By A. Mous
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Extra info for A Short Introduction to Theoretical Mechanics
The particle does not separate from the dome. ) 9. Consider an inclined plane consisting of a right-triangular shaped block that is free to slide on its bottom along a frictionless horizontal surface. If the incline slopes toward the right, the left-hand edge of the block is vertical. Define a coordinate X that keeps track of the position of this vertical side. Now, put another block on the inclined plane so that it slides down the plane (without friction) under the influence of gravity. Keep track of the second block’s position with a coordinate x measured from the top of the inclined plane.
A function only of position and expressible as Fx = − dV , dx show for the one-dimensional case that a formal solution of the equation of motion is, x dx t − t0 = , x0 (2/m)[E − V (x)] where x0 is the value of x at t0 and E is the total energy of the particle, E = T +V. 2. How long would it take an object of mass m that is released at rest from the position of the earth, re , to fall into the sun? The sun represents a fixed center of force that attracts the object according to the inverse square law F (r) = −k/r 2 .
QED. 17 Theorem Theorem: ∂bi /∂q j = ∂bj /∂q i . Proof: This symmetry of pattern merely reflects the fact that the order in which partial derivatives are taken does not matter: ∂bi ∂ ∂r ∂ ∂r ∂bj = j i = i j = . j ∂q ∂q ∂q ∂q ∂q ∂q i QED. Since each of these nine partial derivatives, ∂bi /∂q j are themselves vectors, we can express each as some linear combination of either the unit vectors, reciprocal basis vectors or basis vectors. We choose to express them as a linear combination of basis vectors, ∂bi = Γkij bk .
A Short Introduction to Theoretical Mechanics by A. Mous