# Publications - Computer Vision Laboratory

Curriculum vitae — Professor Dr. Mats I. Pettersson - Blekinge

(4) From (3), it is clear that λ is the outward pointing normal force acting on the particle. The Kane Lagrange equations of … Thus, are the components of the force acting on the first particle, the components of the force acting on the second particle, etc. Using Equation ( 593 ), we can also write. (595) The above expression can be rearranged to give. (596) where.

(6.13) is the tangential F = ma equation, complete with the Coriolis force, ¡2mx_µ_. But never mind about this now. We’ll deal with rotating frames in Chapter 10.2 Remark: After writing down the E-L equations, it is always best to double-check them by trying Review of Lagrange’s equations from D’Alembert’s Principle, Examples of Generalized Forces a way to deal with friction, and other non-conservative forces . If virtual work done by the constraint forces is (=) (from eq.-1), − = D’Alembert’s principle of virtual work where the Lagrange multiplier term accounts for holonomic constraint forces, and FEXCqi includes all additional forces not accounted for by the scalar potential U, or the Lagrange multiplier terms FHCqi. The constraint forces can be included explicitly as generalized … generalized force corresponding to the generalized coordinate q j. Where does it come from?

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Thus, generalized coordinates replace constraint forces from Newtonian mechanics and can be used to easily calculate constrained equations of motion. So, in principle, If we choose our generalized coordinates wisely, we could obtain equations of motion (which implicitly already contain the constraints of the problem) without even using the Lagrange multiplier method.

### Symmetries and conservation laws - DiVA

˙ ψ ez = −. 1. R. ˙x ez =. Euler-Lagrange Equations Recall Newton-Euler Equation for a single rigid body: Generalized force fi and coordinate rate ˙qi are dual to each other in the   q& is its derivative, Qi is the i-th generalized force and UTL. −= is a scalar function called Lagrangian. Clearly, the Lagrangian L is the difference between. 20 Nov 2003 The standard form of Lagrange's equations of motion, ignoring the V and the gradient of the potential V is assumes to be a generalized force.

D’Alembert’s Principle becomes, Lagrange’s Equations! Generalized coordinates qj are independent! Assume forces are conservative jj0 j jj dT T Qq forces also is more convenient by without considering constrained forces.
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(6.4) Thus, we have the familiar ˙pσ = Fσ, also known as Newton’s second law. Note, however, that the {qσ} are generalized coordinates, so pσ may not have dimensions of momentum, nor Fσ of force.

In other words, a generalized force need not necessarily have the dimensions MLT-2. Before going on to describe Lagrange’s equations of motion, let us remind ourselves how we solve problems in mechanics using Newton’s law of motion. nor Fσ of force. For example, if the generalized coordinate in question is an angle φ, then the corresponding generalized momentum is the angular momentum about the axis of φ’s rotation, and the generalized force is the torque.
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