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Lagrange's Equations

Lagrange's Equations - S Widnall 16.07 Dynamics Fall 2009...

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S. Widnall 16.07 Dynamics Fall 2009 Version 3.0 Lecture L20 - Energy Methods: Lagrange’s Equations The motion of particles and rigid bodies is governed by Newton’s law. In this section, we will derive an alternate approach, placing Newton’s law into a form particularly convenient for multiple degree of freedom systems or systems in complex coordinate systems. This approach results in a set of equations called Lagrange’s equations. They are the beginning of a complex, more mathematical approach to mechanics called analytical dynamics. In this course we will only deal with this method at an elementary level. Even at this simplified level, it is clear that considerable simplification occurs in deriving the equations of motion for complex systems. These two approaches–Newton’s Law and Lagrange’s Equations–are totally compatible. No new physical laws result for one approach vs. the other. Many have argued that Lagrange’s Equations, based upon conservation of energy, are a more fundamental statement of the laws governing the motion of particles and rigid bodies. We shall not enter into this debate. Derivation of Lagrange’s Equations in Cartesian Coordinates We begin by considering the conservation equations for a large number (N) of particles in a conservative force field using cartesian coordinates of position x i . For this system, we write the total kinetic energy as M 1 2 T = m i x ˙ (1) 2 i . n =1 where M is the number of degrees of freedom of the system. For particles traveling only in one direction, only one x i is required to define the position of each particle, so that the number of degrees of freedom M = N . For particles traveling in three dimensions, each particle requires 3 x i coordinates, so that M = 3 N . The momentum of a given particle in a given direction can be obtained by di ff erentiating this expression with respect to the appropriate x i coordinate. This gives the momentum p i for this particular particle in this coordinate direction. T = p i (2) x ˙ i The time derivative of the momentum is d T = m i x ¨ i (3) dt x ˙ i 1
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For a conservative force field, the force on a particle is given by the derivative of the potential at the particle position in the desired direction. V F i = x i (4) From Newton’s law we have F i = dp i (5) dt Equating like terms from our manipulations on kinetic energy and the potential of a conservative force field, we write d T V dt x ˙ i = x i (6) Now we make use of the fact that T = 0 (7) x i and V = 0 (8) x ˙ i Using these results, we can rewrite Equation (6) as dt d ( T x ˙ i V ) ( T x i V ) = 0 (9) We now define L = T V : L is called the Lagrangian. Equation (9) takes the final form: Lagrange’s equations in cartesian coordinates.
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Lagrange's Equations - S Widnall 16.07 Dynamics Fall 2009...

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