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Linear equations are the most studied of differential equations and have the most systematic methods for solution. Hence, textbook examples typically center on problems that can be reduced to linear equations. In those problems where the force can be derived as the derivative of a potential function, Fx = −dV /dx, the potential may often be such that it can be expanded in a Taylor’s series, usually about some minimum point. If the series converges rapidly enough, it can be truncated after one or a few terms.
3 Equation: m¨ x + kx = 0 (harmonic oscillator) Method of solution: Substitution of the form Aemt Solution: x = c1 cos(ω0 t) + c2 sin(ω0 t) = A cos(ω0 t − α) where ω0 = conditions. 5 Equation: m¨ x + bx˙ + kx = 0 (particle oscillating in resistive medium) Method of solution: Substitution of general form, Aemt Solution: The general solution is: x = c1 eα1 t + c2 eα2 t where α1 = −(b/2m) + (b/2m)2 − k/m α2 = −(b/2m) − (b/2m)2 − k/m In those cases where α1 and α2 are complex numbers, the solution is oscillatory, but damped.
We are left with the result, ∂bi ∂gik ∂gjk ∂gij + − k = 2 j · bk . j i ∂q ∂q ∂q ∂q 32 4. We may now use our assumption that ∂bi = Γlij bl . ∂q j Then, ∂gik ∂gjk ∂gij + − k = 2[Γlij bl ] · bk = 2Γlij (bl · bk ) = 2Γlij glk . j i ∂q ∂q ∂q 5. It is tempting to solve for Γlij by dividing both sides of this expression by 2glk , but one must remember that the repeated index l indicates a sum of terms, not an isolated term. In fact, what we are dealing with here are twenty-seven equations, one for each of the combinations of i, j, and k!