A Short Introduction to Theoretical Mechanics by A. Mous

By A. Mous

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49 This important superposition principle is valid for systems which obey linear differential equations. The superposition principle breaks down for non-linear differential equations. 9 Equation: gij q¨j + kij q j = 0 (multi-particle, harmonic oscillation) Method of solution: If the kinetic energy and potential energy in a conservative, multi-particle system can be expressed as: 1 T = gij q˙i q˙j 2 1 V = kij q i q j 2 then the above differential equations follow. The equations can also be written in matrix form, ¨ + KQ = 0 GQ A trial solution of the form Q = X cos(ωt − α) can be substituted.

2 Covariant Components of Acceleration 1. To obtain the covariant components of acceleration, we lower the index using Al = glk Ak . Al = glk Ak = glk q¨k + glk Γkij q˙i q˙j . 2. We may use the expression which we have just derived for the Christoffel symbols to write, ∂gin ∂gjn ∂gij 1 + − n glk Γkij = glk g kn 2 ∂q j ∂q i ∂q Observe, glk g kn = δln , 35 . so that we can write, 1 ∂gin ∂gjn ∂gij glk Γkij = δln + − n 2 ∂q j ∂q i ∂q = 1 2 ∂gil ∂gjl ∂gij + − ∂q j ∂q i ∂q l . 3. Thus, Al = glk q¨k + 1 2 ∂gil ∂gjl ∂gij + − ∂q j ∂q i ∂q l q˙i q˙j .

Here there are two masses. In such a case, the Lagrangian is written simply as the sum of the the two independent kinetic energies of the two objects minus the sum of the two independent potential energies. 1 Solutions to differential equations Newton’s Second Law of motion in one dimension, F = m¨ x, is a second order differential equation. In the simplest cases or in the lowest order of approximation, this equation will sometimes reduce to a linear equation. Linear equations are the most studied of differential equations and have the most systematic methods for solution.

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