Hamiltonian systems momenta

O. Gonzales and J. C. Simo. On the stability of symplectic and energy-momentum conserving algorithms for nonlinear Hamiltonian systems with symmetry. Comp. Meth. App. Mech. Engin., 134 197, 1994. 

Figure 3. Spectrum of Lyapunov exponents of a dynamical system of 33 hard spheres of unit diameter and mass at unit temperature and density 0.001. The positive Lyapunov exponents are superposed to minus the negative ones showing that the Lyapunov exponents come in pairs L, —L, as expected in Hamiltonian systems. Eight Lyapunov exponents vanish because the system has four conserved quantities, namely, energy and the three components of momentum and because of the pairing rule. The total number of Lyapunov exponents is equal to 6 x 33 = 198.

The starting point for the study of the symmetry of the superconducting state is the pairing Hamiltonian. For applications to condensed matter systems it is convenient to write this Hamiltonian in momentum space... 

Here s the simplest instance of a Hamiltonian system. Let H p,q) be a smooth, real-valued function of two variables. The variable q is the generalized coordinate and p is the conjugate momentum. (In some physical settings, H could also depend explicitly on time t, but we ll ignore that possibility.) Then a system of the form... 

Various numerical methods have been proposed for collisional Hamiltonian systems [136, 176, 184, 263, 348, 352]. Typically, these schemes rely on the Verlet method to propagate the system between collisions, with collisions detected either (i) by checking for overlap at the end of the step, (ii) checking for overlap during the step, or (iii) approximating the time to collision before the step. Collisions lead to momentum exchange between particles according the principle outlined above. 

An example where the smooth probability density in Euclidean space is not quite the right one is in the setting of conservative (Hamiltonian) systems such as our N-body molecular system, since the evolution is restricted by invariants. The most obvious of these is the energy which we know to be a constant of motion. Therefore we need to work not on open subsets of the phase space R" of our differential equations, but on lower dimensional submanifolds embedded within the phase space, e.g. the energy surface. It will be necessary to assume a density that is defined over the submanifold of constant energy. If other invariants are present, such as fixed total momentum, the discussion would need to be modified to reflect this fact. 

Gonzalez, O., Simo, J. On the stability of symplectic and eneigy-momentum algorithms for non-linear Hamiltonian systems with symmetry. Comput. Methods AppL Mech. Eng. 134, 197-222 (1995). doi 10.1016/0045-7825(96X)1009-2... 

Close inspection of equation (A 1.1.45) reveals that, under very special circumstances, the expectation value does not change with time for any system properties that correspond to fixed (static) operator representations. Specifically, if tlie spatial part of the time-dependent wavefiinction is the exact eigenfiinction ). of the Hamiltonian, then Cj(0) = 1 (the zero of time can be chosen arbitrarily) and all other (O) = 0. The second tenn clearly vanishes in these cases, which are known as stationary states. As the name implies, all observable properties of these states do not vary with time. In a stationary state, the energy of the system has a precise value (the corresponding eigenvalue of //) as do observables that are associated with operators that connmite with ft. For all other properties (such as the position and momentum). 

Initially, we neglect tenns depending on the electron spin and the nuclear spin / in the molecular Hamiltonian //. In this approximation, we can take the total angular momentum to be N(see (equation Al.4.1)) which results from the rotational motion of the nuclei and the orbital motion of the electrons. The components of. m the (X, Y, Z) axis system are given by ... 

For a coupled spin system, the matrix of the Liouvillian must be calculated in the basis set for the spin system. Usually this is a simple product basis, often called product operators, since the vectors in Liouville space are spm operators. The matrix elements can be calculated in various ways. The Liouvillian is the conmuitator with the Hamiltonian, so matrix elements can be calculated from the commutation rules of spin operators. Alternatively, the angular momentum properties of Liouville space can be used. In either case, the chemical shift temis are easily calculated, but the coupling temis (since they are products of operators) are more complex. In section B2.4.2.7. the Liouville matrix for the single-quantum transitions for an AB spin system is presented. 

In the presence of a phase factor, the momentum operator (P), which is expressed in hyperspherical coordinates, should be replaced [53,54] by (P — h. /r ) where VB creates the vector potential in order to define the effective Hamiltonian (see Appendix C). It is important to note that the angle entering the vector potential is shictly only identical to the hyperangle <]> for an A3 system. 

The presence of two angular momenta has as a consequence that only their sum, representing the total angular momentum in the case considered, necessary commutes with the Hamiltonian of the system. Thus only the quantum number K, associated with the sum, N, of and Lj,... 

In a many-eleetron system, one must eombine the spin funetions of the individual eleetrons to generate eigenfunetions of the total Sz = i Sz(i) ( expressions for Sx = i Sx(i) and Sy =Zi Sy(i) also follow from the faet that the total angular momentum of a eolleetion of partieles is the sum of the angular momenta, eomponent-by-eomponent, of the individual angular momenta) and total S2 operators beeause only these operators eommute with the full Hamiltonian, H, and with the permutation operators Pij. No longer are the individual S2(i) and Sz(i) good quantum numbers these operators do not eommute with Pij. 

To construct Nose-Hoover constant-temperature molecular dynamics, an additional coordinate, s, and its conjugate momentum p, are introduced. The Hamiltonian of the extended system of the N particles plus extended degrees of freedom can be expressed... 

The total space of system coordinates consists of a tagged coordinate Q (conjugate momentum P) and a set of mass-scaled bath coordinates q (conjugate momenta p). The Hamiltonian reads... 

For n 0 the above equation must coincide with the equation of motion, hence P0 is the hamiltonian of the system. Similarly one deduoes that the P, (i = 1,2,3) must be the components of the total momentum operator. We call the operator PB the displacement operators for spaoe time translations in the sense that for an arbitrary operator F(x) which is a function of tfi(x) and A (x)... 

Equations (56) and (57) give six constrains and define the BF-system uniquely. The internal coordinates qk(k = 1,2, , 21) are introduced so that the functions satisfy these equations at any qk- In the present calculations, 6 Cartesian coordinates (xi9,X29,xi8,Xn,X2i,X3i) from the triangle Og — H9 — Oi and 15 Cartesian coordinates of 5 atoms C2,C4,Ce,H3,Hy are taken. These 21 coordinates are denoted as qk- Their explicit numeration is immaterial. Equations (56) and (57) enable us to express the rest of the Cartesian coordinates (x39,X28,X38,r5) in terms of qk. With this definition, x, ( i, ,..., 21) are just linear functions of qk, which is convenient for constructing the metric tensor. Note also that the symmetry of the potential is easily established in terms of these internal coordinates. This naturally reduces the numerical effort to one-half. Constmction of the Hamiltonian for zero total angular momentum J = 0) is now straightforward. First, let us consider the metric. 

The Fourier transformation method enables us to immediately write the momentum space equations as soon as the SCF theory used to describe the system under consideration allows us to build one or several effective Fock Hamiltonians for the orbitals to be determined. This includes a rather large variety of situations ... 

Accordingly, the quantum-mechanical Hamiltonian operator H for this system is proportional to the square of the angular momentum operator U-... 

Assuming that the pj (t) and Qj (t) can be interpreted as a TS trajectory, which is discussed later, we can conclude as before that ci = ci = 0 if the exponential instability of the reactive mode is to be suppressed. Coordinate and momentum of the TS trajectory in the reactive mode, if they exist, are therefore unique. For the bath modes, however, difficulties arise. The exponentials in Eq. (35b) remain bounded for all times, so that their coefficients q and q cannot be determined from the condition that we impose on the TS trajectory. Consequently, the TS trajectory cannot be unique. The physical cause of the nonuniqueness is the presence of undamped oscillations, which cannot be avoided in a Hamiltonian setting. In a dissipative system, by contrast, all oscillations are typically damped, and the TS trajectory will be unique. 

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