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Analytical Mechanics. -An Introduction for Chemists. Christian Hedegaard Jensen. Outline. Motivation Generalised Coordinates Potential Energy Kinetic Energy Lagrange’s Equations Hamilton’s Equations Test. Outline. Motivation Generalised Coordinates Potential Energy Kinetic Energy
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Analytical Mechanics -An Introduction for Chemists Christian Hedegaard Jensen
Outline • Motivation • Generalised Coordinates • Potential Energy • Kinetic Energy • Lagrange’s Equations • Hamilton’s Equations • Test
Outline • Motivation • Generalised Coordinates • Potential Energy • Kinetic Energy • Lagrange’s Equations • Hamilton’s Equations • Test
Motivation • Part I. In understanding Molecular Dynamics. • Better understanding of Mechanics • You do not have to think so much!
Outline • Motivation • Generalised Coordinates • Potential Energy • Kinetic Energy • Lagrange’s Equations • Hamilton’s Equations • Test
Generalised Coordinates • Any set of coordinates that describe the system. • Choose coordinates that fits the problem the best.
l Example – Pendulum • The simple choice is to use polar coordinates, where we have a fixed l. • The system has one degree of freedom. I.e. the variable
Outline • Motivation • Generalised Coordinates • Potential Energy • Kinetic Energy • Lagrange’s Equations • Hamilton’s Equations • Test
l Potential Energy • Specify the potential energy as function of the generalised coordinates. V = 0 ( = 0)
Outline • Motivation • Generalised Coordinates • Potential Energy • Kinetic Energy • Lagrange’s Equations • Hamilton’s Equations • Test
Kinetic Energy • Specify the kinetic energy as function of the generalised coordinates.
Outline • Motivation • Generalised Coordinates • Potential Energy • Kinetic Energy • Lagrange’s Equations • Hamilton’s Equations • Test
Lagrange’s Equations • Plug it all into the equations • + Initial conditions
Lagrange’s Equations • Pendulum • I.e.
Lagrange’s Equations • Small angle oscillations • This gives an angular frequency of
Lagrange’s Equations • Particles moving in potential V
Outline • Motivation • Generalised Coordinates • Potential Energy • Kinetic Energy • Lagrange’s Equations • Hamilton’s Equations • Test
Outline • Motivation • Generalised Coordinates • Potential Energy • Kinetic Energy • Lagrange’s Equations • Hamilton’s Equations • Test
l1 1 l2 2 Test • Double Pendulum • Taking 2 to be constant. What is the angular frequency for small angle vibrations (of 1)? • Hint:
References • P. G. Hjorth; A few Concepts from Analytical Mechanics; Lecture Notes; Sep 2003 • H. Goldstein, C. Poole and J. Safko; Classical Mechanics; Addison Wesley 2002