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Applications of Canonical Transformations in Hamiltonian Mechanics

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Applications of Canonical Transformations in Hamiltonian Mechanics Applications of Canonical Transformations in Hamiltonian Mechanics Brian Tu Jan. 14, 2014 Contents 1 Introduction 2 2 Preliminaries 2 3 Poisson Bracket 3 3.1 Characterization of Canonical Transforms . 3 3.2 Application to Integrals of Motion . 5 4 Infinitesimal Canonical Transformations 5 5 Generating Functions 6 1 1 Introduction Canonical transformations were introduced in the theory of Hamiltonians as a class of transformations that preserve the form of the Hamiltonian equations. We explore these transformations in greater detail, looking at topics such as its invariants, generating such transformations, and time evolution of a Hamiltonian system as a canonical transforma- tion. We consider only time-independent Hamiltonians. 2 Preliminaries We begin with a few definitions for review. Definition 2.1. Recall that the Hamiltonian H is a function of 2n coordinates q = (q1; q2; : : : ; qn) and p = (p1; p2; : : : ; pn) that satisfies the 2n relations @H p_i = − @qi @H q_i = ; @pi where 1 ≤ i ≤ n, known as Hamilton's equations. Definition 2.2. A transformation of coordinates is canonical if it preserves the form of Hamiltonians equations [3, p. 14]. Formally, let x = (q1; : : : ; qn; p1; : : : ; pn). Let 0 1 J = −1 0 be a 2n × 2n block matrix, e.g., 0 is an n × n 0 matrix, 1 is In, and -1 is −In. Hamilton's equations then can be written as x_ = JrH; Let y : xi 7! yi(x) be a transformation of coordinates. We have 2n 2n X @yi X @yi @H y_ = x_ = J ; i @x j @x @x j=1 j j=1 j j and hence, y_ = (J JJ T )rH; where (J)ij = @yi=@xj is the Jacobian of y. The form of Hamilton's equation is preserved if J JJ T = J: (1) Hence a transformation is canonical if its Jacobian satisfies (1). Incidentally, such a matrix J is called symplectic. 2 3 Poisson Bracket 3.1 Characterization of Canonical Transforms We saw above that one way to characterize a canonical transformation is by checking (1). Another way of characterizing these transformations is by using an operation called the Poisson bracket. Definition 3.1. A Poisson bracket is a binary operation on two functions f(q; p), g(q; p) defined as n X @f @g @f @g ff; gg = − : @q @p @p @q i=1 i i i i It is immediate from the definition that ff; gg = −{g; fg and that q and p satisfy fqi; qjg = 0; fpi; pjg = 0; fqi; pjg = δij: (2) It is natural to expect to expect that (2) holds for canonical transformations; in fact this is a sufficient condition. We have the following theorem. Theorem 3.1. The Poisson bracket is invariant under canonical transformations. Con- versely, if Q : qi 7! Qi(q; p); P : pi 7! Pi(q; p) is a transformation of q and p such that fQi;Qjg = 0; fPi;Pjg = 0; fQi;Pjg = δij; then the transformation is canonical [4, p. 22]. Proof. For the first part, let f(q; p) and g(q; p) be any two functions. We have n X @f @g @f @g ff; gg = − @q @p @p @q i=1 i i i i @f @g = J : @x @x If we have a transformation y(x) with Jacobian J , then we have @f @f = J @x @y and thus @f @g ff; gg = J JJ T @x @y @f @g = J : @x @y 3 This shows the invariance of the Poisson bracket. For the other direction, note that we can write the Jacobian as 0@Q @Q1 B @q @p C J = B C : @@P @P A @q @p Computing, we have fQ; Qg fQ; P g J JJ T = fP; Qg fP; P g where (fQ; P g)ij = fQi;Pjg; and the other entries are defined similarly. Thus we have J JJ T = J and the transformation is canonical. We develop another method of characterizing canonical transformations. Let Q(q; p) and P (q; p) be a canonical transformation. We have @Q @Q @Q @H @Q @H Q_ = i q_ + i p_ = i − i (3) i @q @p @q @p @p @q and @H @H @q @H @p = + : (4) @Pi @q @Pi @p @Pi Since the transformation is canonical, (3) equals (4), so we have @Q @H @Q @H @H @q @H @p @H @Q @p @H @q @Q i − i = + ) i − = + i : @q @p @p @q @q @Pi @p @Pi @p @q @Pi @q @Pi @p Since @H=@p = q 6= 0 and @H=@q = −p 6= 0, this implies that the quantities in the parentheses are 0. Hence @Q @p @Q @q i = and i = − : (5) @q @Pi @p @Pi _ By equating Pi with @H=@Qi, using a similar argument it can be shown that @P @q @P @p i = and i = − : (6) @p @Qi @q @Qi Hence canonical transformations imply (5) and (6). In fact the converse is true as well. Suppose that (5) and (6) hold for some transformation Q and P not necessarily canonical. We have @H @H @q @H @p @H @Qi @H @Qi _ = + = − + = Qi @Pi @q @Pi @p @Pi @q @p @p @q and @H @H @q @H @p @H @Pi @H @Pi _ = + = + = Pi; @Qi @q @Qi @p @Qi @q @p @p @q and hence the transformation is canonical. Thus we obtain another characterization of canonical transformations [2, p. 9]. 4 3.2 Application to Integrals of Motion The Poisson bracket also neatly characterizes the integrals of motion. A function f is an integral of motion iff ff; Hg = 0 [4]. To see this, note that n n X @f @H @f @H X @f @f df ff; Hg = − = q_ + p_ = ; @q @p @p @q @q i @p i dt i=1 i i i i i=1 i i which is 0 if f is an integral of motion. We can apply this to an integral of motion we discussed this year: Example 3.1 (Conservation of Linear Momentum). Recall that a coordinate qk is cyclic if it does enter into the Hamiltonian; e.g., @H = 0: @qk Suppose that qk is a cyclic coordinate. Then momentum in that direction is conserved: @H fpk;Hg = − = 0: @qk Example 3.2 (Conservation of Energy). We know that that Hamiltonian is the sum of the kinetic and potential energies of a system. We have fH; Hg = 0 from antisymmetry of the Poisson bracket, and hence energy is conserved. 4 Infinitesimal Canonical Transformations Here we consider canonical transformations that are parameterized by a single parameter. We willl see that the time evolution of a Hamiltonian system is a canonical transforma- tion. Suppose that we have a family of canonical transformations indexed by a parameter λ, and suppose that the dependence on λ is continuous (think of them as, say, translations or rotations by a parameter). Furthermore, we can take λ such that at λ = 0 the transformation is the identity. That is, we have Qi(λ) = Qi(q; p; λ);Pi(λ) = Pi(q; p; λ); and Q(0) = q P (0) = p: If λ is very small, we can write Qi(λ) = qi + λdqi;Pi(λ) = pi + λdpi: We wish to know what relationships dq and dp must satisfy for the transformation to be canonical. The Jacobian is I + λ∂dq=@q λ∂dq=@p J = n λ∂dp=@q In + λ∂dp=@p 5 which must satisfy J JJ T = J. This gives us @dq @dp = − : (7) @q @p If we could find a function F such that @F @F = dq and = −dp; (8) @p @q then (7) would be satisfied [5]. Note that the equations in (8) look exactly like Hamilton's equations. Indeed, if we let λ be the time t, the transformation induced by the Hamiltonian H matches the differential equations for the flow of the system. Hence the advance of time in a Hamiltonian system is a canonical transformation. Now that we have established that time evolution of a Hamiltonian system is a canonical transformation, we can give a very quick proof of Liouville's Theorem. Theorem 4.1 (Liouville's Theorem). The phase flow of a Hamiltonian system is volume preserving. Proof. Let t be fixed. Let the evolution of the Hamiltonian system after time t be a canonical transformation taking q to Q(q; p) and p to P (q; p). It suffices to show that the Jacobian of the transformation is 1. Using properties of the Jacobian, we have @(Q; P ) J = @(q; p) @(Q; P ) @(q; p) = @(q; P ) @(q; P ) @Q @p = : @q @P and thus @Q @p = ; @q @P so we have J = 1; as desired. 5 Generating Functions We have seen various characterizations of canonical transformations, and a few examples of them. In fact, in the previous section on infinitesimal transforms we saw an example of a generating function. Here, we explore the construction of canonical transforma- tions. Let F (q; Q) be any function between coordinates. If we define @F pi = ; @qi 6 how can we define Pi so that the resulting transformation is canonical? If we define @F Pi = − @Qi then F will be a canonical transformation. This can be shown using Theorem 3.1. F is known as a generating function of the first kind. There are four other kinds of gener- ating functions that generate canonical transformations, with other coordinates defined as follows [5, 4]: @F @F F = F (q; P ) − QP ; p = 2 ; P = 2 2 @q @P @F @F F = F (p; Q) + qp; q = − 3 ; P = − 3 3 @p @Q @F @F F = F (p; P ) + qp − QP ; q = − 4 ; Q = 4 : 4 @p @P I pledge my honor that this represents my own work in accordance with University pol- icy.
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