Hamiltonian Formalism In modern physics, the Hamiltonian formalism plays a very important role. Apart from being a powerful tool for solving classical-mechanical problems, it is crucial for bridging classical and quantum mechanics, as well as for deriving equilibrium and non-equilibrium statistical description of macro- scopic systems from the first principles. Hamiltonian Function and Hamiltonian Equations Suppose we are dealing with a mechanical system described by a Lagrangian of the general form: L = L(fq;_ qg) : (1) Here f:::g means `the same for each degree of freedom'. [The Lagrangian L can also depend on time, but a presence or absence of time-dependence will not be relevant to our derivation.] For each degree of freedom labeled with the subscript s we introduce a new variable @L ps = ; (2) @q_s which we will be calling generalized momentum. Our next steps will be quite formal. The set of relations (2) allows one to mathematically express all the q_'s as functions of q's and p's: q_s ≡ q_s(fq; pg) : (3) Consider the expression for energy, X @L E = q_s − L; (4) s @q_s and express E as a function of fq; pg rather than a function of fq;_ qg: X E = H(fq; pg) = ps q_s(fq; pg) − L(fq_s(fq; pg); qsg) : (5) s The function H(fq; pg) is called the Hamiltonian function, or just the Hamil- tonian. Note that we introduced a special letter to distinguish the Hamil- tonian function from energy. When speaking of the Hamiltonian we always 1 mean a function of generalized momenta and coordinates rather than just some quantity. Correspondingly, if the energy is expressed as a function of some other variables, it is not the Hamiltonian. Now let us evaluate the partial derivative of the function H with respect to one of the momenta while keeping all the other variables constant: @H X @q_s(fq; pg) @L(fq;_ qg) @q_s(fq; pg) =q _s0 + ps − : (6) @ps0 s @ps0 @q_s @ps0 We see that with our definition of the generalized momentum the expression in the brackets is identically equal to zero. Hence, @H =q _s0 : (7) @ps0 Now we calculate the partial derivative with respect to some coordinate: @H X @q_s(fq; pg) @L(fq;_ qg) @q_s(fq; pg) @L(fq;_ qg) = ps − − : (8) @qs0 s @qs0 @q_s @qs0 @qs0 With our definition of the generalized momentum the expression in the brackets is zero, and @H @L(fq;_ qg) = − : (9) @qs0 @qs0 At this point we recall that we are doing physics and require that apart from abstract mathematical relations between three formally independent groups of variables, fq_g, fqg, and fpg, the equations of motion for q's be satisfied, and thatq _'s have the meaning of the time derivatives of the q's (so far we were treatingq _'s as just some independent of q's variables). In this case the Lagrange equation implies @L(fq;_ qg) d @L(fq;_ qg) = =p _s0 : (10) @qs0 dt @q_s0 We conclude that the Hamiltonian function demonstrates an amazing prop- erty. It yields time derivatives of generalized coordinates and momenta (for each s) @H @H q_s = ; p_s = − : (11) @ps @qs The system of these equations is an alternative (to either Newtonian, or La- grangian) way of describing time evolution of mechanical systems. Given the 2 Hamiltonian and the initial conditions for all q's and p's, one uses Eqs. (11), called Hamiltonian equations, to find the evolution of the system. Let us derive the Hamiltonian of a single particle in an external potential. Starting from the Lagrangian mr_ 2 L = − U(r) ; (12) 2 we find the vector of the generalized momentum: @L p = = mr_ : (13) @r_ The generalized momentum coincides with what we previously introduced as the linear momentum. The Hamiltonian of the system is then p2 H = + U(r) ; (14) 2m and the Hamiltonian equations are @U r_ = p=m ; p_ = − = F(r) : (15) @r The generalization for the system of N particles interacting through pair potentials is as follows. N p2 H = X j + X U (jr − r j) ; (16) 2m ij i j j=1 j i<j X r_ i = pi=mi ; p_ i = Fij ; (17) j6=i where @Uij Fij = − : (18) @ri For a counterintuitive example, note that in magnetic field the generalized momentum, P, does not coincide with the linear momentum p. It is easy to verify that in a uniform (for simplicity) magnetic field B one has e P = p + B × r ; (19) 2 1 e 2 H(P; r) = P − B × r : (20) 2m 2 3 Poisson Brackets. Suppose we are interested in time evolution of some function A(fq; pg) due to the evolution of coordinates and momenta. There is a useful relation A_ = fH; Ag ; (21) where the symbol in the r.h.s. is a shorthand notation|called Poisson bracket|for the following expression @H @A @H @A fH; Ag = X − : (22) s @ps @qs @qs @ps The proof is very simple: The chain rule for dA(fq(t); p(t)g)=dt and then Eqs. (11) forq _s andp _s. Hence, any quantity A(fq; pg) is a constant of motion if, and only if, its Poisson bracket with the Hamiltonian is zero. In particular, the Hamilto- nian itself is a constant of motion (if does not explicitly depend of time), since fH; Hg = 0, and this is nothing else than the conservation of energy. Liouville's Theorem: A Bridge to Statistical Physics Given a mechanical system with m degrees of freedom described by m gener- alized coordinates and corresponding m generalized momenta, we define the notion of phase space as a 2m-dimensional space of points, or, equivalently, vectors of the following form: X = (q1; q2; : : : ; qm; p1; p2; : : : ; pm) : (23) Each point/vector in the phase space represents a state of the mechanical system. If we know X at some time moment, say, t = 0, then the further evolution of X|the trajectory X(t) in the phase space|is unambiguously given by Eqs. (11), since these are the first-order differential equations with respect to the vector function X(t). [Note that this fact immediately im- plies that different trajectories in the phase space cannot intersect.] If we introduce the vector V ≡ V(X) by the formula @H @H @H @H V = ;:::; ; − ;:::; − ; (24) @p1 @pm @q1 @qm then all the Hamiltonian equations can be written in the following vector form X_ = V(X) : (25) 4 The phase space is convenient for statistical description of mechanical system. Suppose that the initial state for a system is known only with a certain finite accuracy. This means that actually we know only the proba- bility density W0(X) of having the point X somewhere in the phase space. If the initial condition is specified in terms of probability density, then the subsequent evolution should be also described probabilistically, that is we have to work with the distribution W (X; t), which should be somehow re- lated to the initial condition W (X; 0) = W0(X). Our goal is to establish this relation. We introduce a notion of a statistical ensemble. Instead of dealing with probability density, we will work with a quantity which is proportional to it, and is more transparent. Namely, we simultaneously take some large number Nens of identical and independent systems distributed in accordance with W (X; t). We call this set of systems statistical ensemble. The j-th member of the ensemble is represented by its point Xj in the phase space. The crucial observation is that the quantity Nens W (X; t) gives the number density (=concentration) of the points fXjg. Hence, to find the evolution of W we just need to describe the evolution of the number density of the points Xj, which is intuitively easier, since each Xj obeys the Hamiltonian equation of motion. A toy model. To get used to the ensemble description, and also to obtain some important insights, consider the following dynamical model with just one degree of freedom: H = (1=4)(p2 + q2)2 : (26) The equations of motion are: q_ = (p2 + q2) p ; (27) p_ = −(p2 + q2) q : (28) The quantity ! = p2 + q2 (29) is a constant of motion, since, up to a numeric factor, it is a square root of energy. We thus have a linear system of equations q_ = !p ; (30) p_ = −!q ; (31) 5 which is easily solved: q(t) = q0 cos !t + p0 sin !t ; (32) p(t) = p0 cos !t − q0 sin !t ; (33) 2 2 where q0 ≡ q(0), p0 ≡ p(0), and ! = p0 + q0. We see that our system is a non-linear harmonic oscillator. It performs harmonic oscillations, but in contrast to a linear harmonic oscillator, the frequency of oscillations is a function of energy. Now we take Nens = 1000 replicas of our system and uniformly dis- tribute them within the square 0:75 ≤ q ≤ 1:25, −0:25 ≤ p ≤ 0:25 of the two-dimensional phase space. Then we apply the equations of motion (32)- (33) to each points and trace the evolution. Some characteristic snapshots are presented in Fig. 1. In accordance with the equations of motion, each q 2 2 point rotates along corresponding circle of the radius p0 + q0. Since our oscillators are non-linear, points with larger radii rotate faster, and this leads to the formation of the spiral structure. The number of the spiral windings increases with time. With a fixed number of points in the ensemble, at some large enough time it becomes simply impossible to resolve the spiral struc- ture.