Lagrangian Mechanics

Anh Le van is Professor at the University of Nantes, France, and teaches structural mechanics at the Faculty of Science. His research at the GeM laboratory (Institute for Research in Civil and Mechanical Engineering) focuses on membrane structures in large deformations.Rabah Bouzidi is Associate Professor at the University of Nantes, France. He also teaches structural mechanics, and researches membrane structures in large deformations at the GeM laboratory.

• Preface xi1 Kinematics 11.1 Observer – Reference frame 11.2 Time 21.2.1 Date postulate 21.2.2 Date change postulate 21.3 Space 31.3.1 Physical space 31.3.2 Mathematical space 41.3.3 Position postulate 41.3.4 Typical operations on the mathematical space E 61.3.5 Position change postulate 71.3.6 The common reference frame R0 91.3.7 Coordinate system of a reference frame 121.3.8 Fixed point and fixed vector in a reference frame 141.4 Derivative of a vector with respect to a reference frame 151.5 Velocity of a particle 171.6 Angular velocity 171.7 Reference frame defined by a rigid body: Rigid body defined by a reference frame 191.8 Point attached to a rigid body: Vector attached to a rigid body 191.9 Velocities in a rigid body 201.10 Velocities in a mechanical system 221.11 Acceleration 241.11.1 Acceleration of a particle 241.11.2 Accelerations in a mechanical system 241.12 Composition of velocities and accelerations 241.12.1 Composition of velocities 241.12.2 Composition of accelerations 251.13 Angular momentum: Dynamic moment 252 Parameterization and Parameterized Kinematics 272.1 Position parameters 272.1.1 Position parameters of a particle 272.1.2 Position parameters for a rigid body 282.1.3 Position parameters for a system of rigid bodies 322.2 Mechanical joints 332.3 Constraint equations 332.4 Parameterization 372.5 Dependence of the rotation tensor of the reference frame on the retained parameters 392.6 Velocity of a particle 412.7 Angular velocity 442.8 Velocities in a rigid body 452.9 Velocities in a mechanical system 472.10 Parameterized velocity of a particle 482.10.1 Definition 482.10.2 Practical calculation of the parameterized velocity 492.11 Parameterized velocities in a rigid body 502.12 Parameterized velocities in a mechanical system 512.13 Lagrange’s kinematic formula 522.14 Parameterized kinetic energy 543 Efforts 573.1 Forces 573.2 Torque 593.3 Efforts 603.4 External and internal efforts 613.4.1 External effort 613.4.2 Internal effort 623.5 Given efforts and constraint efforts 623.6 Moment field 644 Virtual Kinematics 674.1 Virtual derivative of a vector with respect to a reference frame 674.2 Virtual velocity of a particle 704.3 Virtual angular velocity 754.4 Virtual velocities in a rigid body 814.4.1 The virtual velocity field (VVF) associated with a parameterization 814.4.2 Virtual velocity field (VVF) in a rigid body 824.5 Virtual velocities in a system 834.5.1 VVF associated with a parameterization 834.5.2 VVF on each rigid body of a system 844.5.3 Virtual velocity of the center of mass 844.6 Composition of virtual velocities 854.6.1 Composition of virtual velocities of a particle 854.6.2 Composition of virtual angular velocities 864.6.3 Composition of VVFs in rigid bodies 874.7 Method of calculating the virtual velocity at a point 885 Virtual Powers 915.1 Principle of virtual powers 915.2 VP of efforts internal to each rigid body 925.3 VP of efforts 925.4 VP of efforts exerted on a rigid body 945.4.1 General expression 945.4.2 VP of zero moment field efforts exerted upon a rigid body 945.4.3 Dependence of the VP of efforts on the reference frame 945.5 VP of efforts exerted on a system of rigid bodies 955.5.1 General expression 955.5.2 Dependence of the VP of the efforts on the reference frame 965.5.3 VP of zero moment field efforts exerted on a system of rigid bodies 965.5.4 VP of inter-efforts between the rigid bodies of a system 965.5.5 The specific case of the inter-efforts between two rigid bodies 975.6 Summary of the cases where the VV and VP are independent of the reference frame 985.7 VP of efforts expressed as a linear form of the qi∗995.8 Potential 1015.8.1 Definition 1015.8.2 Examples of potential 1015.9 VP of the quantities of acceleration 1086 Lagrange’s Equations 1116.1 Choice of the common reference frame R0 1116.2 Lagrange’s equations 1126.3 Review and the need to model joints 1156.4 Existence and uniqueness of the solution 1166.5 Equations of motion 1186.6 Example 1 1186.7 Example 2 1206.7.1 Reduced parameterization 1206.7.2 Total parameterization 1216.7.3 Comparing the two parameterizations 1226.8 Example 3 1236.8.1 Independent parameterization 1246.8.2 Total parameterization 1266.8.3 Comparing the two parameterizations 1276.9 Working in a non-Galilean reference frame 1277 Perfect Joints 1317.1 VFs compatible with a mechanical joint 1327.1.1 Definition 1327.1.2 Generalizing the definition of a compatible VVF 1347.1.3 Example 1 for VVFs compatible with a mechanical joint 1357.1.4 Example 2 1367.1.5 Example 3: particle moving along a hoop rotating around a fixed axis 1377.2 Invariance of the compatible VVFs with respect to the choice of the primitive parameters 1397.2.1 Context 1397.2.2 Relationships between the real quantities resulting from the two parameterizations 1407.2.3 Relationships between the virtual quantities resulting from the two parameterizations 1417.2.4 Identity between the VVFs associated with the two parameterizations and compatible with a mechanical joint 1437.2.5 Example 1457.3 Invariance of the compatible VVFs with respect to the choice of the retained parameters 1487.3.1 Context 1487.3.2 Relationships between the real quantities resulting from the two parameterizations 1507.3.3 Relationships between the virtual quantities resulting from the two parameterizations 1517.3.4 Identity between the VVFs associated with the two paramaterizations and compatible with a mechanical joint 1537.3.5 Example 1 1557.3.6 Example 2 1567.3.7 Example 3: particle moving along a hoop rotating around a fixed axis 1567.4 Invariance of the compatible VVFs with respect to the choice of the parameterization 1577.5 Perfect joints 1587.5.1 Definition of a perfect joint 1587.5.2 Example 1 1607.5.3 Example 2 1617.5.4 Example 3 1627.5.5 Example 4 1657.6 Example: a perfect compound joint 1677.6.1 Perfect combined joint 1687.6.2 Superimposition of two perfect elementary joints 1698 Lagrange’s Equations in the Case of Perfect Joints 1738.1 Lagrange’s equations in the case of perfect joints and an independent parameterization 1748.1.1 Lagrange’s equations 1748.1.2 Review 1748.1.3 Particular case 1758.2 Lagrange’s equations in the case of perfect joints and in the presence of complementary constraint equations 1758.2.1 Lagrange’s equations with multipliers 1768.2.2 Practical calculation using Lagrange’s multipliers 1788.2.3 Review 1808.2.4 Remarks 1808.3 Example: particle on a rotating hoop 1818.3.1 Independent parameterization 1828.3.2 Reduced parameterization no. 1 1838.3.3 Reduced parameterization no. 2 1848.3.4 Calculation of the engine torque 1858.4 Example: rigid body connected to a rotating rod by a spherical joint (no. 1) 1878.5 Example: rigid body connected to a rotating rod by a spherical (joint no. 2) 1898.5.1 Total parameterization 1898.5.2 Independent parameterization 1918.6 Example: rigid body subjected to a double contact 1918.6.1 Preliminary analysis 1928.6.2 Independent parameterization 1948.6.3 Reduced parameterization 1969 First Integrals 1999.1 Painlevé’s first integral 1999.1.1 Painlevé’s lemma 1999.1.2 Painlevé’s first integral 2019.2 The energy integral: conservative systems 2039.2.1 Energy considerations in addition to the energy integral 2049.3 Example: disk rolling on a suspended rod 2059.4 Example: particle on a rotating hoop 2089.5 Example: a rigid body connected to a rotating rod by a spherical joint (no. 1) 2089.5.1 First integrals via Newtonian mechanics 2099.6 Example: rigid body connected to a rotating rod by a spherical joint (no. 2) 2099.7 Example: rigid body subjected to a double contact 2109.7.1 Using Newtonian mechanics to find a first integral 21110 Equilibrium 21310.1 Definitions 21310.1.1 Absolute equilibrium 21310.1.2 Parametric equilibrium 21410.2 Equilibrium equations 21510.2.1 List of equations and unknowns 21810.2.2 The explicit presence of time in equilibrium equations 21810.3 Equilibrium equations in the case of perfect joints and independent parameterization 21910.3.1 List of equations and unknowns 22010.4 Equilibrium equations in the case of perfect joints and in the presence of complementary constraint equations 22110.4.1 List of equations and unknowns 22210.5 Stability of an equilibrium 22310.6 Example: equilibrium of a jack 22410.7 Example: equilibrium of a lifting platform 22510.8 Example: equilibrium of a rod in a gutter 22710.9 Example: existence of ranges of equilibrium positions 22910.10 Example: relative equilibrium with respect to a rotating reference frame 23010.11 Example: equilibrium in the presence of contact inequalities 23210.12 Calculating internal efforts 23610.13 Example: internal efforts in a truss 23610.13.1 Tension force in bar A’A 23610.13.2 Tension force in bar B’B 23810.14 Example: internal efforts in a tripod 24011 Revision Problems 24311.1 Equilibrium of two rods 24311.1.1 Analysis using Newton’s law 24411.2 Equilibrium of an elastic chair 24511.3 Equilibrium of a dump truck 24611.4 Equilibrium of a set square 24811.5 Motion of a metronome 25011.5.1 Equation of motion 25011.5.2 First integral 25111.5.3 The case of small oscillations 25111.5.4 Case where the base S may slide without friction on the table T 25111.5.5 First integral 25211.6 Analysis of a hemispherical envelope 25211.6.1 Studying the static equilibrium 25311.6.2 Studying the oscillatory motion 25411.7 A block rolling on a cylinder 25411.7.1 Studying the equilibrium 25611.7.2 Dynamic analysis 25711.7.3 Case of small oscillations 25711.8 Disk welded to a rod 25711.8.1 First parameterization 25811.8.2 First integral 25911.8.3 Second parameterization 25911.8.4 Third parameterization 26111.9 Motion of two rods 26111.9.1 Equations of motion 26211.9.2 Relative equilibrium 26411.10 System with a perfect wire joint 26411.10.1 Lagrange’s equations 26511.10.2 First integral 26711.11 Rotating disk–rod system 26811.11.1 Independent parameterization 26811.11.2 Total parameterization 26911.11.3 Engine torque 27111.12 Dumbbell 27111.12.1 Equations set 27211.12.2 First integrals 27311.12.3 Analysis with specific initial conditions 27511.12.4 Relative equilibrium 27511.13 Dumbbell under engine torque 27511.13.1 Independent parameterization 27611.13.2 Painlevé’s first integral 27711.13.3 Total parameterization 27711.14 Rigid body with a non-perfect joint 27811.14.1 Equations set 27911.14.2 Solving the equations set 28111.14.3 Power of the engine and work dissipated through friction 281Appendix 1 283Appendix 2 287Bibliography 299Index 301