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Notes1 - Module I Review of Classical Mechanics 1...

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Module I Review of Classical Mechanics 1. Coordinates and Trajectories Two of the primary goals of classical physics is to describe the location of an object in space and how the object's location will change with time. A. Coordinates A set of numbers that uniquely describes mathematically the spatial location of an object is called the object's coordinates. Some important points about coordinates: a ) An object has an actual physical location independent of our describing it mathematically. Thus, no particular coordinate system is required by the physical world. We choose the coordinate system to simplify mathematical calculations. b) Between any two coordinate systems there are a unique set of equations (transformation equations) that allows us to take measurements made in one system to predict measurements made in the other system. c) Since there is no preferred coordinate system, the true mathematical expressions of the laws of the physical world should be unchanged in form under coordinate transformations (i.e. Invariant under coordinate changes). d) The origin for any coordinate system is arbitrary. Since the value for a coordinate depends on the origin chosen, you should specify your origin in every problem.

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B. Trajectories The set of values that an object's coordinates take over an interval of time is called a trajectory. The central problem in classical mechanics is to find the equations of motion and then solve these equations to find the object's trajectory (i.e. how the bodies position changes with time). We will see later that the very concept of a trajectory is impossible in quantum mechanics (Chapter 2). This leads us to the following questions (answered in Chapter 3): 1. What quantities should be used to describe a quantum system? 2. What do we use instead of trajectories to mathematically describe the time evolution of a quantum system? 3. What replaces the equations of motion in solving quantum systems? 4. What is the connection that links the results of quantum mechanics to those of classical physics? C. Constraints In many physics problems, the values of an object coordinates are limited in some manner. An example is a particle moving in the x-y plane. In this case, the particle's z-coordinate is limited to some constant values. Such a condition is known as a constraint condition. Constraints will reduce a system's degrees of freedom and the number of independent coordinates needed to specify the system. Holonomic and Non-Holonomic Constraint If the constraint condition can be written as an equation with the right hand side equal to zero then the constraint is said to be holonomic otherwise the constraint is said to be non-holonomic.
EXAMPLE: A particle moves in the x-y plane with z = 3. The constraint condition can be written as 0 3 z = - so it is a holonomic constraint. EXAMPLE: A particle slides along the outer surface of a cylinder of radius R before

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Notes1 - Module I Review of Classical Mechanics 1...

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