Classical Mechanics - An undergraduate text by R. Douglas Gregory

By R. Douglas Gregory

Gregory's Classical Mechanics is an important new textbook for undergraduates in arithmetic and physics. it's a thorough, self-contained and hugely readable account of a subject matter many scholars locate tough. The author's transparent and systematic kind promotes a great knowing of the topic; every one notion is inspired and illustrated through labored examples, whereas challenge units offer lots of perform for knowing and procedure. machine assisted difficulties, a few appropriate for initiatives, also are integrated. The publication is based to make studying the topic effortless; there's a usual development from middle themes to extra complicated ones and difficult issues are taken care of with specific care. A topic of the publication is the significance of conservation rules. those look first in vectorial mechanics the place they're proved and utilized to challenge fixing. They reappear in analytical mechanics, the place they're proven to be concerning symmetries of the Lagrangian, culminating in Noether's theorem.

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In other words, r, θ are vector functions of the scalar variable θ . We will now evaluate the two derivatives d r/dθ , d θ /dθ . These will be needed when we derive the formulae for the velocity and acceleration of P in polar co-ordinates. First we expand† r, θ in terms of the Cartesian basis vectors {i, j }. 6) θ = − sin θ i + cos θ j . 7) Since r, θ are now expressed in terms of the constant vectors i, j , the differentiations with respect to θ are simple and give ∗ If this is not clear, sketch the directions of the polar unit vectors for P in a few different positions.

Computer assisted problems 2 . 22 Dog chasing a hare; another pursuit problem. 15 shows a dog with position vector r D and velocity v D chasing a hare with position vector r H and velocity v H . The dog’s strategy is to run directly towards the current position of the hare. Given the motion of the hare and the speed of the dog, what path does the dog follow? Since the dog runs directly towards the hare, its velocity v D must satisfy rH − rD vD . = H D v |r − r D | In terms of the position vector of the dog relative to the hare, given by R = r D − r H , this equation becomes R R˙ = − v D − v H .

The hare then runs along the positive x-axis and is chased by the dog. Show that the hare gets caught if v D > v H , but when v D = v H the dog always misses (unless he starts directly in the path of the hare). This remarkable result can be proved analytically. 6 Problems 49 (ii) The hare runs in a circle (like the lion problem). In this case, with v D = v H , the dog seems to miss no matter where he starts. Try some examples of your own and see if you can find interesting paths taken by the dog.

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