Physics
Lagrangian Constraints
Lagrangian constraints in physics refer to restrictions placed on the motion of a system, typically due to external forces or conditions. These constraints are incorporated into the Lagrangian formalism, which is a mathematical framework used to describe the dynamics of physical systems. By accounting for these constraints, the Lagrangian approach allows for a more comprehensive and accurate analysis of the system's behavior.
Written by Perlego with AI-assistance
Related key terms
1 of 5
9 Key excerpts on "Lagrangian Constraints"
eBook - PDF- T. M. Helliwell, V. V. Sahakian(Authors)
- 2020(Publication Date)
- Cambridge University Press(Publisher)
PART II 6 Constraints and Symmetries An enormous advantage of using Lagrangian methods in mechanics is the simpli- fications that can occur when a system is constrained or if there are symmetries of some kind in the environment of the system. Constraints can be used to reduce the number of generalized coordinates so that solutions become more practicable. In this chapter we will illustrate this fact using the example of contact forces, and we will demonstrate the use of Lagrange multipliers to learn about the contact forces themselves. Constraints are also typically associated with the breaking of symmetries. Lagrangian mechanics allows us to efficiently explore the relationship between symmetries in a physical situation and dynamical quantities that are conserved. These properties are nicely summarized in a theorem by the German mathematician Emmy Noether (1882–1935), and provide us with deep insight into the physics – in addition to helping us make important technical simplifications while solving problems. We first discuss constraints and contact forces, and then symmetries and conservation laws. 6.1 Contact Forces A square block of mass M rests on a horizontal floor, as shown in Figure 6.1. Newtonian mechanics tells us that the block experiences two forces: the downward pull of gravity Mg and the upward push of the floor, called the normal force N. A static scenario implies that N = Mg, so the forces sum to zero and there is no acceleration. Hence, the normal force adjusts its strength as needed to counteract the gravitational pull Mg. If the balance succeeds, the block stays put on the floor. If, however, the normal push of the floor is not enough because the block is too heavy, the floor would disintegrate and the block would fall through. The normal force is an example of a contact force. It arises by virtue of the physical contact between two objects. Contact forces are always electromagnetic in origin.
eBook - PDF- Patrick Hamill(Author)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
4 For the particle on the tabletop, the constraint is z = constant. A pendulum bob is constrained by the string to remain a constant distance from the point of support. That is, the bob is constrained to the surface of a sphere whose radius equals the length of the string. A bar of soap slipping in a sink is constrained to the surface of the sink. If the sink is a hemispherical bowl of radius a, the equation of constraint is x 2 + y 2 + z 2 = a 2 . This relationship indicates that there are only two independent coordinates because the third coordinate (say z) is related to x and y through z = a 2 − x 2 − y 2 . Constraints are important to us because each constraint reduces the number of degrees of freedom by one. Going back to the Lagrangian procedure, recall that there is one Lagrange equation of motion for each coordinate; see Equation (4.9). It is obviously beneficial to reduce the number of equations. You can do this by using constraints to get rid of as many nonindependent coordinates as possible. When you describe the system in terms of independent coordinates (say q 1 ,q 2 , . . . ,q n ), you have minimized the number of Lagrange equations. These independent coordinates are referred to as generalized coordinates and are usually denoted by q i . A system with n degrees of freedom can be described in terms of the n generalized coordinates q 1 , . . . ,q n . For such a system, the Lagrangian will be a function of the generalized coordinates, the generalized velocities, and possibly the time. That is, L = L(q 1 ,q 2 , . . . q n ; ˙ q 1 , ˙ q 2 , . . . ˙ q n ; t). In this case, there will be n Lagrange equations of motion. 5 4.6 Generalized Momentum Consider again the problem of a mass m connected to a spring of constant k as illustrated in Figure 4.3. If the mass is moving in the x direction with speed ˙ x it has momentum p x = m ˙ x . We have seen that the Lagrangian for this system is L = 1 2 m ˙ x 2 − 1 2 kx 2 .
eBook - PDF- R. Douglas Gregory(Author)
- 2006(Publication Date)
- Cambridge University Press(Publisher)
. . , P N . The particles of S may have interconnections of various kinds (light strings, springs and so on) and also be subject to external connections and constraints. These could include features such as a particle being forced to remain on a fixed surface or suspended from a ∗ Joseph-Louis Lagrange (Giuseppe Lodovico Lagrangia), (1736–1813). Although Lagrange is often con- sidered to be French, he was in fact born in Turin, Italy and did not move to Paris until 1787. Lagrange had a long career in Turin and Berlin during which time he made major contributions to mechanics, fluid mechanics and the calculus of variations. His famous book M´ ecanique Analitique, published in Paris in 1788, is a definitive account of his contributions to mechanics. This work transformed mechanics into a branch of mathematical analysis. Perhaps to emphasise this, there is not a single diagram in the whole book! 324 Chapter 12 Lagrange’s equations and conservation principles FIGURE 12.1 The general mechanical system S consists of any number of particles { P i } (i = 1 . . . , N ). The typical particle P i has mass m i , position vector r i and velocity v i . F S i is the specified force and F C i the constraint force acting on P i . O v i r i P i F i S F i C S fixed point by a light inextensible string. The pendulum, the spinning top, the bicycle and the solar system are examples of mechanical systems. Unconstrained systems If the particles of S are free to move anywhere in space independently of each other then S is said to be an unconstrained system. In this special case, the equations of motion for S are simply Newton’s equations for the N individual particles. Suppose that the typical particle P i has mass m i , position vector r i and velocity v i . Then the equations of motion for the system S are m i ˙ v i = F i (i = 1 . . . , N ), where F i is the force acting on the particle P i .
eBook - PDF- John L. Bohn(Author)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
They were a useful means of guaranteeing that we dealt only with motions allowed by the constraints, but by now that’s automatically taken care of by the very choice of the generalized coordinates. By invoking the energies, rather than the applied forces, Lagrange’s equa-tions can be formulated without considering the applied forces and their directions in space. In this sense, all the geometry has been excised from the problem; we never have to consider the directions of any forces, applied or constraining. This is the point where mechanics takes the leap from being phrased geometrically, in terms of the directions of forces, to being phrased analytically, in terms of formulas relating energies. It is perhaps significant that, in the book where Lagrange invents analytical mechanics, he chose not to include a single diagram, as everything can be done analytically, in terms of formulas. 76 4 Lagrangian Mechanics In the next chapter we will concern ourselves with applications of Lagrange’s equations, to see both how they work and how they illuminate various problems in physics. But right now, let’s briefly return to the springulum problem from Chapter 3 . There we considered this problem as an application of d’Alembert’s principle. Now instead we note the following. The potential energy is given by contributions from gravitational potential energy, plus the energy of stretching the spring: V ( r , φ) = − mgr cos φ + 1 2 k ( r − l ) 2 . The kinetic energy is T ( r , ˙ r , ˙ φ) = 1 2 m ( ˙ r 2 + r 2 ˙ φ 2 ) . From this we construct the Lagrangian L = T − V = 1 2 m ( ˙ r 2 + r 2 ˙ φ 2 ) + mgr cos φ − 1 2 k ( r − l ) 2 . For the φ coordinate we have ∂ L ∂φ = − mgr sin φ d dt ∂ L ∂ ˙ φ = d dt mr 2 ˙ φ = mr 2 ¨ φ + 2 mr ˙ r ˙ φ . Setting these equal gives the equation of motion (assuming r is never = 0) r ¨ φ + 2 ˙ r ˙ φ = − g sin φ .
eBook - PDF- Donald T. Greenwood(Author)
- 2006(Publication Date)
- Cambridge University Press(Publisher)
There are, however, qualitative differences between ordinary kinematic con-straints and energy constraints. For a given configuration and time, an energy constraint is actually a constraint on the kinetic energy, and is represented in velocity space by a closed 116 Lagrange’s and Hamilton’s equations surface which surrounds the origin. This means that an energy constraint does not restrict the direction of ˙ q , but for any given direction, its magnitude is specified. On the other hand, we found that linear catastatic constraints restrict the possible directions of ˙ q but not the magnitude. As a natural application of velocity space methods, let us consider the derivation of Jourdain’s principle. We start with Lagrange’s principle as expressed in (2.33), namely, n i = 1 d dt ∂ T ∂ ˙ q i − ∂ T ∂ q i − Q i δ q i = 0 (2.292) where the δ q s satisfy the instantaneous constraints expressed in (2.280) or (2.281). Lagrange’s principle applies to holonomic or nonholonomic systems and leads to ( n − m ) differential equations of motion if there are m constraints. A comparison of (2.281) which constrains the δ q s in configuration space with (2.290) which constrains the δw s in velocity space shows that the corresponding δ q s and δw s are proportional. Hence, we can substitute δw i for δ q i in (2.292). We obtain n i = 1 d dt ∂ T ∂ ˙ q i − ∂ T ∂ q i − Q i δw i = 0 (2.293) This is Jourdain’s principle . The variation δ w is considered to take place in velocity space at an operating point P on the common intersection of the constraint planes. By choosing ( n − m ) independent sets of δw s which satisfy the constraints expressed in (2.290), one can use Jourdain’s principle to obtain ( n − m ) second-order differential equations of motion. In addition, there are m constraint equations, making a total of n equations to solve for the n q s. Example 2.12 Let us apply Jourdain’s principle to the nonholonomic system (Fig. 2.3) which was studied previously in Example 2.3 on page 83.
- Patrick Hamill(Author)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
Conse- quently, the term λf can be interpreted as the potential energy related to the force of constraint. Thus, the x i component of the reaction force is F i = − ∂(λf ) ∂x i = −λ ∂f ∂x i − ∂λ ∂x i f. But f = 0, so F i = −λ ∂f ∂x i . We conclude that a holonomic constraint is maintained by forces that can be derived from a scalar function. As we have seen, some non-holonomic constraints can also be dealt with using the Lagrangian λ-method; however, these forces (friction is an example) cannot be derived from a scalar function. Example 3.2 You have probably proved by using calculus (perhaps in a previ- ous mechanics course) that the curve formed by a hanging chain or heavy rope is a catenary. That problem can also be solved using variational techniques. The quantity to be minimized is the potential energy, V = V (y). The problem is to determine y = y(x) subject to the constraint that the length of the chain is a given constant. In variational terms, we want to determine y = y(x) given that δV = 0 and subject to the constraint x 2 x 1 ds = constant = length = l. Solution 3.2 The constraint can be expressed as l = x 2 x 1 1 + y 2 dx. 86 3 Lagrangian dynamics The potential energy of a length of chain ds is dV = (dm)gh = ρgyds. Dropping unnecessary constants, V = x 2 x 1 yds = x 2 x 1 y 1 + y 2 dx. The Lagrange λ-method leads to δ x 2 x 1 (y + λ) 1 + y 2 dx = 0. Determining the equation of the curve is left as an exercises. Exercise 3.6 The functional in the preceding example does not depend on x. Reformulate the problem as discussed in Section 2.2.2 and show that the curve is a catenary. Exercise 3.7 Consider a simple pendulum composed of a mass m con- strained by a wire of length l to swing in an arc. Assume r and θ are both variables. Obtain the Lagrange equations and using the λ-method, deter- mine the tension in the wire.
eBook - PDF- Thomas R. Kurfess(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
The main difference between the two approaches is in dealing with constraints. While Newton’s equations treat each rigid body separately and explicitly model the constraints through the forces required to enforce them, Lagrange and d’Alembert provided systematic procedures for eliminating the constraints from the dynamic equations, typically yielding a simpler system of equations. Constraints imposed by joints and other mechanical components are one of the defining features of robots so it is not surprising that the Lagrange’s formalism is often the method of choice in robotics literature. 5.2 Preliminaries The approach and the notation in this section follow [21], and we refer the reader to that text for additional details. A starting point in describing a physical system is the formalism for describing its motion. Since we will be concerned with robots consisting of rigid links, we start by describing rigid body motion. Formally, a rigid body O is a subset of R 3 where each element in O corresponds to a point on the rigid body. The defining property of a rigid body is that the distance between arbitrary two points on the rigid body remains unchanged as the rigid body moves. If a body-fixed coordinate frame B is attached to O , an 0-8493-1804-1/05 $ 0.00+ $ 1.50 © 2005 by CRC Press, LLC 5 -1 5 -2 Robotics and Automation Handbook arbitrary point p ∈ O can be described by a fixed vector p B . As a result, the position of any point in O is uniquely determined by the location of the frame B. To describe the location of B in space we choose a global coordinate frame S. The position and orientation of the frame B in the frame S is called the configuration of O and can be described by a 4 × 4 homogeneous matrix g SB : g SB = R SB d SB 0 1 , R SB ∈ R 3 × 3 , d SB ∈ R 3 , R T SB R SB = I 3 , det( R SB ) = 1 (5.1) Here I n denotes the identity matrix in R n × n .
eBook - PDFIntermediate Dynamics for Engineers
Newton-Euler and Lagrangian Mechanics
- Oliver M. O'Reilly(Author)
- 2020(Publication Date)
- Cambridge University Press(Publisher)
Our developments emphasized that these equations are equivalent to the Newtonian balance laws of linear momenta for each particle. This important feature enables us to confidently calculate the forces K = ∑ N i=1 F i · ∂ r i ∂ q K that appear on the right-hand side of Lagrange’s equations of motion: d dt ∂ T ∂ ˙ q K − ∂ T ∂ q K = K (R = 1, . . . , 3N) . We are also able to introduce integrable and nonintegrable constraints and the constraint forces associated with them. On a deeper level, if we wish to consider the system of particles as a single par- ticle moving on a configuration manifold M that is embedded in a 3N-dimensional Euclidean space, then Casey’s construction of the representative particle enables us to do this in a straightforward manner. In the next chapter, we consider examples in which Lagrange’s equations of motion for several systems of particles are established and discussed. 4.14 Exercises Exercise 4.1: What are the configuration manifolds M, generalized coordinates, and kinematical line elements ds of the following systems? (i) a particle attached to a fixed point by a spring; (ii) a particle attached to a fixed point by a rod of length (t); (iii) a harmonic oscillator consisting of a single particle; (iv) a planar double pendulum; (v) a spherical double pendulum. Exercise 4.2: Under which circumstances do constraint forces perform no work? 168 Lagrange’s Equations of Motion for a System of Particles Exercise 4.3: It is typical to assume that the kinetic energy of a system of particles is a positive definite function of the velocities ˙ q K . As T = 3N I =1 3N J=1 m 2 a IJ ˙ q I ˙ q J , this assumption is equivalent to stating that the mass matrix formed by the components m 2 a IJ is positive definite. 25 Using the spherical pendulum as an example, show that certain representations of T are not always positive definite.
eBook - PDF- Nivaldo A. Lemos(Author)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
1.5 Applications of Lagrange’s Equations In order to write down Lagrange’s equations associated with a given mechanical system, the procedure to be followed is rather simple. First, generalised coordinates q 1 , . . . , q n must be chosen. Next, the kinetic and potential energies relative to an inertial reference frame must be expressed exclusively in terms of the qs and ˙ qs, so that the Lagrangian L = T − V is also expressed only in terms of the generalised coordinates and velocities. Finally, all one has to do is compute the relevant partial derivatives of L, insert them into Eqs. (1.100) and the process of constructing the equations of motion for the system is finished. Let us see some examples of this procedure. Example 1.17 A bead slides on a smooth straight massless rod which rotates with constant angular velocity on a horizontal plane. Describe its motion by Lagrange’s formalism. Solution Let xy be the horizontal plane containing the rod and let us use polar coordinates to locate the bead of mass m (see Fig. 1.7). The two variables r, θ cannot be taken as generalised coordinates because θ is restricted to obey θ −ωt = 0, which is a holonomic constraint (ω is the rod’s constant angular velocity, supposed known). The system has 25 Applications of Lagrange’s Equations x m w r y q = w t Fig. 1.7 Bead on horizontal rotating rod. only one degree of freedom associated with the radial motion and we can choose q 1 = r as generalised coordinate. According to Example 1.16, the kinetic energy can be put in the form T = m 2 ( ˙ r 2 + r 2 ˙ θ 2 ) = m 2 ( ˙ r 2 + ω 2 r 2 ) , (1.111) where ˙ θ = ω has been used. Setting the plane of motion as the zero level of the gravitational potential energy, the Lagrangian for the system reduces to the kinetic energy: L = T − V = m 2 ( ˙ r 2 + ω 2 r 2 ) .
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
