An Introduction to the Use of Generalized Coördinates in Mechanics and Physics |
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actual forces actual motion angle angular velocity axis Bertrand's Theorem center of gravity components configuration conservative forces constant constraints coördinates 91 coördinates q corresponding cos² degrees of freedom differential equations displacement dt dt dx dy dz effective forces electromotive force equal equations of motion expressed in terms fixed geometrical equations Hamilton's principle Hamiltonian equations hanging particle Hence homogeneous quadratic hypothetical motion ignored coördinates impressed forces impulsive forces kinetic energy Lagrangian equation Lagrangian expression Lagrangian function liquid ma² mk² modified Lagrangian momenta momentum moving system number of degrees P₁ P₂ plane potential energy problem of Art q₁ rectangular coördinates remaining coördinates rigid body sin² single coördinate solved sphere string Theorem tractrix Whence wire ÿ² ат аф дак ду дяк Օզ ᎧᎢ
Popular passages
Page 21 - M, m are connected by a string which passes through a hole in a smooth horizontal plane, the mass m hanging vertically. Shew that M describes on the plane a curve whose differential equation is m\d2u mg 1 Prove also that the tension of the string is Mm M+m
Page 15 - ... spatial coordinates are included in the meaning of the term); but they must be such that, when the values of the q's at any instant are given, the state of the system at that instant is completely fixed. If n is chosen to be as small as possible, consistently with this condition being fulfilled, it is called the number of degrees of freedom of the system. The motion, or change of state, of the system at the same instant is then completely described by the n quantities ql, ..., qn.
Page 88 - ... of conservation of energy and angular momentum. According to the former, the sum of the kinetic energy and the potential energy is constant.
Page 15 - To the extent possible, the problem is decomposed using the facts that the total kinetic energy of the system is the sum of the kinetic energies of the...
Page 24 - If an inclined plane is just rough enough to insure the rolling of a homogeneous cylinder, show that a thin hollow drum will roll and slip, the rate of slipping at any instant being one half the linear velocity. Let x be the distance the axis of the cylinder has moved down the incline...
Page 111 - S(.Rr).0, (48) that is the sum of the moments of the impressed forces about the axis multiplied into the angular velocity of the body is equal to the rate of increase of the kinetic energy.
Page 85 - Cl (L + m dt] = 0 (6) where the Lagrangian function L is the difference between the kinetic energy T and the electric enthalpy H.10 The work term, W, includes the potential of all applied mechanical loads and the electrical charges. Thus, using the definition of electric enthalpy:10 L = TH...
Page 63 - OA-GO,^- (2) 28. Illustrative Examples, (a) A lamina of mass m rests on a smooth horizontal table and is acted on by an impulsive force of magnitude P in the plane of the lamina. Find the initial motion. Let...
Page 23 - Let h be the distance of the center of gravity from the axis, and k the radius of gyration of the body about a horizontal axis through the center of gravity. Then T = !2 (A
Page 6 - ... may be taken as its coordinates ql and qt, provided that by the equation of the given surface in rectangular coordinates and the equations formed by writing ql and qt equal to their values in terms of x, y, and z the last-named coordinates may be uniquely obtained as explicit functions of ql and q^.