as may also be seen directly from formula (49.11) and the expression (49.6) for the period. Conversely, if we express q and P, or any one-valued function F(p, q) of them, in terms of the canonical variables, then they remain unchanged when W increases by 2nd (with I constant). That is, any one-valued function F(p, q), when expressed in terms of the canonical variables, is a periodic function of W with period 2. $50. General properties of motion in S dimensions Let us consider a system with any number of degrees of freedom, executing a motion finite in all the co-ordinates, and assume that the variables can be completely separated in the Hamilton-Jacobi treatment. This means that, when the co-ordinates are appropriately chosen, the abbreviated action can be written in the form (50.1) as a sum of functions each depending on only one co-ordinate. Since the generalised momenta are Pi = aso/dqi = dSi/dqi, each function Si can be written (50.2) These are many-valued functions. Since the motion is finite, each co-ordinate can take values only in a finite range. When qi varies “there and back” in this range, the action increases by (50.3) where (50.4) the integral being taken over the variation of qi just mentioned. Let us now effect a canonical transformation similar to that used in 49, for the case of a single degree of freedom. The new variables are “action vari- ables” Ii and “angle variables” w(a(q (50.5) + It should be emphasised, however, that this refers to the formal variation of the co- ordinate qi over the whole possible range of values, not to its variation during the period of the actual motion as in the case of motion in one dimension. An actual finite motion of a system with several degrees of freedom not only is not in general periodic as a whole, but does not even involve a periodic time variation of each co-ordinate separately (see below). §50 General properties of motion in S dimensions 159 where the generating function is again the action expressed as a function of the co-ordinates and the Ii. The equations of motion in these variables are Ii = 0, w = de(I)/I, which give I=constant, (50.6) + constant. (50.7) We also find, analogously to (49.13), that a variation “there and back” of the co-ordinate qi corresponds to a change of 2n in Wi: Awi==2m (50.8) In other words, the quantities Wi(q, I) are many-valued functions of the co- ordinates: when the latter vary and return to their original values, the Wi may vary by any integral multiple of 2. This property may also be formulated as a property of the function Wi(P, q), expressed in terms of the co-ordinates and momenta, in the phase space of the system. Since the Ii, expressed in terms of P and q, are one-valued functions, substitution of Ii(p, q) in wi(q, I) gives a function wilp, q) which may vary by any integral multiple of 2n (including zero) on passing round any closed path in phase space. Hence it follows that any one-valued function F(P, q) of the state of the system, if expressed in terms of the canonical variables, is a periodic function of the angle variables, and its period in each variable is 2nr. It can be expanded as a multiple Fourier series: (50.9) ls== where l1, l2, ls are integers. Substituting the angle variables as functions of time, we find that the time dependence of F is given by a sum of the form (50.10) lg== Each term in this sum is a periodic function of time, with frequency (50.11) Since these frequencies are not in general commensurable, the sum itself is not a periodic function, nor, in particular, are the co-ordinates q and momenta P of the system. Thus the motion of the system is in general not strictly periodic either as a whole or in any co-ordinate. This means that, having passed through a given state, the system does not return to that state in a finite time. We can say, t Rotational co-ordinates (see the first footnote to 49) are not in one-to-one relation with the state of the system, since the position of the latter is the same for all values of differing by an integral multiple of 2nr. If the co-ordinates q include such angles, therefore, these can appear in the function F(P, q) only in such expressions as cos and sin , which are in one-to-one relation with the state of the system. 160 The Canonical Equations §50 however, that in the course of a sufficient time the system passes arbitrarily close to the given state. For this reason such a motion is said to be conditionally periodic. In certain particular cases, two or more of the fundamental frequencies Wi = DE/DI are commensurable for arbitrary values of the Ii. This is called degeneracy, and if all S frequencies are commensurable, the motion of the system is said to be completely degenerate. In the latter case the motion is evidently periodic, and the path of every particle is closed. The existence of degeneracy leads, first of all, to a reduction in the number of independent quantities Ii on which the energy of the system depends. If two frequencies W1 and W2 are such that (50.12) where N1 and N2 are integers, then it follows that I1 and I2 appear in the energy only as the sum n2I1+n1I2. A very important property of degenerate motion is the increase in the number of one-valued integrals of the motion over their number for a general non-degenerate system with the same number of degrees of freedom. In the latter case, of the 2s-1 integrals of the motion, only s functions of the state of the system are one-valued; these may be, for example, the S quantities I The remaining S - 1 integrals may be written as differences (50.13) The constancy of these quantities follows immediately from formula (50.7), but they are not one-valued functions of the state of the system, because the angle variables are not one-valued. When there is degeneracy, the situation is different. For example, the rela- tion (50.12) shows that, although the integral WIN1-W2N2 (50.14) is not one-valued, it is so except for the addition of an arbitrary integral multiple of 2nr. Hence we need only take a trigonometrical function of this quantity to obtain a further one-valued integral of the motion. An example of degeneracy is motion in a field U = -a/r (see Problem). There is consequently a further one-valued integral of the motion (15.17) peculiar to this field, besides the two (since the motion is two-dimensional) ordinary one-valued integrals, the angular momentum M and the energy E, which hold for motion in any central field. It may also be noted that the existence of further one-valued integrals leads in turn to another property of degenerate motions: they allow a complete separation of the variables for several (and not only one+) choices of the co- t We ignore such trivial changes in the co-ordinates as q1’ = q1’(q1), q2’ = 92’(92). §50 General properties of motion in S dimensions 161 ordinates. For the quantities Ii are one-valued integrals of the motion in co-ordinates which allow separation of the variables. When degeneracy occurs, the number of one-valued integrals exceeds S, and so the choice of those which are the desired I is no longer unique. As an example, we may again mention Keplerian motion, which allows separation of the variables in both spherical and parabolic co-ordinates. In §49 it has been shown that, for finite motion in one dimension, the action variable is an adiabatic invariant. This statement holds also for systems with more than one degree of freedom. Here we shall give a proof valid for the general case. Let X(t) be again a slowly varying parameter of the system. In the canonical transformation from the variables P, q to I, W, the generating function is, as we know, the action So(q, I). This depends on A as a parameter and, if A is a func- tion of time, the function So(q, I; X(t)) depends explicitly on time. In such a case the new Hamiltonian H’ is not the same as H, i.e. the energy E(I), and by the general formulae (45.8) for the canonical transformation we have H’ E(I)+asoldt = E(I)+A, where A III (aso/ad)r. Hamilton’s equations give ig = - (50.15) We average this equation over a time large compared with the fundamental periods of the system but small compared with the time during which the parameter A varies appreciably. Because of the latter condition we need not average 1 on the right-hand side, and in averaging the quantities we may regard the motion of the system as taking place at a constant value of A and therefore as having the properties of conditionally periodic motion described above. The action So is not a one-valued function of the co-ordinates: when q returns to its initial value, So increases by an integral multiple of 2I. The derivative A = (aso/ax), is a one-valued function, since the differentiation is effected for constant Ii, and there is therefore no increase in So. Hence A, expressed as a function of the angle variables Wr, is periodic. The mean value of the derivatives of such a function is zero, and therefore by (50.15) we have also which shows that the quantities Ii are adiabatic invariants. Finally, we may briefly discuss the properties of finite motion of closed systems with S degrees of freedom in the most general case, where the vari- ables in the Hamilton-Jacobi equation are not assumed to be separable. The fundamental property of systems with separable variables is that the integrals of the motion Ii, whose number is equal to the number of degrees + To simplify the formulae we assume that there is only one such parameter, but the proof is valid for any number. 162 The Canonical Equations §50 of freedom, are one-valued. In the general case where the variables are not separable, however, the one-valued integrals of the motion include only those whose constancy is derived from the homogeneity and isotropy of space and time, namely energy, momentum and angular momentum. The phase path of the system traverses those regions of phase space which are defined by the given constant values of the one-valued integrals of the motion. For a system with separable variables and S one-valued integrals, these conditions define an s-dimensional manifold (hypersurface) in phase space. During a sufficient time, the path of the system passes arbitrarily close to every point on this hypersurface. In a system where the variables are not separable, however, the number of one-valued integrals is less than S, and the phase path occupies, completely or partly, a manifold of more than S dimensions in phase space. In degenerate systems, on the other hand, which have more than S integrals of the motion, the phase path occupies a manifold of fewer than S dimensions. If the Hamiltonian of the system differs only by small terms from one which allows separation of the variables, then the properties of the motion are close to those of a conditionally periodic motion, and the difference between the two is of a much higher order of smallness than that of the additional terms in the Hamiltonian. PROBLEM Calculate the action variables for elliptic motion in a field U = -a/r. SOLUTION. In polar co-ordinates r, in the plane of the motion we have ‘max = 1+av(m2)E) Hence the energy, expressed in terms of the action variables, is E = It depends only on the sum Ir+I, and the motion is therefore degenerate; the two funda- mental frequencies (in r and in b) coincide. The parameters P and e of the orbit (see (15.4)) are related to Ir and I by p= Since Ir and I are adiabatic invariants, when the coefficient a or the mass m varies slowly the eccentricity of the orbit remains unchanged, while its dimensions vary in inverse propor- tion to a and to m. INDEX Acceleration, 1 Coriolis force, 128 Action, 2, 138ff. Couple, 109 abbreviated, 141 Cross-section, effective, for scattering, variable, 157 49ff. Additivity of C system, 41 angular momentum, 19 Cyclic co-ordinates, 30 energy, 14 integrals of the motion, 13 d’Alembert’s principle, 124 Lagrangians, 4 Damped oscillations, 74ff. mass, 17 Damping momentum, 15 aperiodic, 76 Adiabatic invariants, 155, 161 coefficient, 75 Amplitude, 59 decrement, 75 complex, 59 Degeneracy, 39, 69, 160f. Angle variable, 157 complete, 160 Angular momentum, 19ff. Degrees of freedom, 1 of rigid body, 105ff. Disintegration of particles, 41ff. Angular velocity, 97f. Dispersion-type absorption, 79 Area integral, 31n. Dissipative function, 76f. Dummy suffix, 99n. Beats, 63 Brackets, Poisson, 135ff. Eccentricity, 36 Eigenfrequencies, 67 Canonical equations (VII), 131ff. Elastic collision, 44 Canonical transformation, 143ff. Elliptic functions, 118f. Canonical variables, 157 Elliptic integrals, 26, 118 Canonically conjugate quantities, 145 Energy, 14, 25f. Central field, 21, 30 centrifugal, 32, 128 motion in, 30ff. internal, 17 Centrally symmetric field, 21 kinetic, see Kinetic energy Centre of field, 21 potential, see Potential energy Centre of mass, 17 Equations of motion (I), 1ff. system, 41 canonical (VII), 131ff. Centrifugal force, 128 integration of (III), 25ff. Centrifugal potential, 32, 128 of rigid body, 107ff. Characteristic equation, 67 Eulerian angles, 110ff. Characteristic frequencies, 67 Euler’s equations, 115, 119 Closed system, 8 Collisions between particles (IV), 41ff. Finite motion, 25 elastic, 44ff. Force, 9 Combination frequencies, 85 generalised, 16 Complete integral, 148 Foucault’s pendulum, 129f. Conditionally periodic motion, 160 Frame of reference, 4 Conservation laws (II), 13ff. inertial, 5f. Conservative systems, 14 non-inertial, 126ff. Conserved quantities, 13 Freedom, degrees of, 1 Constraints, 10 Frequency, 59 equations of, 123 circular, 59 holonomic, 123 combination, 85 Co-ordinates, 1 Friction, 75, 122 cyclic, 30 generalised, 1ff. Galilean transformation, 6 normal, 68f. Galileo’s relativity principle, 6 163 164 Index General integral, 148 Mechanical similarity, 22ff. Generalised Molecules, vibrations of, 70ff. co-ordinates, 1ff. Moment forces, 16 of force, 108 momenta, 16 of inertia, 99ff. velocities, 1ff. principal, 100ff. Generating function, 144 Momentum, 15f. angular, see Angular momentum Half-width, 79 generalised, 16 Hamiltonian, 131f. moment of, see Angular momentum Hamilton-Jacobi equation, 147ff. Multi-dimensional motion, 158ff. Hamilton’s equations, 132 Hamilton’s function, 131 Hamilton’s principle, 2ff. Newton’s equations, 9 Holonomic constraint, 123 Newton’s third law, 16 Nodes, line of, 110 Impact parameter, 48 Non-holonomic constraint, 123 Inertia Normal co-ordinates, 68f. law of, 5 Normal oscillations, 68 moments of, 99ff. Nutation, 113 principal, 100ff. principal axes of, 100 One-dimensional motion, 25ff., 58ff. tensor, 99 Oscillations, see Small oscillations Inertial frames, 5f. Oscillator Infinite motion, 25 one-dimensional, 58n. Instantaneous axis, 98 space, 32, 70 Integrals of the motion, 13, 135 Jacobi’s identity, 136 Particle, 1 Pendulums, 11f., 26, 33ff., 61, 70, 95, Kepler’s problem, 35ff. 102f., 129f. Kepler’s second law, 31 compound, 102f. Kepler’s third law, 23 conical, 34 Kinetic energy, 8, 15 Foucault’s, 129f. of rigid body, 98f. spherical, 33f. Perihelion, 36 Laboratory system, 41 movement of, 40 Lagrange’s equations, 3f. Phase, 59 Lagrangian, 2ff. path, 146 for free motion, 5 space, 146 of free particle, 6ff. Point transformation, 143 in non-inertial frame, 127 Poisson brackets, 135ff. for one-dimensional motion, 25, 58 Poisson’s theorem, 137 of rigid body, 99 Polhodes, 117n. for small oscillations, 58, 61, 66, 69, 84 Potential energy, 8, 15 of system of particles, 8ff. centrifugal, 32, 128 of two bodies, 29 effective, 32, 94 Latus rectum, 36 from period of oscillation, 27ff. Least action, principle of, 2ff. Potential well, 26, 54f. Legendre’s transformation, 131 Precession, regular, 107 Liouville’s theorem, 147 L system, 41 Rapidly oscillating field, motion in, 93ff. Reactions, 122 Mass, 7 Reduced mass, 29 additivity of, 17 Resonance, 62, 79 centre of, 17 in non-linear oscillations, 87ff. reduced, 29 parametric, 80ff. Mathieu’s equation, 82n. Rest, system at, 17 Maupertuis’ principle, 141 Reversibility of motion, 9 Index 165 Rigid bodies, 96 Space angular momentum of, 105ff. homogeneity of, 5, 15 in contact, 122ff. isotropy of, 5, 18 equations of motion of, 107ff. Space oscillator, 32, 70 motion of (VI), 96ff. Rolling, 122 Time Rotator, 101, 106 homogeneity of, 5, 13ff. Rough surface, 122 isotropy of, 8f. Routhian, 134f. Top Rutherford’s formula, 53f. asymmetrical, 100, 116ff. “fast”, 113f. spherical, 100, 106 Scattering, 48ff. symmetrical, 100, 106f., 111f. cross-section, effective, 49ff. Torque, 108 Rutherford’s formula for, 53f. Turning points, 25, 32 small-angle, 55ff. Two-body problem, 29 Sectorial velocity, 31 Separation of variables, 149ff. Uniform field, 10 Similarity, mechanical, 22ff. Sliding, 122 Variation, 2, 3 Small oscillations, 22, (V) 58ff. first, 3 anharmonic, 84ff. Velocity, 1 damped, 74ff. angular, 97f. forced, 61ff., 77ff. sectorial, 31 free, 58ff., 65ff. translational, 97 linear, 84 Virial, 23n. non-linear, 84ff. theorem, 23f. normal, 68 Smooth surface, 122 Well, potential, 26, 54f. 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