# Momenta Sentence Examples

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.AdvertisementLet 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**.AdvertisementFor 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.