Constantin Udriste, PhD, is Professor Emeritus of Mathematics-Informatics at the University Politehnica of Bucharest, Romania. He received his PhD in Mathematics from the University Babes-Bolyai of Cluj-Napoca, Romania. Ionel Tevy, PhD, is a Professor of Mathematics at the University Politehnica of Bucharest, Romania. He received his PhD in Mathematical Sciences from the Faculty of Mathematics and Mechanics of the University of Bucharest, Romania.

• Preface ix  1 Extrema of Differentiable Functionals 11.1 Differentiable Functionals 11.2 Extrema of Differentiable Functionals 61.3 Second Variation; Sufficient Conditions for Extremum 141.4 Optimum with Constraints; the Principle of Reciprocity 171.4.1 Isoperimetric Problems 181.4.2 The Reciprocity Principle 191.4.3 Constrained Extrema: The Lagrange Problem 191.5 Maple Application Topics 21  2 Variational Principles 232.1 Problems with Natural Conditions at the Boundary 232.2 Sufficiency by the Legendre-Jacobi Test 272.3 Unitemporal Lagrangian Dynamics 302.3.1 Null Lagrangians 312.3.2 Invexity Test 322.4 Lavrentiev phenomenon 332.5 Unitemporal Hamiltonian Dynamics 352.6 Particular Euler–Lagrange ODEs 372.7 Multitemporal Lagrangian Dynamics 382.7.1 The Case of Multiple Integral Functionals 382.7.2 Invexity Test 402.7.3 The Case of Path-Independent Curvilinear Integral Functionals 412.7.4 Invexity Test 442.8 Multitemporal Hamiltonian Dynamics 452.9 Particular Euler–Lagrange PDEs 472.10 Maple Application Topics 48  3 Optimal Models Based on Energies 533.1 Brachistochrone Problem 533.2 Ropes, Chains and Cables 543.3 Newton’s Aerodynamic Problem 563.4 Pendulums 593.4.1 Plane Pendulum 593.4.2 Spherical Pendulum 603.4.3 Variable Length Pendulum 613.5 Soap Bubbles 623.6 Elastic Beam 633.7 The ODE of an Evolutionary Microstructure 633.8 The Evolution of a Multi-Particle System 643.8.1 Conservation of Linear Momentum 653.8.2 Conservation of Angular Momentum 663.8.3 Energy Conservation 673.9 String Vibration 673.10 Membrane Vibration 703.11 The Schrödinger Equation in Quantum Mechanics 733.11.1 Quantum Harmonic Oscillator 733.12 Maple Application Topics 74  4 Variational Integrators 794.1 Discrete Single-time Lagrangian Dynamics 794.2 Discrete Hamilton’s Equations 844.3 Numeric Newton’s Aerodynamic Problem 874.4 Discrete Multi-time Lagrangian Dynamics 884.5 Numerical Study of the Vibrating String Motion 924.5.1 Initial Conditions for Infinite String 944.5.2 Finite String, Fixed at the Ends 954.5.3 Monomial (Soliton) Solutions 964.5.4 More About Recurrence Relations 1004.5.5 Solution by Maple via Eigenvalues 1014.5.6 Solution by Maple via Matrix Techniques 1024.6 Numerical Study of the Vibrating Membrane Motion 1044.6.1 Monomial (Soliton) Solutions 1054.6.2 Initial and Boundary Conditions 1084.7 Linearization of Nonlinear ODEs and PDEs 1094.8 Von Neumann Analysis of Linearized Discrete Tzitzeica PDE 1134.8.1 Von Neumann Analysis of Dual Variational Integrator Equation 1154.8.2 Von Neumann Analysis of Linearized Discrete Tzitzeica Equation 1164.9 Maple Application Topics 119  5 Miscellaneous Topics 1235.1 Magnetic Levitation 1235.1.1 Electric Subsystem 1235.1.2 Electromechanic Subsystem 1245.1.3 State Nonlinear Model 1245.1.4 The Linearized Model of States 1255.2 The Problem of Sensors 1255.2.1 Simplified Problem 1265.2.2 Extending the Simplified Problem of Sensors 1285.3 The Movement of a Particle in Non-stationary Gravito-vortex Field 1285.4 Geometric Dynamics 1295.4.1 Single-time Case 1295.4.2 The Least Squares Lagrangian in Conditioning Problems 1305.4.3 Multi-time Case 1335.5 The Movement of Charged Particle in Electromagnetic Field 1345.5.1 Unitemporal Geometric Dynamics Induced by Vector Potential A 1355.5.2 Unitemporal Geometric Dynamics Produced by Magnetic Induction B 1365.5.3 Unitemporal Geometric Dynamics Produced by Electric Field E 1365.5.4 Potentials Associated to Electromagnetic Forms 1375.5.5 Potential Associated to Electric 1-form E 1385.5.6 Potential Associated to Magnetic 1-form H 1385.5.7 Potential Associated to Potential 1-form A 1385.6 Wind Theory and Geometric Dynamics 1395.6.1 Pendular Geometric Dynamics and Pendular Wind 1415.6.2 Lorenz Geometric Dynamics and Lorenz Wind 1425.7 Maple Application Topics 143  6 Nonholonomic Constraints 1476.1 Models With Holonomic and Nonholonomic Constraints 1476.2 Rolling Cylinder as a Model with Holonomic Constraints 1516.3 Rolling Disc (Unicycle) as a Model with Nonholonomic Constraint 1526.3.1 Nonholonomic Geodesics 1526.3.2 Geodesics in Sleigh Problem 1556.3.3 Unicycle Dynamics 1566.4 Nonholonomic Constraints to the Car as a Four-wheeled Robot 1576.5 Nonholonomic Constraints to the N-trailer 1586.6 Famous Lagrangians 1606.7 Significant Problems 1606.8 Maple Application Topics 163  7 Problems: Free and Constrained Extremals 1657.1 Simple Integral Functionals 1657.2 Curvilinear Integral Functionals 1697.3 Multiple Integral Functionals 1717.4 Lagrange Multiplier Details 1747.5 Simple Integral Functionals with ODE Constraints 1757.6 Simple Integral Functionals with Nonholonomic Constraints 1817.7 Simple Integral Functionals with Isoperimetric Constraints 1847.8 Multiple Integral Functionals with PDE Constraints 1867.9 Multiple Integral Functionals With Nonholonomic Constraints 1887.10 Multiple Integral Functionals With Isoperimetric Constraints 1897.11 Curvilinear Integral Functionals With PDE Constraints 1917.12 Curvilinear Integral Functionals With Nonholonomic Constraints 1937.13 Curvilinear Integral Functionals with Isoperimetric Constraints 1957.14 Maple Application Topics 197 Bibliography 203Index 209

Constantin Udriste, PhD, is Professor Emeritus of Mathematics-Informatics at the University Politehnica of Bucharest, Romania. He received his PhD in Mathematics from the University Babes-Bolyai of Cluj-Napoca, Romania. Ionel Tevy, PhD, is a Professor of Mathematics at the University Politehnica of Bucharest, Romania. He received his PhD in Mathematical Sciences from the Faculty of Mathematics and Mechanics of the University of Bucharest, Romania.

• Preface ix  1 Extrema of Differentiable Functionals 11.1 Differentiable Functionals 11.2 Extrema of Differentiable Functionals 61.3 Second Variation; Sufficient Conditions for Extremum 141.4 Optimum with Constraints; the Principle of Reciprocity 171.4.1 Isoperimetric Problems 181.4.2 The Reciprocity Principle 191.4.3 Constrained Extrema: The Lagrange Problem 191.5 Maple Application Topics 21  2 Variational Principles 232.1 Problems with Natural Conditions at the Boundary 232.2 Sufficiency by the Legendre-Jacobi Test 272.3 Unitemporal Lagrangian Dynamics 302.3.1 Null Lagrangians 312.3.2 Invexity Test 322.4 Lavrentiev phenomenon 332.5 Unitemporal Hamiltonian Dynamics 352.6 Particular Euler–Lagrange ODEs 372.7 Multitemporal Lagrangian Dynamics 382.7.1 The Case of Multiple Integral Functionals 382.7.2 Invexity Test 402.7.3 The Case of Path-Independent Curvilinear Integral Functionals 412.7.4 Invexity Test 442.8 Multitemporal Hamiltonian Dynamics 452.9 Particular Euler–Lagrange PDEs 472.10 Maple Application Topics 48  3 Optimal Models Based on Energies 533.1 Brachistochrone Problem 533.2 Ropes, Chains and Cables 543.3 Newton’s Aerodynamic Problem 563.4 Pendulums 593.4.1 Plane Pendulum 593.4.2 Spherical Pendulum 603.4.3 Variable Length Pendulum 613.5 Soap Bubbles 623.6 Elastic Beam 633.7 The ODE of an Evolutionary Microstructure 633.8 The Evolution of a Multi-Particle System 643.8.1 Conservation of Linear Momentum 653.8.2 Conservation of Angular Momentum 663.8.3 Energy Conservation 673.9 String Vibration 673.10 Membrane Vibration 703.11 The Schrödinger Equation in Quantum Mechanics 733.11.1 Quantum Harmonic Oscillator 733.12 Maple Application Topics 74  4 Variational Integrators 794.1 Discrete Single-time Lagrangian Dynamics 794.2 Discrete Hamilton’s Equations 844.3 Numeric Newton’s Aerodynamic Problem 874.4 Discrete Multi-time Lagrangian Dynamics 884.5 Numerical Study of the Vibrating String Motion 924.5.1 Initial Conditions for Infinite String 944.5.2 Finite String, Fixed at the Ends 954.5.3 Monomial (Soliton) Solutions 964.5.4 More About Recurrence Relations 1004.5.5 Solution by Maple via Eigenvalues 1014.5.6 Solution by Maple via Matrix Techniques 1024.6 Numerical Study of the Vibrating Membrane Motion 1044.6.1 Monomial (Soliton) Solutions 1054.6.2 Initial and Boundary Conditions 1084.7 Linearization of Nonlinear ODEs and PDEs 1094.8 Von Neumann Analysis of Linearized Discrete Tzitzeica PDE 1134.8.1 Von Neumann Analysis of Dual Variational Integrator Equation 1154.8.2 Von Neumann Analysis of Linearized Discrete Tzitzeica Equation 1164.9 Maple Application Topics 119  5 Miscellaneous Topics 1235.1 Magnetic Levitation 1235.1.1 Electric Subsystem 1235.1.2 Electromechanic Subsystem 1245.1.3 State Nonlinear Model 1245.1.4 The Linearized Model of States 1255.2 The Problem of Sensors 1255.2.1 Simplified Problem 1265.2.2 Extending the Simplified Problem of Sensors 1285.3 The Movement of a Particle in Non-stationary Gravito-vortex Field 1285.4 Geometric Dynamics 1295.4.1 Single-time Case 1295.4.2 The Least Squares Lagrangian in Conditioning Problems 1305.4.3 Multi-time Case 1335.5 The Movement of Charged Particle in Electromagnetic Field 1345.5.1 Unitemporal Geometric Dynamics Induced by Vector Potential A 1355.5.2 Unitemporal Geometric Dynamics Produced by Magnetic Induction B 1365.5.3 Unitemporal Geometric Dynamics Produced by Electric Field E 1365.5.4 Potentials Associated to Electromagnetic Forms 1375.5.5 Potential Associated to Electric 1-form E 1385.5.6 Potential Associated to Magnetic 1-form H 1385.5.7 Potential Associated to Potential 1-form A 1385.6 Wind Theory and Geometric Dynamics 1395.6.1 Pendular Geometric Dynamics and Pendular Wind 1415.6.2 Lorenz Geometric Dynamics and Lorenz Wind 1425.7 Maple Application Topics 143  6 Nonholonomic Constraints 1476.1 Models With Holonomic and Nonholonomic Constraints 1476.2 Rolling Cylinder as a Model with Holonomic Constraints 1516.3 Rolling Disc (Unicycle) as a Model with Nonholonomic Constraint 1526.3.1 Nonholonomic Geodesics 1526.3.2 Geodesics in Sleigh Problem 1556.3.3 Unicycle Dynamics 1566.4 Nonholonomic Constraints to the Car as a Four-wheeled Robot 1576.5 Nonholonomic Constraints to the N-trailer 1586.6 Famous Lagrangians 1606.7 Significant Problems 1606.8 Maple Application Topics 163  7 Problems: Free and Constrained Extremals 1657.1 Simple Integral Functionals 1657.2 Curvilinear Integral Functionals 1697.3 Multiple Integral Functionals 1717.4 Lagrange Multiplier Details 1747.5 Simple Integral Functionals with ODE Constraints 1757.6 Simple Integral Functionals with Nonholonomic Constraints 1817.7 Simple Integral Functionals with Isoperimetric Constraints 1847.8 Multiple Integral Functionals with PDE Constraints 1867.9 Multiple Integral Functionals With Nonholonomic Constraints 1887.10 Multiple Integral Functionals With Isoperimetric Constraints 1897.11 Curvilinear Integral Functionals With PDE Constraints 1917.12 Curvilinear Integral Functionals With Nonholonomic Constraints 1937.13 Curvilinear Integral Functionals with Isoperimetric Constraints 1957.14 Maple Application Topics 197 Bibliography 203Index 209

Constantin Udriste, PhD, is Professor Emeritus of Mathematics-Informatics at the University Politehnica of Bucharest, Romania. He received his PhD in Mathematics from the University Babes-Bolyai of Cluj-Napoca, Romania. Ionel Tevy, PhD, is a Professor of Mathematics at the University Politehnica of Bucharest, Romania. He received his PhD in Mathematical Sciences from the Faculty of Mathematics and Mechanics of the University of Bucharest, Romania.

• Preface ix  1 Extrema of Differentiable Functionals 11.1 Differentiable Functionals 11.2 Extrema of Differentiable Functionals 61.3 Second Variation; Sufficient Conditions for Extremum 141.4 Optimum with Constraints; the Principle of Reciprocity 171.4.1 Isoperimetric Problems 181.4.2 The Reciprocity Principle 191.4.3 Constrained Extrema: The Lagrange Problem 191.5 Maple Application Topics 21  2 Variational Principles 232.1 Problems with Natural Conditions at the Boundary 232.2 Sufficiency by the Legendre-Jacobi Test 272.3 Unitemporal Lagrangian Dynamics 302.3.1 Null Lagrangians 312.3.2 Invexity Test 322.4 Lavrentiev phenomenon 332.5 Unitemporal Hamiltonian Dynamics 352.6 Particular Euler–Lagrange ODEs 372.7 Multitemporal Lagrangian Dynamics 382.7.1 The Case of Multiple Integral Functionals 382.7.2 Invexity Test 402.7.3 The Case of Path-Independent Curvilinear Integral Functionals 412.7.4 Invexity Test 442.8 Multitemporal Hamiltonian Dynamics 452.9 Particular Euler–Lagrange PDEs 472.10 Maple Application Topics 48  3 Optimal Models Based on Energies 533.1 Brachistochrone Problem 533.2 Ropes, Chains and Cables 543.3 Newton’s Aerodynamic Problem 563.4 Pendulums 593.4.1 Plane Pendulum 593.4.2 Spherical Pendulum 603.4.3 Variable Length Pendulum 613.5 Soap Bubbles 623.6 Elastic Beam 633.7 The ODE of an Evolutionary Microstructure 633.8 The Evolution of a Multi-Particle System 643.8.1 Conservation of Linear Momentum 653.8.2 Conservation of Angular Momentum 663.8.3 Energy Conservation 673.9 String Vibration 673.10 Membrane Vibration 703.11 The Schrödinger Equation in Quantum Mechanics 733.11.1 Quantum Harmonic Oscillator 733.12 Maple Application Topics 74  4 Variational Integrators 794.1 Discrete Single-time Lagrangian Dynamics 794.2 Discrete Hamilton’s Equations 844.3 Numeric Newton’s Aerodynamic Problem 874.4 Discrete Multi-time Lagrangian Dynamics 884.5 Numerical Study of the Vibrating String Motion 924.5.1 Initial Conditions for Infinite String 944.5.2 Finite String, Fixed at the Ends 954.5.3 Monomial (Soliton) Solutions 964.5.4 More About Recurrence Relations 1004.5.5 Solution by Maple via Eigenvalues 1014.5.6 Solution by Maple via Matrix Techniques 1024.6 Numerical Study of the Vibrating Membrane Motion 1044.6.1 Monomial (Soliton) Solutions 1054.6.2 Initial and Boundary Conditions 1084.7 Linearization of Nonlinear ODEs and PDEs 1094.8 Von Neumann Analysis of Linearized Discrete Tzitzeica PDE 1134.8.1 Von Neumann Analysis of Dual Variational Integrator Equation 1154.8.2 Von Neumann Analysis of Linearized Discrete Tzitzeica Equation 1164.9 Maple Application Topics 119  5 Miscellaneous Topics 1235.1 Magnetic Levitation 1235.1.1 Electric Subsystem 1235.1.2 Electromechanic Subsystem 1245.1.3 State Nonlinear Model 1245.1.4 The Linearized Model of States 1255.2 The Problem of Sensors 1255.2.1 Simplified Problem 1265.2.2 Extending the Simplified Problem of Sensors 1285.3 The Movement of a Particle in Non-stationary Gravito-vortex Field 1285.4 Geometric Dynamics 1295.4.1 Single-time Case 1295.4.2 The Least Squares Lagrangian in Conditioning Problems 1305.4.3 Multi-time Case 1335.5 The Movement of Charged Particle in Electromagnetic Field 1345.5.1 Unitemporal Geometric Dynamics Induced by Vector Potential A 1355.5.2 Unitemporal Geometric Dynamics Produced by Magnetic Induction B 1365.5.3 Unitemporal Geometric Dynamics Produced by Electric Field E 1365.5.4 Potentials Associated to Electromagnetic Forms 1375.5.5 Potential Associated to Electric 1-form E 1385.5.6 Potential Associated to Magnetic 1-form H 1385.5.7 Potential Associated to Potential 1-form A 1385.6 Wind Theory and Geometric Dynamics 1395.6.1 Pendular Geometric Dynamics and Pendular Wind 1415.6.2 Lorenz Geometric Dynamics and Lorenz Wind 1425.7 Maple Application Topics 143  6 Nonholonomic Constraints 1476.1 Models With Holonomic and Nonholonomic Constraints 1476.2 Rolling Cylinder as a Model with Holonomic Constraints 1516.3 Rolling Disc (Unicycle) as a Model with Nonholonomic Constraint 1526.3.1 Nonholonomic Geodesics 1526.3.2 Geodesics in Sleigh Problem 1556.3.3 Unicycle Dynamics 1566.4 Nonholonomic Constraints to the Car as a Four-wheeled Robot 1576.5 Nonholonomic Constraints to the N-trailer 1586.6 Famous Lagrangians 1606.7 Significant Problems 1606.8 Maple Application Topics 163  7 Problems: Free and Constrained Extremals 1657.1 Simple Integral Functionals 1657.2 Curvilinear Integral Functionals 1697.3 Multiple Integral Functionals 1717.4 Lagrange Multiplier Details 1747.5 Simple Integral Functionals with ODE Constraints 1757.6 Simple Integral Functionals with Nonholonomic Constraints 1817.7 Simple Integral Functionals with Isoperimetric Constraints 1847.8 Multiple Integral Functionals with PDE Constraints 1867.9 Multiple Integral Functionals With Nonholonomic Constraints 1887.10 Multiple Integral Functionals With Isoperimetric Constraints 1897.11 Curvilinear Integral Functionals With PDE Constraints 1917.12 Curvilinear Integral Functionals With Nonholonomic Constraints 1937.13 Curvilinear Integral Functionals with Isoperimetric Constraints 1957.14 Maple Application Topics 197 Bibliography 203Index 209