Three variables

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Lagrangian mechanics

Action and Symmetry, Ch Multipliers, non-holonomic constraints and forces outside the potential

Problem

Three variables (x,y,z)(x,y,z), all pure numbers, are restricted in turn by each of two constraints on their changes: (i)x dy−y dx+dz=0,(ii)yz dx+xz dy+2xy dz=0.\text{(i)}\quad x\,\mathrm{d}y-y\,\mathrm{d}x+\mathrm{d}z=0, \qquad\qquad \text{(ii)}\quad yz\,\mathrm{d}x+xz\,\mathrm{d}y+2xy\,\mathrm{d}z=0. ((a)) For each constraint, write it as cx dx+cy dy+cz dz=0c_x\,\mathrm{d}x+c_y\,\mathrm{d}y+c_z\,\mathrm{d}z=0, compute c⋅(∇×c)\mathbf{c}\cdot(\nabla\times\mathbf{c}), and say whether the constraint is equivalent to an equation between the coordinates. ((b)) For the constraint that passes the test, find an integrating factor μ\mu and a function ff with μc=∇f\mu\mathbf{c}=\nabla f, and state the equation between coordinates it is equivalent to. ((c)) A point obeying constraint (ii) starts at (x,y,z)=(1.0,1.0,1.0)(x,y,z)=(1.0,1.0,1.0) and later has x=2.0x=2.0, y=1.0y=1.0, with zz still positive. Find zz there.

Related theory

Browse the problem library