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This is called a transformation, going from one set of dependent variablesto another set of independent variables. Use independent variables, eliminate dependent coordi-nates. Forces are not known beforehand, and must be obtained from solution.įor holonomic constraints introduce generalized coordinates. Equations of motion are not all independent, because coordinates are nolonger all independentĢ. scleronomous constraints: equations of contraint are NOT explicitly de-pendent on time.example: bead on rigid curved wire fixed in spaceġ. rheonomous constraints: time is an explicit variable.example: bead onmoving wireĢ. Nonholonomic constraints: think walls of a gas container, think particleplaced on surface of a sphere because it will eventually slide down part ofthe way but will fall off, not moving along the curve of the sphere.ġ. , t) = 0, thinka particle constrained to move along any curve or on a given surface. Holonomic constraints: think rigid body, think f(r1, r2, r3. For a rigid body the internalforces do no work and the internal potential energy remains constant.
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For rigid bodiesthe internal potential energy will be constant. The term on the right is called the internal potential energy. If the external and internal forces are both derivable from potentials it ispossible to define a total potential energy such that the total energy T + V isconserved.
#GOLDSTEIN CLASSICAL MECHANICS SOLUTIONS PDF PLUS#
obtained ifall the mass were concentrated at the center of mass, plus the K.E. Kinetic energy, like angular momentum, has two parts: the K.E. W12 = T2 T1where T is the total kinetic energy of the system: T = 12 If the center of mass is at rest wrt the origin then theangular momentum is independent of the point of reference. Total angular momentum about a point O is the angular momentum of mo-tion concentrated at the center of mass, plus the angular momentum of motionabout the center of mass.
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Linear Momentum Conservation requires weak law of action and reaction.Īngular Momentum Conservation requires strong law of action and reaction. Torque is also called the moment of the external force about the given point.Ĭonservation Theorem for Total Angular Momentum: L is constant in timeif the applied torque is zero. Then the time derivative of angular momentumis the total external torque: The strong law of action and reaction is the condition that the internal forcesbetween two particles, in addition to being equal and opposite, also lie alongthe line joining the particles. This is how rockets work in space.Ĭonservation Theorem for the Linear Momentum of a System of Particles:If the total external force is zero, the total linear momentum is conserved. Internal forces that obeyNewtons third law, have no effect on the motion of the center of mass. It is called the weak law of action and reaction.Ĭenter of mass moves as if the total external force were acting on the entiremass of the system concentrated at the center of mass. Newtons third law of motion, equal and opposite forces, does not hold for allforces. The Conservation Theorem for the Angular Momentum of a Particle statesthat angular momentum, L, is conserved if the total torque T, is zero. The Conservation Theorem for the Linear Momentum of a Particle statesthat linear momentum, p, is conserved if the total force F, is zero. The change is -V.Įnergy Conservation Theorem for a Particle: If forces acting on a particleare conservative, then the total energy of the particle, T + V, is conserved. This quantity is potential energy.Work is now V1 V2. To express workin a way that is independent of the path taken, a change in a quantity thatdepends on only the end points is needed. Independence of W12 on the particular path implies that thework done around a closed ciruit is zero:į dr = 0If friction is present, a system is non-conservative.į = V (r).The capacity to do work that a body or system has by viture of is position T = r F.Torque is the time derivative of angular momentum:į dr.In most cases, mass is constant and work simplifies to:ĢThe work is the change in kinetic energy.Ī force is considered conservative if the work is the same for any physicallypossible path. Newtons second law of motion holds in a reference frame that is inertial orGalilean. In most cases, mass is constant and force is simplified: Classical refers to the con-tradistinction to quantum mechanics. Classical mechanics incorporates special relativity.