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Hamiltonian Mechanics

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Hamiltonian Mechanics Part I Hamiltonian Mechanics Both Newtonian and Lagrangian formalisms operate with systems of second-order differential equations for time-dependent generalized coordinates,q ¨i = :::. For a system with N degrees of freedom, N such equations can be reformulated as systems of 2N first-order differential equations if one considers velocities vi =q _i as additional dynamical variables. This system of equations has the formq _i = vi; v_i = ::: that is non-symmetric with respect to qi and vi: Hamiltonian formalism uses qi and pi as dynamical variables, where pi are generalized momenta defined by @L pi = : (0.1) @q_i The resulting 2N Hamiltonian equations of motion for qi and pi have an elegant symmetric form that is the reason for calling them canonical equations. Although for most of mechanical problems Hamiltonian formalism is of no practical advantage, it is worth studying because of the similarity between its mathematical structure and that of quantum mechanics. In fact, a significant part of quantum mechanics using matrix and operator algebra grew out of Hamiltonian mechanics. The latter is invoked in constructing new field theories. Hamiltonian formalism finds application in statistical physics, too. 1 Hamiltonian function and equations Hamiltonian equations can be obtained from Lagrange equations that can be written in the form @L p_i = (1.1) @qi using a Legendre transformation. The differential of the Lagrange function has the form X @L @L dL = dq + dq_ (1.2) @q i @q_ i i i i that with the help of Eqs. (0.1) and (1.1) can be rewritten as X dL = (_pidqi + pidq_i) : (1.3) i Let us intoduce the Hamiltonian function, or the Hamiltonian, defined by X H = piq_i − L: (1.4) i With the help of the above its differential becomes X dH = (_qidpi − p_idqi) : (1.5) i One can see that the natural variables of H are q and p and @H @H q_i = ; p_i = − (1.6) @pi @qi that are Hamiltonian equations of motion for q and p: 1 As the Lagrange function is bilinear inq _i; 1 X L = E (q; q_) − U(q);E = a (q)_q q_ ; (1.7) k k 2 ij i j ij the first term in Eq. (1.4) is the double kinetic energy, X X @L p q_ = q_ = 2E : (1.8) i i @q_ i k i i i Now one can see that H is the full energy, H = Ek + U: (1.9) In most cases the kinetic energy is written in a natural way in terms ofq _i; so that expressing it in terms of pi may require work. For unconstrained particles there is the linear relation pi = mvi and reexpressing H in terms of pi is not a problem. In particular, for a system of particles one obtains X p2 H = i + U: (1.10) 2m i i On the other hand, for systems with constraints, such as double pendulum, different generalized velocitiesq _i couple since aij ≡ (A)ij in Eq. (1.7) is a non-diagonal matrix. Eq. (1.7) can be written in the matrix-vector form 1 E = _qT · · q_ ; (1.11) k 2 A T where _q = (q _1; q_2;:::) is a column and the transposed _q is a row. Further, from Eqs. (0.1) and (1.7) one obtains p = A · q_ (1.12) and thus −1 T −1 T T −1T T −1 q_ = A · p; _q = A ·p = p · A = p · A ; (1.13) −1 where we have taken into account that A ; as well as A; is a symmetric matrix. Plugging this result into Eq. (1.11) one obtains 1 H = pT · −1·p + U: (1.14) 2 A Although inverting kinetic-energy matrix A is possible, at least numerically, it makes Hamiltonian formalism less appropriate. For this reason it is never used for systems with constraints relevant in engineering, exactly where Lagrangian approach shows its strength. Let us, as an illustration, find the Hamiltonian of a particle in spherical coordinates. Kinetic energy has the form mv2 m m 2 2 E = = v2 + v2 + v2 = r_2 + rθ_ + r sin θ φ_ : (1.15) k 2 2 r θ φ 2 Thus the generalized momenta (0.1) are given by @E p = k = mr_ r @r_ @E k 2 _ pθ = = mr θ @θ_ @E k 2 2 _ pφ = = mr sin θ φ. (1.16) @φ_ Solving these equations forr; _ θ;_ and φ_ yields p p p r_ = r ; θ_ = θ ; φ_ = φ : (1.17) m mr2 mr2 sin2 θ 2 Inserting the results into Eq. (1.15) one obtains ! 1 p2 p2 H = E + U = p2 + θ + φ + U(r; θ; φ): (1.18) k 2m r r2 r2 sin2 θ One can see that Eqs. (1.17) are Hamilton equations @H @H @H r_ = ; θ_ = ; φ_ = : (1.19) @pr @pθ @pφ 2 Hamiltonian equations from the least-action principle As Lagrange equations follow from the least-action principle and Hamiltonian equations can be derived from Lagrange equations, one can obtain Hamiltonian equations from the least-action principle directly. The calculation starts from the action written with the help of Eq. (1.4) in the form ! ! t2 t2 X t2 X S = dtL(q; q;_ t) = dt p q_ − H = p dq − Hdt : (2.1) ˆ ˆ i i ˆ i i t1 t1 i t1 i Here, in contrast to Lagrange formalism, one considers both pi and qi as independent dynamical variables and makes a variation qi ) qi + δqi; pi ) pi + δpi: (2.2) The corresponding variation of the action has the form X t2 @H @H δS = δp dq + p dδq − δq dt − δp dt : (2.3) ˆ i i i i @q i @p i i t1 i i Integrating the second term by parts and using that δqi = 0 at the begining and at the end, one can rewrite this as X t2 @H @H δS = δp dq − dt + δq −dp − dt : (2.4) ˆ i i @p i i @q i t1 i i Since for the actual trajectories δS = 0 for arbitrary and independent δqi and δpi; one concludes that both expressions in round brackets are zero that leads to Hamilton equations (1.6). 3 Poisson brackets and conservation Time derivative of any quantity f = f(q; p; t) has the form df @f X @f @f = + q_ + p_ : (3.1) dt @t @q i @p i i i i With the help of Eqs. (1.6) it can be rewritten as df @f = + ff; Hg ; (3.2) dt @t where ff; Hg is a Poisson bracket defined by X @f @g @f @g ff; gg ≡ − : (3.3) @q @p @p @q i i i i i 3 One can see that if f does not depend on time explicitly and its Poisson bracket with the Hamiltonian is zero, f is an integral of motion. In particular, obviously fH; Hg = 0; so that in the absence of explicit time dependence the energy is an integral of motion, H = E = const. Note the similarity between Eq. (3.2) and the quantum mechanical equation of motion for an operator in Heisenberg representation where the h i commutator f; H^ replaces Poisson bracket. Poisson brackets satisfy obvious relations such as fg; fg = − ff; gg ; ff; g1 + g2g = ff; g1g + ff; g2g ; (3.4) etc. There is also the nontrivial Jacobi identity ff; fg; hgg + fh; ff; ggg + fg; fh; fgg = 0 (3.5) that we quote without proof here. It follows then that if both f and g are integrals of motion, then ff; gg is also an integral of motion. For the proof we take h = H, then Jacobi identity takes the form 8 9 8 9 > > > > <> => <> => f; fg; Hg + fH; ff; ggg + g; fH; fg = fH; ff; ggg = 0; (3.6) > | {z } > > | {z } > :> 0 ;> :> 0 ;> thus ff; gg = const: In principle, this could be used to obtain new integrals of motion from already known \old" integrals of motion. However, in most cases new integrals of motion ff; gg are trivial, such as functions of the old integrals of motion. The straightforwardly obtainable Poisson brackets fqi; qjg = 0; fqi; pjg = δij fpi; qjg = −δij; fpi; pjg = 0 (3.7) are called fundamental Poisson brackets. If the Hamiltonian depends on one generalized coordinate and the corresponding generalized momentum as a combination that does not contain other variables, this combination is an integral of motion. That is, for H (fqi; pig) = H fqi; pigi6=1 ; F (q1; p1) (3.8) F (q1; p1) is an integral of motion. This can be proven with the help of Poisson brackets. Since differentiation with respect to the variables i 6= 1 gives zero, one obtains @F @H @F @H @H @F @F @F @F @H F_ = fF; Hg = − = − = fF; F g = 0: (3.9) @q1 @p1 @p1 @q1 @F @q1 @p1 @p1 @q1 @F This result can be generalized for nested dependences of the type H (fqi; pig) = H fqi; pigi6=1;2 ; F2(q2; p2; F1(q1; p1)) : (3.10) Here F1(q1; p1) = α1;F2(q2; p2; α1) = α2 (3.11) are integrals of motion. For F2 one obtains @F2 @H @F2 @H @F2 @H @F2 @H F_2 = fF2; Hg = − + − @q1 @p1 @p1 @q1 @q2 @p2 @p2 @q2 2 @F2 @H @H = fF1;F1g + fF2;F2g = 0: (3.12) @F1 @F2 @F2 4 3.1 Example: Particle's motion in spherical coordinates As an example of nontrivial integrals of motion detectable with the help of Poisson brackets consider a particle moving in the potential b(θ) c(φ) U(r; θ; φ) = a(r) + + (3.13) r2 r2 sin2 θ in the spherical coordinates.
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