MechanicalSystems - 1.6 Mechanical Systems 1.6 49...

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1.6 Mechanical Systems 49 1.6 Mechanical Systems Mechanics provides an important and large class of systems for both moti- vating the ideas of dynamical systems and to which the ideas of dynamical systems apply. We saw some simple examples of Newtonian mechanical systems in the introductory section. To illustrate and motivate the introduction of addi- tional structures governing mechanical systems, we will develop two exam- ples as we go along. Example 1. Consider a particle moving in Euclidean 3 space, R 3 , subject to potential forces. As a variant of this example, also consider an object whose mass is time-varying and is subject to external forces (e.g. a rocket burning fuel and generating propulsion). Example 2. Consider a circular hoop, rotating along the z - axis with a certain Fxed angular velocity ω , as shown in ±igure 1.6.1 . Consider a particle with mass m moving in this hoop. R Figure 1.6.1. A particle moving in a rotating hoop. Confguration Space Q . The conFguration space of a mechanical sys- tem is a space whose points determine the spatial positions of the system. Although this space is generally parametrized by generalized coordinates, denoted ( q 1 ,...,q n ), the space Q itself need not be an Euclidean space. It can be rather thought as a conFguration manifold . Since we will be working
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50 Introduction with coordinates for this space, it is ok to think of Q as Euclidean space for purposes of this section. However, the distinction turns out to be an important general issue. Example 1. Here Q = R 3 since a point in space determines where our sys- tem is; the coordinates are simply standard Euclidean coordinates: ( x,y,z )= ( q 1 ,q 2 3 ). Example 2. Here Q = S 1 , the circle of radius R since the position of the particle is completely determined by where it is in the hoop. Note that the hoop’s position in space is already determined as it has a prescribed angular velocity. The Lagrangian L ( q,v ) . The Lagrangian is a function of 2 n variables, if n is the dimension of the conFguration space. These variables are the positions and velocities of the mechanical system. We write this as follows L ( q 1 ,...,q n ,v 1 ,...,v n )) = L ( q 1 n , ˙ q 1 ,..., ˙ q n ) . (1.6.1) At this stage, the ˙ q i s are not time derivatives yet (since L is just a function of 2 n variables, but as soon as we introduce time dependence so that the q i s are functions of time, then we will require that the v i s to be the time derivatives of the q i s. In many (but not all) of our examples we will set L = K E - P E , i.e. as the di±erence between the kinetic and potential energies. The sign in front of the potential energy in this deFnition of L is very important. ²or instance, consider a particle with constant mass, moving in a potential Feld V , which generates a force F = -∇ V . We will see shortly that the equation F = ma is a particular case of the basic equations associated to L , namely the Euler Lagrange equations, therefore necessarily we need the minus sign before the P E . There are other deeper reasons why the minus sign that involve relativistic invariance. 4 Recall also that the
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This note was uploaded on 01/04/2012 for the course CDS 140A taught by Professor Marsden during the Fall '09 term at Caltech.

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MechanicalSystems - 1.6 Mechanical Systems 1.6 49...

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