Constant of the motion

Regardless of the nature of the intramolecular dynamics of the reactant A, there are two constants of the motion in a nnimolecular reaction, i.e. the energy E and the total angular momentum j. The latter ensures the rotational quantum number J is fixed during the nnimolecular reaction and the quantum RRKM rate constant is specified as k E, J). 

In other words, if we look at any phase-space volume element, the rate of incoming state points should equal the rate of outflow. This requires that be a fiinction of the constants of the motion, and especially Q=Q i). Equilibrium also implies d(/)/dt = 0 for any /. The extension of the above equations to nonequilibriiim ensembles requires a consideration of entropy production, the method of controlling energy dissipation (diennostatting) and the consequent non-Liouville nature of the time evolution [35]. 

Here the friction coefficient is completely detemiined by the instantaneous values of the coordhiates and momenta. It is easy to see that the kinetic energy jc = Y.ia constant of the motion ... 

One property of the exact trajectory for a conservative system is that the total energy is a constant of the motion. [12] Finite difference integrators provide approximate solutions to the equations of motion and for trajectories generated numerically the total energy is not strictly conserved. The exact trajectory will move on a constant energy surface in the 61V dimensional phase space of the system defined by. 

In our opening remarks in this section, we mentioned that an analogy with dynamical Ising models can only be carried so far since there is no known conserved energy for the Life rule. However, Schulman and Seiden were able to discover a possible constant of the motion , namely a normalized entropy. 

Equation (9-392) together with (9-394) and (9-395) are the proofs of the assertions that x is the position operator in the Foldy-Wouthuysen representation.16 (Note also that x commutes with /J the sign of the energy.) We further note that in the FTP-representation the operators x x p and Z commute with SFW separately and, hence, are constants of the motion. In the F W-representation the orbital and spin angular momentum operators are thus separately constants of the motion. The fact that... 

The total momentum operator P, as well as the total angular momentum operator M, commute with H and hence are constants of the motion. However, they do not commute with another, their commutator being equal to... 

We first inquire as to the constants of the motion in this situation. Since h is invariant under the group of spatial rotations, and under spatial inversions, the total angular momentum and the parity operator are constants of the motion. The total angular momentum operator is... 

We have exhibited two constants of the motion that can be diagonalized simultaneously with h. A third constant of motion is the parity operator... 

The theory behind body-fixed representations and the associated angular momentum function expansions of the wavefunction (or wave packet) in terms of bases parameterized by the relevant constants of the motion and approximate constants of the motion is highly technical. Some pertinent results will simply be stated. The two good constants of the motion are total angular momentum, J, and parity, p = +1 or 1. An approximate constant of the motion is K, the body-fixed projection of total angular momentum on the body-fixed axis. For simplicity, we will restrict attention to the helicity-decoupled or centrifugal sudden (CS) approximation in which K can be assumed to be a constant of the motion. In terms of aU its components, and the iteration number k, the real wave packet is taken to be [21]... 

It is now shown how the abrupt changes in the eigenvalue distribution around the central critical point relate to changes in the classical mechanics, bearing in mind that the analog of quantization in classical mechanics is a transformation of the Hamiltonian from a representation in the variables pR, p, R, 0) to one in angle-action variables (/, /e, Qr, 0) such that the transformed Hamiltonian depends only on the actions 1r, /e) [37]. Hamilton s equations diR/dt = (0///00 j), etc.) then show that the actions are constants of the motion, which are related to the quantum numbers by the Bohr correspondence principle [23]. In the present case,... 

Just as K, the Hamiltonian // depends on APX and AQ only through the action variable 7, which is a constant of the motion. 

Since [P, H] = 0, P is a constant of the motion, which means that a system of particles represented by either ips or will keep that symmetry for all time. The particles of Nature that fall into the two classes with either symmetrical or antisymmetrical states, are known as bosons and fermions respectively. 

In order for an ensemble to represent a system in equilibrium, the density of phase must remain constant in time. This means that Liouville s equation is satisfied, which requires that g is constant with respect to the coordinates q and p, or that g = g(a), where a = a(q,p) is a constant of the motion. Constants of motion are properties of an isolated system which do not change with time. Familiar examples are energy, linear momentum and angular momentum. For constants of motion H,a = 0. Hence, if g = g a) and a is a constant of motion, then... 

Equation (31) is known as Heisenberg s equation of motion and is the quantum-mechanical analogue of the classical equation (17). The commutator of two quantum-mechanical operators multiplied by 2mfh) is the analogue of the classical Poisson bracket. In quantum mechanics a dynamical quantity whose operator commutes with the Hamiltonian, [A, H] = 0, is a constant of the motion. 

For conservative systems with time-independent Hamiltonian the density operator may be defined as a function of one or more quantum-mechanical operators A, i.e. g= tp( A). This definition implies that for statistical equilibrium of an ensemble of conservative systems, the density operator depends only on constants of the motion. The most important case is g= [Pg.463]

For each Lie algebra, one can construct a set of operators, called invariant (or Casimir, 1931) operators after the name of the physicist who first introduced them in connection with the rotation group. These operators play a very important role since they are associated with constants of the motion. They are defined as those operators that commute with all the elements of the algebra... 

Thus the Casimir operator of SO(3) is the familiar square of the angular momentum (a constant of the motion when the Hamiltonian is invariant under rotation). One can show that SO(3) has only one Casimir operator, and it is thus an algebra of rank one. Multiplication of C by a constant a, which obviously satisfies (2.7), does not count as an independent Casimir operator, nor do powers of C (i.e., C2,...) count. Casimir operators can be constructed directly from the algebra. This construction has been done for the large majority of algebras used in physics. 

The problem of time evolution for a Hamiltonian bilinear in the generators (Levine, 1982) has been extensively discussed. The proposed solutions include the use of variational principles (Tishby and Levine, 1984), mean-field self-consistent methods (Meyer, Kucar, and Cederbaum, 1988), time-dependent constants of the motion (Levine, 1982), and numerous others, which we hope to discuss in detail in a sequel to this volume. 

The Vlasov-Newton equation has many steady solutions describing a self-gravitating cluster. This is easy to show in the spherically symmetric case (the situation we shall restrict in this work, except for a few remarks at the end of this section). If one assumes a given r(r) in the steady state, the general steady solution of Eq. (4) is a somewhat arbitrary function of the constants of the motion of a single mass in this given external held, namely a funchon/(E, I ) where niE is the total energy of a star in a potenhal (r) such that r(r) = —(r/r) [d r)/dr] and where — (r.v) is the square of the... 

As an example take a gas in a cylindrical vessel. In addition to the energy there is one other constant of the motion the angular momentum around the cylinder axis. The 6A/-dimensional phase space is thereby reduced to subshells of 6N-2 dimensions. Consider a small sub volume in the vessel and let Y(t) be the number of molecules in it. According to III.2 Y(t) is a stochastic function, with range n = 0,1,2,. .., N. Each value Y = n delineates a phase cell ) one expects that Y(t) is a Markov process if the gas is sufficiently dilute and that pi is approximately a Poisson distribution if the subvolume is much smaller than the vessel. 

Moreover, since the equilibrium distribution is a function of the constants of the motion, which must also be even functions of the velocities, one has... 

As pointed out by Solov ev,8 if the magnetic field is low enough that the coulomb force is dominant, then there exist approximate constants of the motion in addition to Lz and parity. The first, A, is given by8-10... 

A very important property is that the magnetic and electric lines of an electromagnetic knot are the level curves of the scalar fields 4>(r, t) and 0(r, f), respectively. Another is that the magnetic and the electric helicities are topological constants of the motion, equal to the common Hopf index of the corresponding pair of dual maps constant with dimensions of action times velocity. 

Furthermore, it was shown in Section II.C that the semisum of the two helicities = (/im + he) = na, which we call the electromagnetic helicity, is a constant of the motion for any standard electromagnetic field in empty space ... 

The linear Maxwell equations appear in the model as the linearization by change of variables of nonlinear equations that refer to the scalars < ),0. This introduces a subtle form of nonlinearity that we call hidden nonlinearity. For this reason, the linearity of Maxwell s equations is compatible with the existence of topological constants of the motion. 

One of these topological constants of the motion is the electromagnetic helicity, defined as the semisum of the magnetic and electric helicities, which is equal to the linking number of the force lines... 

By analogy with non-exchanging spin systems the superoperators which commute with both the super-Hamiltonian and the superoperator T in composite Liouville space may be called the constants of the motion. In some instances there may be additional constants of the motion which result from the conservation of some molecular symmetry in the exchange, from the magnetic equivalence of some nuclei, and from weak spin-spin coupling. (15, 52) For example,... 

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