Theoretical Mechanics: Lagrangian dynamics

Slides:

Advertisements

Similar presentations

Sect. 1.6: Simple Applications of the Lagrangian Formulation

Advertisements

Physics 430: Lecture 17 Examples of Lagrange’s Equations

Physics 430: Lecture 16 Lagrange’s Equations with Constraints

Lagrangian and Hamiltonian Dynamics

Work. Generalized Coordinates  Transformation rules define alternate sets of coordinates. 3N Cartesian coordinates x i f generalized coordinates q m.

Coordinates. Cartesian Coordinates  Cartesian coordinates form a linear system. Perpendicular axesPerpendicular axes Right-handed systemRight-handed.

Forward Kinematics.

Hamiltonian Dynamics. Cylindrical Constraint Problem  A particle of mass m is attracted to the origin by a force proportional to the distance.  The.

Simple Harmonic Motion

Lagrangian. Using the Lagrangian  Identify the degrees of freedom. One generalized coordinate for eachOne generalized coordinate for each Velocities.

ME451 Kinematics and Dynamics of Machine Systems Initial Conditions for Dynamic Analysis Constraint Reaction Forces October 23, 2013 Radu Serban University.

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the.

STATIC EQUILIBRIUM [4] Calkin, M. G. “Lagrangian and Hamiltonian Mechanics”, World Scientific, Singapore, 1996, ISBN Consider an object having.

Constrained Motion of Connected Particles

Revision Previous lecture was about Generating Function Approach Derivation of Conservation Laws via Lagrangian via Hamiltonian.

CONSTRAINTS. TOPICS TO BE DISCUSSED  DEFINITION OF CONSTRAINTS  EXAMPLES OF CONSTRAINTS  TYPES OF CONSTRAINTS WITH EXAMPLES.

Dr. Wang Xingbo Fall ， 2005 Mathematical & Mechanical Method in Mechanical Engineering.

A PPLIED M ECHANICS Lecture 02 Slovak University of Technology Faculty of Material Science and Technology in Trnava.

Energy Transformations and Conservation of Mechanical Energy 8

Hamilton’s Principle Lagrangian & Hamiltonian Dynamics Newton’s 2 nd Law: F = (dp/dt) –This is a 100% correct description of particle motion in an Inertial.

Classical Mechanics 420 J. D. Gunton Lewis Lab 418

9/11/2013PHY 711 Fall Lecture 71 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 7: Continue reading.

ME451 Kinematics and Dynamics of Machine Systems Basic Concepts in Planar Kinematics 3.1, 3.2 September 18, 2013 Radu Serban University of Wisconsin-Madison.

In the Hamiltonian Formulation, the generalized coordinate q k & the generalized momentum p k are called Canonically Conjugate quantities. Hamilton’s.

Ch. 8: Hamilton Equations of Motion Sect. 8.1: Legendre Transformations Lagrange Eqtns of motion: n degrees of freedom (d/dt)[(∂L/∂q i )] - (∂L/∂q i )

D’Alembert’s Principle the sum of the work done by

Sect. 1.3: Constraints Discussion up to now  All mechanics is reduced to solving a set of simultaneous, coupled, 2 nd order differential eqtns which.

Lagrange’s Equations with Undetermined Multipliers Marion, Section 7.5 Holonomic Constraints are defined as those which can be expressed as algebraic.

1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.

MA4248 Weeks 6-7. Topics Work and Energy, Single Particle Constraints, Multiple Particle Constraints, The Principle of Virtual Work and d’Alembert’s Principle.

LAGRANGE EQUATION x i : Generalized coordinate Q i : Generalized force i=1,2,....,n In a mechanical system, Lagrange parameter L is called as the difference.

Phy 303: Classical Mechanics (2) Chapter 3 Lagrangian and Hamiltonian Mechanics.

ME451 Kinematics and Dynamics of Machine Systems

The Hamiltonian method

Canonical Equations of Motion -- Hamiltonian Dynamics

Advanced Computer Graphics Spring 2014 K. H. Ko School of Mechatronics Gwangju Institute of Science and Technology.

The state of a system of n particles & subject to m constraints connecting some of the 3n rectangular coordinates is completely specified by s = 3n –

Differentially Constrained Dynamics Wayne Lawton Department of Mathematics National University of Singapore (65)

Equivalence of Lagrange’s & Newton’s Equations Section 7.6 The Lagrangian & the Newtonian formulations of mechanics are 100% equivalent! –As we know,

PHYSICS 361 (Classical Mechanics) Dr. Anatoli Frishman Web Page:

Syllabus Note : Attendance is important because the theory and questions will be explained in the class. II ntroduction. LL agrange’s Equation. SS.

Work Done by a Constant Force The work done by a constant force is defined as the distance moved multiplied by the component of the force in the direction.

Chapter 4 Dynamic Analysis and Forces 4.1 INTRODUCTION In this chapters …….  The dynamics, related with accelerations, loads, masses and inertias. In.

Purdue Aeroelasticity

A Simple Example Another simple example: An illustration of my point that the Hamiltonian formalism doesn’t help much in solving mechanics problems. In.

Introduction to Lagrangian and Hamiltonian Mechanics

ME451 Kinematics and Dynamics of Machine Systems Dynamics of Planar Systems November 4, 2010 Chapter 6 © Dan Negrut, 2010 ME451, UW-Madison TexPoint fonts.

Classical Mechanics Lagrangian Mechanics.

Canonical Quantization

Theoretical Mechanics: Lecture 1

Mathematical & Mechanical Method in Mechanical Engineering

Figure 1. Spring characteristics

Continuum Mechanics (MTH487)

Figure 1. Spring characteristics

Hamiltonian Mechanics

Preview of Ch. 7 Hamilton’s Principle

Variational Time Integrators

Variational Calculus: Euler’s Equation

Sect. 1.4: D’Alembert’s Principle & Lagrange’s Eqtns

Advanced Computer Graphics Spring 2008

Continuous Systems and Fields

Momentum Conservation: Review

© Sharif University of Technology - CEDRA By: Professor Ali Meghdari

Differential Equation of the Mechanical Oscillator

Figure 1. Spring characteristics

Physics 319 Classical Mechanics

Physics 451/551 Theoretical Mechanics

Physics 319 Classical Mechanics

Presentation transcript:

Theoretical Mechanics: Lagrangian dynamics Martikainen Jani-Petri

Often simpler way to present Learning goals Constraints generalized coordinates (D’Alembert’s principle) Lagrange equations Often simpler way to present Newton’s laws

Constrained motion Often we have constraints on particles Examples? The number of degrees of freedom is reduced Want to work only with independent coordinates Constraints imply forces of constraint on the particles to ensure that constraint is satisfied Forces of constraints can be complicated and you might not be interested in them. Get rid of them! We will now learn how to achieve these goals simultaneously

Example of a constraint

Examples of constraints Here forces of constraint become too much!

Holonomic constraints N-particles  n=3N degrees of freedom Holonomic constraints depend on coordinates and (possibly) time Non-holonomic constraints can also exist: mass sliding on a sphere….why? Answer: no geometric constraint at all times since particle will leave the surface eventually Non-holonomic nonintegrable constraints

Holonomic constraints Examples

Examples of other constraints? Non-holonomic: z_j>0 …for example (other examples?) No explicit time dependence: scleronomic constraint (constant length of a pendulum) Rheonomic constraint: Opposite to scleronomic (pendulum whose length changes in a predetermined way length-kt^2=0 for example) Non-holonomic

Constraint:sometimes you have it, sometimes…

Constraints Constraints in the lecture….

Generalized coordinates 3N-k=n-k independent degrees of freedom Choose that completely specify the system These are generalized coordinates Note: if the constraint depends on time…just add the dependence in.

Choosing generalized coordinates Choose wisely

Virtual displacement What is it? Infinitesimal and instantaneous displacement of coordinates consistent with constraints General differential of course

D’Alembert’s principle “Reaction forces, or forces of constraint, do no work under a virtual displacement.” For i:th particle We have The middle term drops out due to D’Alembert’s principle If xi are independent, each term must vanish. Generally more complicated…see notes. Note: this means also that for a dropping particle kinetic energy ONLY comes from gravity and not forces of constraints…!!

D’Alembert’s principle implies what? See notes elsewhere…

Lagrange’s equations In terms of Kinetic energy T and generalized forces Q For conservative forces: L=T-V (V=potential energy)

Examples On the blackboard or somewhere…

Recipe Choose generalized coordinates Degrees of freedom How many degrees of freedom (n-k) you have Think about the system! Here you make your life easier! How are the positions related to generalized coordinates? Form kinetic energy…

Recipe (continues) Potential: L=T-V Calculate derivatives: Lagrange’s equations: Enjoy!

Puzzle Usually energy conservation, for example, Can you think of a conservation law as a constraint? Conservation of energy for example? Is there a difference? Usually energy conservation, for example, implies non-holonomic constraint. It mixes coordinates and velocities.

Next lecture Calculus of variations Hamilton’s principle Forces of constraint About pages 60-77 in F&W