Phys 325 Spring 2011 Lecture 20 - Physics 325 Lecture 20...

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Physics 325 Lecture 20 Generalized Forces and Momenta In several of the examples that we have done in the past two lectures, we have come upon situations in which the derivative of the Lagrangian with respect to one of the generalized coordinates is zero 0 i L q (20.1) Given the Lagrange equations 0 ii L d L q dt q   Equation (20.1) leads to the conservation law 0 i dL dt q (20.2) or constant i L q In Cartesian coordinates we have i i LT xx mx p Thus 0 i L x leads to conservation of linear momentum along the i x direction. Now, in Cartesian coordinates we also have i LU F  which is the component of the force along the x i direction. In the situation where Equation (20.1) is satisfied, then, we have 0 i F and the component of the momentum along the i x direction is a constant in time. This is good.
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For the puck in a bowl problem from the last lecture, we found that the Lagrangian had no explicit dependence ( 0 L    ), and this led to conservation of the z -component of angular momentum, as expected from the vanishing of the z -component of the net torque on the system. Inspired by these discoveries, we define the generalized momentum, j p , associated with the generalized coordinate q i as j j L p q and we note that p j is conserved whenever 0 j Lq . In a conservative system, the potential energy is independent of the velocity. Therefore 0 and j j j U L T q q q  We now defined the generalized force . If the system is conservative, we have F i   U x i , so i i j i j j i i i j x UU q x q Q x F q   where we have defined the generalized force as j j i i i j U Q q x F q (20.3) Note that if j q is a length, then the partial derivative is dimensionless and j Q is a force. If q j is an angle, then the partial derivative has the dimensions of length ( x j ) and Q j is a torque (force length). Let’s see.
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In polar coordinates we have 12 xy yx x cos x sin Q = F F FF sin cos Q F F where 0 0 z x r y r xx r y r x i j k           Systems with Holonomic Constraints We consider a system of n particles ( 3n coordinates) with m equations of constraint:   , 0 1,.
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Phys 325 Spring 2011 Lecture 20 - Physics 325 Lecture 20...

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