx(t) of x(t), subject to x(t) fixed at the end-points, so that We take L to be given by Eq.(4) and we therefore have δs = = t2 t 1 t2 δx(t 1 ) = 0 = δx(t 2 ) (5) dt (mẋδẋ V (x)δx) t 1 dt (mẍ + V (x)) δx(t) + [mẋδx(t)] t 2 t 1 (6) 1

2 The end-point term vanishes by the end-point conditions, Eq.(5). Requiring that δs = 0 for all δx(t) satisfying the end-point conditions immediately leads to the classical equations of motion, Eq.(1). Hamiltonian Form It is very useful to pass to a different description of the classical dynamics in which the position x and momentum p = mẋ enter as independent dynamical variables. This form, the Hamiltonian form, is essential for the passage to quantum mechanics. We define the momentum by p = L = mẋ (7) ẋ and we seek a transformation from a description in terms of (x, ẋ) to a description in terms of (x, p). (This is called a Legendre transformation and such transformation are used extensively in thermodynamics). The way to proceed is to define the Hamiltonian, H(x, p) = pẋ L(x, ẋ) (8) This is a function of independent variables (x, p) because using the definition of p. Since we also have it follows that dh = dp ẋ + p dẋ L L dx x ẋ dẋ = dp ẋ L dx (9) x dh = H p ẋ = H p, H dp + dx (10) x L x = H x Finally, the Euler-Lagrange equation (i.e. the classical equations of moton) and the definition of p shows that L x = ṗ (12) and we arrive at the Hamiltonian form of the equations of motion (11) ẋ = H p, ṗ = H x (13) 2

3 (Note the minus sign in the second equation). Here, the Hamiltonian H is deduced, for a given Lagrangian, from Eq.(8) and for the point particle system considered here is H = p2 + V (x) (14) 2m which is recognizable as the energy of the system. It is readily confirmed that Eqs.(13), (14) give the classical equation of motion Eq.(1). The Hamilton equations may also be derived from the action S = t2 t 1 dt (pẋ H(p, x)) (15) similar to the Lagrangian form, except that variations are taken with respect to both x and p, as independent variables, with δx(t) satisfying the conditions Eq.(5), but δp(t) does not need to satisfy any conditions at the end-points. The Hamilton equations of motion may be usefully rewritten using the Poisson bracket between any pair of phase space functions A(x, p), B(x, p), defined by {A, B} = A B x p B A x p (16) It has a number of properties that make it almost identical to the commutator in quantum mechanics, in particular, {x, p} = 1 (17) Note also that {A, B} = {B, A} {A, BC} = B{A, C} + {A, B}C Hamilton s equation may be rewritten in terms of the Poisson bracket as ẋ = {x, H}, ṗ = {p, H} (18) (Note that neither equation has a minus sign this is taken care of by the definition of the Poisson bracket). In fact, the equations of motion are concisely summarized by the statement that any function on phase space F (x, p) evolves in time according to df dt = {F, H} (19) We will encounter the analogue of this equation in quantum mechanics. 3

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11 CARTESIAN TENSORS The three Cartesian coordinates (x, y, z) of 3-dimensional Euclidean space are denoted x i, where i = 1, 2, 3, x 1 = x, x 2 = y, x 3 = z. Cartesian tensors are objects which transform according to a definite transformation law under rotations of axes, x i x i = j R ij x j where R ij are the components of a rotation matrix R, which satisfies the properties R T = R 1, and detr = 1. A vector is defined to be a quantity v whose components v i transform under rotations according to the transformation law v i = R ijv j We adopt a common abuse of nomenclature, which is to use the word vector when we really mean the components of a vector. Hence we speak of the vector v i. We use the summation convention automatic summation over all indices that appear twice. Hence the above is shorthand for 3 v i = R ij v j j=1 Also, the convention for the positioning of indices in matrix mutliplication is, for matrices A, B, (AB) ij = A ik B kj So for example, the rotation matrix R ij satisfies R ki R kj = Rik T R kj = δ ij where δ ij is the Kronecker delta, equal to diag(1, 1, 1). The indices on Cartesian tensors are, by convention, always written as subscripts, i.e., downstairs. (In special relativity, we sometimes have occasion to write a spatial index upstairs, as in v i but this may differ by a sign, depending on the signature convention used).

12 A tensor T ijk (here a third rank tensor) is defined to be an object whose transformation law is T ijk = R ilr jm R kn T lmn A tensor S ij is symmetric if S ij = S ji. A tensor A ij is antisymmetric if A ij = A ji. The contraction of a symmetric tensor S ij with an antisymmetric tensor A ij gives zero, S ij A ij = 0. tensor. The Kronecker delta δ ij is readily shown to be a symmetric second rank The alternating tensor or permutation tensor ɛ ijk is defined by { 1, if ijk are an even permutation of 123; ɛ ijk = 1, if ijk are an odd permuation of 123; 0, otherwise. The even permutations of 123 are 231 and 312. The odd permutations are 213, 321 and 132. The important thing to remember is that an even permutation is one in which adjacent pairs of indices are exchanged an even number of times, and similarly for odd. This rule still holds for the four-dimensional permutation tensor discussed later. We say that the alternating tensor is totally antisymmetric, meaning that it changes sign under exchange of any pair of adjacent indices. The determinant of a 3 3 matrix M is given by detm ɛ lmn = M il M jm M kn ɛ ijk Setting M equal to the rotation matrix R, and using detr = 1, it follows that ɛ ijk is a third rank tensor (strictly, a pseudotensor), with the special property that its components have the same numerical value in all frames. The scalar product between two vectors A and B is defined by The vector product A B is defined by A B = A i B i (A B) i = ɛ ijk A j B k In quantum mechanics the notation is extended to the Pauli matrices, so that, for example, a σ means a σ = a i σ i = a 1 σ 1 + a 2 σ 2 + a 3 σ 3 = a x σ x + a y σ y + a z σ z 2

13 The ɛ δ identities are are useful for working out properties of vector products. They are ɛ ijk ɛ ilm = δ jl δ km δ jm δ kl ɛ ijk ɛ ijm = 2δ km The gradient of a scalar φ is a vector, defined by ( φ) i = φ x i = i φ where we have introduced the notation i = / x i. The divergence of a vector v is a scalar defined by v = i v i The curl of a vector v is a vector, defined by ( v) i = ɛ ijk j v k The Laplacian a scalar φ is a scalar defined by 2 φ = i i φ From these definitions it is straightforwardly shown that ( v) = 0, ( φ) = 0 Also, if V = 0, then there exists a vector U such that V = U; and, if U = 0, then there exists a scalar φ such that U = φ. 3

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