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momenta

P. be the corresponding momenta.

If the system is supposed to obey the conservation of energy and to move solely under its own internal forces, the changes in the co-ordinates and momenta can be found from the Hamiltonian equations aE aE qr = 49 - 1 57., gr where q r denotes dg r ldt, &c., and E is the total energy expressed as a function of pi, qi,.

Thus after a time dt the values of the coordinates and momenta of the small group of systems under consideration will lie within a range such that pi is between pi +pidt and pi +dp,+(pi+ap?dpi) dt „ qi +gidt „ qi+dqi+ (qi +agLdgi) dt, Thus the extension of the range after the interval dt is dp i (i +aidt) dq i (I +?gidt).

Since the values of the co-ordinates and momenta at any instant during the motion may be treated as " initial " values, it is clear that the " extension " of the range must remain constant throughout the whole motion.

This result at once disposes of the possibility of all the systems acquiring any common characteristic in the course of their motion through a tendency for their co-ordinates or momenta to concentrate about any particular set, or series of sets, of values.

Let us imagine that the systems had the initial values of their co-ordinates and momenta so arranged that the number of systems for which the co-ordinates and momenta were within a given range was proportional simply to the extension of the range.

Then the result proves that the values of the coordinates and momenta remain distributed in this way throughout the whole motion of the systems. Thus, if there is any characteristic which is common to all the systems after the motion has been in progress for any interval of time, this same characteristic must equally have been common to all the systems initially.

Ow are any momenta or functions of the co-ordinates and momenta or co-ordinates alone which are subject only to the condition that they do not enter into the coefficients a 1, a 2, &c.

The aggregate amount of these pressures is clearly the sum of the momenta, normal to the boundary, of all molecules which have left dS within a time dt, and this will be given by expression (pp), integrated with respect to u from o to and with respect to v and w from - oo to +oo, and then summed for all kinds of molecules in the gas.

It is fair in dealing with Schelling's development to take into account the indications of his own opinion regarding its more significant momenta.

Kinetics of a System of Discrete Particles.The momenta of the several particles constitute a system of localized vectors which, for purposes of resolving and taking moments, may be reduced like a system of forces in statics (~ 8).

Secondly, we have an angular momentum whose components are ~{m(y~z3)}, ~lm(z~xb)1, ~{m(xi?yi~)}, (2) these being the sums of the moments of the momenta of the several particles about the respective axes.

At the instant t+t5t the momenta of the system are equivalent to a linear momentum represented by a localized vector ~(m).(U+U) in a line through G tangential to the path of G, together with a certain angular momentum.

In the time ~5t a certain impulse is given to the first particle in the direction (say) from P to Q, whilst an equal and opposite impulse is given to the second in the, direction from Q to P. Since these impulses produce equal and opposite momenta in the two particles, the resultant lineal momentumof the system is unaltered.

For in time t the mutual action between two particles at P and Q produces equal and opposite momenta in the line PQ, and these will have equal and opposite moments about the fixed axis.

by 7 (5); hence, if ~, u, v be now used to denote the component angular momenta about the co-ordinate axes, we have X=~tm(pyqx)ym(rxpz)zl, with two similar formulae, or x= ApHqGr=~, 1

L, M, N) denotes the system of extraneous forces referred (like the momenta) to the mass-centre as base, the co-ordinate axes being of course fixed in direction.

j To prove these, we may take fixed axes Ox, Oy, Oz coincident with the moving axes at time t, and compare the linear and angular momenta E+E, ~ ~ ?~+~X, u+u, v+~v relative to the new position of the axes, Ox, Oy, Oz at time t+t with the original momenta ~, ~ ~, A, j~i, v relative to Ox, Oy, Oz at time t.

of which the last two express the constancy of the momenta ~s, p. Hence AdA sin OcosO~+v sinoI=MghsinO, ~ 2

angular momentum DWBA (Distorted Wave Born Approximation) cross sections for a range of transferred angular momenta.

Restrictions: In the Born approximation angular momenta lar momenta l 10 are used.

Restrictions: In the Born approximation angular momenta lass="ex">angular momenta l 10 are used.

momentumegion is characterized by small values of the transverse momenta of the particles produced in the collision.

momentumpose to generate rapidity variable instead of longitudinal momenta, or some kind of intermediate masses.

P. be the corresponding momenta.

If the system is supposed to obey the conservation of energy and to move solely under its own internal forces, the changes in the co-ordinates and momenta can be found from the Hamiltonian equations aE aE qr = 49 - 1 57., gr where q r denotes dg r ldt, &c., and E is the total energy expressed as a function of pi, qi,.

Thus after a time dt the values of the coordinates and momenta of the small group of systems under consideration will lie within a range such that pi is between pi +pidt and pi +dp,+(pi+ap?dpi) dt „ qi +gidt „ qi+dqi+ (qi +agLdgi) dt, Thus the extension of the range after the interval dt is dp i (i +aidt) dq i (I +?gidt).

Since the values of the co-ordinates and momenta at any instant during the motion may be treated as " initial " values, it is clear that the " extension " of the range must remain constant throughout the whole motion.

This result at once disposes of the possibility of all the systems acquiring any common characteristic in the course of their motion through a tendency for their co-ordinates or momenta to concentrate about any particular set, or series of sets, of values.

Let us imagine that the systems had the initial values of their co-ordinates and momenta so arranged that the number of systems for which the co-ordinates and momenta were within a given range was proportional simply to the extension of the range.

Then the result proves that the values of the coordinates and momenta remain distributed in this way throughout the whole motion of the systems. Thus, if there is any characteristic which is common to all the systems after the motion has been in progress for any interval of time, this same characteristic must equally have been common to all the systems initially.

Ow are any momenta or functions of the co-ordinates and momenta or co-ordinates alone which are subject only to the condition that they do not enter into the coefficients a 1, a 2, &c.

The aggregate amount of these pressures is clearly the sum of the momenta, normal to the boundary, of all molecules which have left dS within a time dt, and this will be given by expression (pp), integrated with respect to u from o to and with respect to v and w from - oo to +oo, and then summed for all kinds of molecules in the gas.

It is fair in dealing with Schelling's development to take into account the indications of his own opinion regarding its more significant momenta.

Kinetics of a System of Discrete Particles.The momenta of the several particles constitute a system of localized vectors which, for purposes of resolving and taking moments, may be reduced like a system of forces in statics (~ 8).

Secondly, we have an angular momentum whose components are ~{m(y~z3)}, ~lm(z~xb)1, ~{m(xi?yi~)}, (2) these being the sums of the moments of the momenta of the several particles about the respective axes.

At the instant t+t5t the momenta of the system are equivalent to a linear momentum represented by a localized vector ~(m).(U+U) in a line through G tangential to the path of G, together with a certain angular momentum.

In the time ~5t a certain impulse is given to the first particle in the direction (say) from P to Q, whilst an equal and opposite impulse is given to the second in the, direction from Q to P. Since these impulses produce equal and opposite momenta in the two particles, the resultant lineal momentumof the system is unaltered.

For in time t the mutual action between two particles at P and Q produces equal and opposite momenta in the line PQ, and these will have equal and opposite moments about the fixed axis.

by 7 (5); hence, if ~, u, v be now used to denote the component angular momenta about the co-ordinate axes, we have X=~tm(pyqx)ym(rxpz)zl, with two similar formulae, or x= ApHqGr=~, 1

L, M, N) denotes the system of extraneous forces referred (like the momenta) to the mass-centre as base, the co-ordinate axes being of course fixed in direction.

j To prove these, we may take fixed axes Ox, Oy, Oz coincident with the moving axes at time t, and compare the linear and angular momenta E+E, ~ ~ ?~+~X, u+u, v+~v relative to the new position of the axes, Ox, Oy, Oz at time t+t with the original momenta ~, ~ ~, A, j~i, v relative to Ox, Oy, Oz at time t.

of which the last two express the constancy of the momenta ~s, p. Hence AdA sin OcosO~+v sinoI=MghsinO, ~ 2

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