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Ⅰ Fundamental Concepts in Constrained Mechanical Systems
1 Constraints and Their Classification
1.1 Constraints
1.2 Equations of Constraint
1.3 Classification of Constraints
1.3.1 Holonomic Constraints and Nonholonomic Constraints
1.3.2 Stationary Constraints and Non-stationary Constraints
1.3.3 Unilateral Constraints and Bilateral Constraints
1.3.4 Passive Constraints and Active Constraints
1.4 Integrability Theorem of Differential Constraints
1.5 Generalization of the Concept of Constraints
1.5.1 First Integral as Nonholonomic Constraints
1.5.2 Controllable System as Holonomic or Nonholonomic System
1.5.3 Nonholonomic Constraints of Higher Order
1.5.4 Restriction on Change of Dynamical Properties as Constraint
1.6 Remarks
2 Generalized Coordinates
2.1 Generalized Coordinates
2.2 Generalized Velocities
2.3 Generalized Accelerations
2.4 Expression of Equations of Nonholonomic Constraints in Terms of Generalized Coordinates and Generalized Velocities
2.5 Remarks
3 Quasi-Velocities and Quasi-Coordinates
3.1 Quasi-Velocities
3.2 Quasi-Coordinates
3.3 Quasi-Accelerations
3.4 Remarks
4 Virtual Displacements
4.1 Virtual Displacements
4.1.1 Concept of Virtual Displacements
4.1.2 Condition of Constraints Exerted on Virtual Displacements
4.1.3 Degree of Freedom
4.2 Necessary and Sufficient Condition Under Which Actual Displacement Is One of Virtual Displacements
4.3 Generalization of the Concept of Virtual Displacement
4.4 Remarks
5 Ideal Constraints
5.1 Constraint Reactions
5.2 Examples of Ideal Constraints
5.3 Importance and Possibility of Hypothesis of Ideal Constraints
5.4 Remarks
6 Transpositional Relations of Differential and Variational Operations
6.1 Transpositional Relations for First Order Nonholonomic Systems
6.1.1 Transpositional Relations in Terms of Generalized Coordinates
6.1.2 Transpositional Relations in Terms of Quasi-Coordinates
6.2 Transpositional Relations of Higher Order Nonholonomic Systems
6.2.1 Transpositional Relations in Terms of Generalized Coordinates
6.2.2 Transpositional Relations in Terms of Quasi-Coordinates
6.3 Vujanovic Transpositional Relations
6.3.1 Transpositional Relations for Holonomic Nonconservative Systems
6.3.2 Transpositional Relations for Nonholonomic Systems
6.4 Remarks
Ⅱ Variational Principles in Constrained Mechanical Systems
7 Differential Variational Principles
7.1 DAlembert-Lagrange Principle
7.1.1 DAlembert Principle
7.1.2 Principle of Virtual Displacements
7.1.3 DAlembert-Lagrange Principle
7.1.4 DAlembert-Lagrange Principle in
Terms of Generalized Coordinates
7.2 Jourdain Principle
7.2.1 Jourdain Principle
7.2.2 Jourdain Principle in Terms of Generalized Coordinates
7.3 Gauss Principle
7.3.1 Gauss Principle
7.3.2 Gauss Principle in Terms of Generalized Coordinates
7.4 Universal DAlerabert Principle
7.4.1 Universal DAlembert Principle
7.4.2 Universal DAlembert Principle in
Terms of Generalized Coordinates
7.5 Applications of Gauss Principle
7.5.1 Simple Applications
7.5.2 Application of Gauss Principle in Robot Dynamics
7.5.3 Application of Gauss Principle in Study Approximate Solution of Equations of Nonlinear Vibration
7.6 Remarks
8 Integral Variational Principles in Terms of Generalized Coordinates for Holonomic Systems
8.1 Hamiltons Principle
8.1.1 Hamiltons Principle
8.1.2 Deduction of Lagrange Equations
by Means of Hamiltons Principle
8.1.3 Character of Extreme of Hamiltons Principle
8.1.4 Applications in Finding Approximate Solution
8.1.5 Hamiltons Principle for General Holonomic Systems
8.2 Lagranges Principle
8.2.1 Non-contemporaneous Variation
8.2.2 Lagranges Principle
8.2.3 Other Forms of Lagranges Principle
8.2.4 Deduction of Lagrangcs Equations by Means of Lagranges Principle
8.2.5 Generalization of Lagranges Principle to Non-conservative Systems and Its Application
8.3 Remarks
9 Integral Variational Principles in Terms of Quasi-Coordinates for Holonomic Systems
9.1 Hamiltons Principle in Terms of Quasi-Coordinates
9.1.1 Hamiltons Principle
9.1.2 Transpositional Relations
9.1.3 Deduction of Equations of Motion in Terms of Quasi-Coordinates by Means of Hamiltons Principle
9.1.4 Hamiltons Principle for General Holonomic Systems
9.2 Lagranges Principle in Terms of Quasi-Coordinates
9.2.1 Lagranges Principle
9.2.2 Deduction of Equations of Motion in Terms of Quasi-Coordinates by Means of Lagranges Principle
9.3 Remarks
l0 Integral Variational Principles for Nonholonomic Systems
10.1 Definitions of Variation
10.1.1 Necessity of Definition of Variation of Generalized Velocities for Nonholonomic Systems
10.1.2 Suslovs Definition
10.1.3 HSlders Definition
10.2 Integral Variational Principles in Terms of Generalized Coordinates for Nonholonomic Systems
10.2.1 Hamiltons Principle for Nonholonomic Systems
10.2.2 Necessary and Sufficient Condition Under Which Hamiltons Principle for Nonholonomic Systems Is Principle of Stationary Action
10.2.3 Deduction of Equations of Motion for Nonholonomie Systems by Means of Hamiltons Principle
10.2.4 General Form of Hamiltons Principle for Nonhononomic Systems
10.2.5 Lagranges Principle in Terms of Generalized Coordinates for Nonholonomic Systems
10.3 Integral Variational Principle in Terms of QuasiCoordinates for Nonholonomic Systems
10.3.1 Hamiltons Principle in Terms of Quasi-Coordinates
10.3.2 Lagranges Principle in Terms of Quasi-Coordinates
10.4 Remarks
11 Pfaff-Birkhoff Principle
11.1 Statement of Pfaff-Birkhoff Principle
11.2 Hamiltons Principle as a Particular Case of Pfaff-Birkhoff Principle
11.3 Birkhoffs Equations
11.4 Pfaff-Birkhoff-dAlembert Principle
11.5 Remarks
III Differential Equations of Motion of Constrained Mechanical
Systems
12 Lagrange Equations of Holonomic Systems
12.1 Lagrange Equations of Second Kind
12.2 Lagrange Equations of Systems with Redundant Coordinates
12.3 Lagrange Equations in Terms of Quasi-Coordinates
12.4 Lagrange Equations with Dissipative Function
12.5 Remarks
13 Lagrange Equations with Multiplier for Nonholonomic Systems
13.1 Deduction of Lagrange Equations with Multiplier
13.2 Determination of Nonholonomic Constraint Forces
13.3 Remarks
14 Mac Millan Equations for Nonholonomie Systems
14.1 Deduction of Mac Millan Equations
14.2 Application of Mac MiUan Equations
14.3 Remarks
15 Volterra Equations for Nonholonomic Systems
15.1 Deduction of Generalized Volterra Equations
15.2 Volterra Equations and Their Equivalent Forms
15.2.1 Volterra Equations of First Form
15.2.2 Volterra Equations of Second Form
15.2.3 Volterra Equations of Third Form
15.2.4 Volterra Equations of Fourth Form
15.3 Application of Volterra Equations
15.4 Remarks
16 Chaplygin Equations for Nonholonomic Systems
16.1 Generalized Chaplygin Equations
16.2 Voronetz Equations
16.3 Chaplygin Equations
16.4 Chaplygin Equations in Terms of Quasi-Coordinates
16.5 Application of Chaplygin Equations
16.6 Remarks
……
Ⅳ Special Problems in Constrained Mechanical Systems
Ⅴ Integration Methods in Constrained Mechanical Systems
Ⅵ Symmetries and Conserved Quantities in Constrained Mechanical Systems
