# Generalized: Swedish translation, definition, meaning

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The D'Alembert-Lagrange  These n equations are known as the Euler–Lagrange equations. Some- times we only the generalized coordinates, and generalized forces conjugate to them,. Rayleigh dissipation function. Й = -. F. Х here Й is the component of the generalized force due to friction - gravity is incorporated into Д. The Lagrange equations  method presented here also allows us to nd the constraint forces.

∂qj ∂qj Example: Cart with Pendulum, Springs, and Dashpots Figure 6: The system contains a cart that has a spring (k) and a dashpot (c) attached to it. On the cart is a pendulum that has … What you do is to compute the work done, $W(q)$, by the force as a function of how $q$ (the generalized position) changes. Then the modification to the Euler-Lagrange equations is:  \frac d{dt}\left(\frac{\partial L}{\partial \dot q}\right) - \frac{\partial L}{\partial q} … 1992-01-01 That is, this leads to Euler-Lagrange equations of motion for the generalized forces. As discussed in chapter when holonomic constraint forces apply, it is possible to reduce the system to independent generalized coordinates for which Equation applies. In Leibniz proposed minimizing the time integral of his “vis viva", which equals That is, The differential/algebraic equations of motion of the system can be derived using Lagrange's equations with Lagrange multipliers.

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"=−EF" (& = −EF" $"$%& # " =− $F"$%& # " =− %& The generalized forces are defined as F i = (∂L/∂q i) These forces must be defined in terms of the Lagrangian rather than the Hamiltonian. The dynamics of a physical system are given by the system of n equations: (dp i /dt) = F i – If the generalized coordinate corresponds to an angle, for example, the generalized momentum associated with it will be an angular momentum • With this definition of generalized momentum, Lagrange’s Equation of Motion can be written as: j 0 j j j L d p q dt L p q ∂ − = ∂ ∂ = ∂ Just like Newton’s Laws, if we call a “generalized force” j L q ∂ ∂ • Lagrange’s Equation: sin qr dLL mq kq mg Q dt q q ∂∂ −=+− = ∂∂ && & θ • To handle friction force in the generalized force term, need to know the normal force Æ Lagrange approach does not indicate the value of this force. Fs mg Fd N Ff mq&& o Look at the free body diagram.

### Nationalbibliografin april 2006 - StudyLib

2.1 Generalized Coordinates and Forces . will be shown in the following sections, the Lagrange's equation derived from this new formalism The corresponding generalized forces of constraints can be. Related terms: Diffusion · Lagrange Equation · Brownian Particle · Entropy Production · Fluid Velocity · Generalized Flux · Hamiltonians  The only external force is gravity.

1 k m q j jk kk j d K K Fa dt q q O §·ww ¨¸ ©¹ww ¦ ( 1, , )kn (2) Here, K is the kinetic energy of the system, q k F is the generalized force associated with the generalized coordinate q k, O j is the Lagrange force balance that exists at each mass due to the deﬂection of the springs as was done in Lecture 19. The deﬂection of springs 1 and 3 are inﬂuenced by the boundary condition at either end of the slot; in this case the deﬂection is zero. The governing equations can also be obtained by direct application of Lagrange’s Equation. This equation, complete with the centrifugal force, m(‘+x)µ_2.
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So we see that eqs. (6.1) and (6.3) together say exactly the same thing that F = ma says, when using a Cartesian coordinate in one dimension (but this result is in fact quite general, as we’ll see in Section 6.4). Note that LAGRANGE’S EQUATIONS FOR IMPULSIVE FORCES .

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The generalized coordinate is the variable η=η(x,t). If the continuous system were three-dimensional, then we would have η=η(x,y,z,t), where x,y,z, and twould be completely independent of each other. We can generalize the Lagrangian for the three-dimensional system as. L=∫∫∫Ldxdydz, (4.160) Lagrange’s equation is d dt @L @q˙ j @L @q j = Q j where , and is the generalized velocity and is the nonconservative generalized force corresponding to the generalized coordinate j =1, 2,,n q˙ j = @q j Q @t j q j Lagrange’s equation from D’Alembert’s principle 7 78 $C$%9& − $C$%& %& # & = (& %& # & 7 78 $C$%9& − $C$%& −(& %& # & =0 D’Alembert’s principle in generalized coordinates becomes Since generalized coordinates %&are all independent each term in the summation is zero 7 78 $C$%9& − $C$%& =(& If all the forces are conservative, then ! "=−EF" (& = −EF" $"$%& # " =− $F"$%& # " =− %& The generalized forces are defined as F i = (∂L/∂q i) These forces must be defined in terms of the Lagrangian rather than the Hamiltonian. The dynamics of a physical system are given by the system of n equations: (dp i /dt) = F i – If the generalized coordinate corresponds to an angle, for example, the generalized momentum associated with it will be an angular momentum • With this definition of generalized momentum, Lagrange’s Equation of Motion can be written as: j 0 j j j L d p q dt L p q ∂ − = ∂ ∂ = ∂ Just like Newton’s Laws, if we call a “generalized force” j L q ∂ ∂ • Lagrange’s Equation: sin qr dLL mq kq mg Q dt q q ∂∂ −=+− = ∂∂ && & θ • To handle friction force in the generalized force term, need to know the normal force Æ Lagrange approach does not indicate the value of this force.