University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 2015 16. Canonical Transformations Gerhard Müller University of Rhode Island, [email protected] Creative Commons License This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License. Follow this and additional works at: http://digitalcommons.uri.edu/classical_dynamics Abstract Part sixteen of course materials for Classical Dynamics (Physics 520), taught by Gerhard Müller at the University of Rhode Island. Documents will be updated periodically as more entries become presentable. Recommended Citation Müller, Gerhard, "16. Canonical Transformations" (2015). Classical Dynamics. Paper 6. http://digitalcommons.uri.edu/classical_dynamics/6 This Course Material is brought to you for free and open access by the Physics Course Materials at DigitalCommons@URI. It has been accepted for inclusion in Classical Dynamics by an authorized administrator of DigitalCommons@URI. For more information, please contact [email protected]. Contents of this Document [mtc16] 16. Canonical Transformations • Point transformations (of coordinates in configuration space) [mln88] • Effect of point transformation on Hamiltonian [mex80] • Effect of point transformation on canonical equations [mex82] • Hamiltonian of free particle in rotating frame [mex193] • Canonical transformations (of coordinates in phase space) [mln89] • Canonicity and volume preservation [mln90] • Determine canonicity and generating function I [mex87] • Determine canonicity and generating function II [mex90] • Determine canonicity and generating function III [mex194] • Determine canonicity and generating function IV [mex198] • Infinitesimal canonical transformations [mln91] • Canonicity of time evolution: Liouville theorem [tln45], [tln46] • Canonicity of gauge transformation [mex195] • Electromagnetic gauge transformation [mex196] • Check the canonicity of coordinate transformations [mex84] • Time-dependent generating functions [mex83] • Canonical transformation from rest frame to moving frame [mex85] • Canonical transformation applied to harmonic oscillator [mex86] Point transformations [mln88] Point transformations qi → Qi are invertible coordinate transformations in configuration space. Given a set of transformation relations between generalized coordinates: qi = qi(Q1,...,Qn; t), i = 1, . , n. X ∂qj ∂qj Generalized velocities:q ˙ (Q ,...,Q ; Q˙ ,..., Q˙ ; t) = Q˙ + . i 1 n 1 n ∂Q j ∂t j j Transformed Lagrangian via substitution of transformation relations: ˜ ˙ ˙ L(Q1,...,Qn; Q1,..., Qn; t) = L(q1, . , qn;q ˙1,..., q˙n; t). Invariance of Lagrange equations under point transformations [mex79]: ∂L d ∂L ∂L˜ d ∂L˜ − = 0 ⇔ − = 0. ˙ ∂qi dt ∂q˙i ∂Qi dt ∂Qi . ∂L . ∂L˜ Canonical momenta: p = ,P = . i i ˙ ∂q˙i ∂Qi X ∂qj Relation between canonical momenta [mex80]: P = p . i j ∂Q j i Relation between Hamiltonians [mex80]: X ∂qj H˜ (Q ,...,Q ,P ,...,P , t) = H(q , . , q , p , . , p , t) − p . 1 n 1 n 1 n 1 n j ∂t j Invariance of canonical equations under point transformations [mex82]: ˜ ˜ ∂H ∂H ˙ ∂H ˙ ∂H q˙i = , p˙i = − ⇔ Qi = , Pi = − . ∂pi ∂qi ∂Pi ∂Qi [mex80] Effect of point transformation on Hamiltonian Consider the point transformation qi = qi(Q1;:::;Qn; t); i = 1; : : : ; n between two sets of gen- eralized coordinates. The original and transformed Lagrangians are L(q1; : : : ; qn; q_1;:::; q_n; t) = L~(Q1;:::;Qn; Q_ 1;:::; Q_ n; t). By comparing the differentials dL and dL~ show that the following re- lations hold between the canonical momenta fPig; fpig and between the Hamiltonians H; H~ before and after the transformation: X @qj P (Q ;:::;Q ; p ; : : : ; p ; t) = p ; i 1 n 1 n j @Q j i X @qj H~ (Q ;:::;Q ;P ;:::;P ; t) = H(q ; : : : ; q ; p ; : : : ; p ; t) − p : 1 n 1 n 1 n 1 n j @t j Solution: [mex82] Effect of point transformation on canonical equations Consider the point transformation qi = qi(Q1;:::;Qn; t); i = 1; : : : ; n between two sets of general- ized coordinates. Use the results stated in [mex80] to show (by substitution of coordinates) that the structure of the canonical equations is the same before and after the point transformation: @H @H @H~ @H~ q_i = ; p_i = − , Q_ i = ; P_i = − : @pi @qi @Pi @Qi Solution: [mex193] Hamiltonian of free particle in rotating frame Consider a particle of mass m that is free to move in the xy-plane. (a) Find the Hamiltonian H(r; θ; pr; pθ), where x = r cos θ; y = r sin θ. (b) Convert the resulting canonical equations into two 2nd-order ODEs for r and θ. (c) Perform a point transformation R = r; φ = θ + !t to a rotating frame with ! = const. Find the transformed Hamiltonian H~ (r; φ, pR; pφ) following the prescription derived in [mex80] and convert the resulting canonical equations into two 2nd-order ODEs for R and φ. (d) Derive the equations of motion found in (c) from those found in (b) through direct substitution of the transformation relations. Solution: Canonical Transformations [mln89] Canonical transformations (q; p) → (Q; P ) operate in phase space. Notation: (q; p) ≡ (q1, . , qn; p1, . , pn) etc. Not every transformation qi = qi(Q; P ; t), pi = pi(Q; P ; t) preserves the structure of the canonical equations. Canonicity of transformation (q; p) → (Q; P ) hinges on relation between Hamiltonians H(q; p; t) and K(Q; P ; t) such that ∂H ∂H ˙ ∂K ˙ ∂K q˙i = , p˙i = − ⇒ Qi = , Pi = − . ∂pi ∂qi ∂Pi ∂Qi Canonicity enforced via modified Hamilton’s principle [mln83]: " # " # Z t2 Z t2 X X ˙ δ dt pjq˙j − H(q; p; t) = δ dt PjQj − K(Q; P ; t) = 0. t1 j t1 j X X dF ⇒ p q˙ − H(q; p; t) = P Q˙ − K(Q; P ; t) + . j j j j dt j j Generating function F (x; Y ; t) depends on n old and n new coordinates. For example: (x; Y ) ≡ (q; Q), (q; P ), (p; Q), (p; P ). Total time derivative of F has vanishing variation: t Z 2 dF h it2 δ dt = δF = 0. t t1 dt 1 Different generating functions for the same canonical transformation are re- lated to each other via Legendre transform. The four basic types of generating functions are X F1(q; Q; t) = F2(q; P ; t) − PjQj j X = F3(p; Q; t) + pjqj j X X = F4(p; P ; t) − PjQj + pjqj. j j 1 Implementation of canonical transformation specified by F1(q; Q; t): d X F (q; Q; t) = (p q˙ − P Q˙ ) − [H(q; p; t) − K(Q; P ; t)] . dt 1 j j j j j X ∂F1 ∂F1 ∂F1 X ⇒ dq + dQ + dt = (p dq − P dQ ) ∂q j ∂Q j ∂t j j j j j j j j − [H(q; p; t) − K(Q; P ; t)] dt. Comparison of coefficients yields ∂F1 ∂F1 ∂F1 pj = ,Pj = − ,K − H = . ∂qj ∂Qj ∂t Transformation relations: • Invert relations Pj(q; Q; t) into qj(Q; P ; t). • Combine relations pj(q; Q, t) with qj(Q; P ; t) to get pj(Q; P ; t). Transformed Hamiltonian: ∂ • K(Q; P ; t) = H(q; p; t) + F (q; Q; t). ∂t 1 **** generating transformation transformation function of coordinates of Hamiltonian ∂F1 ∂F1 ∂F1 F1(q, Q, t) pj = Pj = − K = H + ∂qj ∂Qj ∂t ∂F2 ∂F2 ∂F2 F2(q, P, t) pj = Qj = K = H + ∂qj ∂Pj ∂t ∂F3 ∂F3 ∂F3 F3(p, Q, t) qj = − Pj = − K = H + ∂pj ∂Qj ∂t ∂F4 ∂F4 ∂F4 F4(p, P, t) qj = − Qj = K = H + ∂pj ∂Pj ∂t 2 Canonicity and Volume Preservation [mln90] Illustration for one degree of freedom (2D phase space). Consider transformation (q, p) → (Q, P ). Area preservation: Jacobian determinant D = 1 or, equivalently, area inside any closed path C is invariant. Canonicity: There exists a generating function, e.g. F1(q, Q). Canonicity implies area preservation: • Given canonicity specified by F1(q, Q). ∂F ∂F • ⇒ p = 1 ,P = − 1 . ∂q ∂Q ∂Q ∂P • ⇒ = 0 (Q and q are independent); is finite, in general. ∂q ∂p ∂P ∂2F ∂Q−1 • ⇒ = − 1 = − . ∂q ∂Q∂q ∂p . ∂(Q, P ) ∂Q/∂q ∂Q/∂p ∂Q ∂P ∂Q ∂P • ⇒ D = = = − = 1. ∂(q, p) ∂P/∂q ∂P/∂p ∂q ∂p ∂p ∂q Area preservation implies canonicity: • Transformation: p = p(q, Q) P = P (q, Q). I I • Area inside closed path C: pdq = P dQ. C C I • ⇒ [p(q, Q)dq − P (q, Q)dQ] = 0 for arbitrary closed paths C. C • ⇒ Integrand must be perfect differential dF1(q, Q): ∂F ∂F pdq − P dQ = dF = 1 dq + 1 dQ. 1 ∂q ∂Q ∂F ∂F • ⇒ p(q, Q) = 1 ,P (q, Q) = − 1 . ∂q ∂Q [mex87] Determine canonicity and generating functions I Consider the following transformation from a set of canonical coordinates (q; p) to a new set of coordinates (Q; P ): sin p Q = ln ;P = q cot p: q (a) Verify that this transformation is canonical by investigating its Jacobian determinant. (b) De- termine the generating function F3(p; Q) by integration of the total differential dF3. (c) Determine the generating function F2(q; P ) from F3(p; Q) via Legendre transform. Solution: [mex90] Determine canonicity and generating functions II Consider the following transformation from a set of canonical coordinates (q; p) to a new set of coordinates (Q; P ): 1 q Q = (q2 + p2);P = − arctan : 2 p (a) Verify that this transformation is canonical by investigating its Jacobian determinant. (b) De- termine the generating function F4(p; P ) by integration of the total differential dF4. (c) Determine the generating function F1(q; Q) from F4(p; P ) via Legendre transform. Solution: [mex194] Determine canonicity and generating functions III Consider the following transformation from a set of canonical coordinates (q; p) to a new set of coordinates (Q; P ): Q = ln p; P = −qp: (a) Verify that this transformation is canonical by investigating its Jacobian determinant. (b) De- termine the generating function F1(q; Q) by integration of the total differential dF1.