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LA Intro RCED/PLR1 Least Action, Lagrangian & Hamiltonian Mechanics A very brief introduction to some very powerful ideas –‘foolproof’ way to find the equations of motion for complicated dynamical systems –equivalent to Newton’s equations –provides a framework for relating conservation laws to symmetries –the ideas may be extended to most areas of fundamental physics (special and general relativity, electromagnetism, quantum mechanics, quantum field theory…) NOT FOR EXAMINATION

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LA Intro RCED/PLR2 Lagrange and Hamilton Joseph-Louis Lagrange (1736-1810) [Giuseppe Lodovico Lagrangia] Sir William Rowan Hamilton (1805-1865)

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LA Intro RCED/PLR3 Why? Newton’s laws are vector relations For complicated situations maybe hard to identify all the forces, especially if there are constraints

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LA Intro RCED/PLR4 Hints We have already exploited the energy conservation equation, especially for conservative forces Note that the energy equation relates scalar quantities E U(x) xBxB xAxA or

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LA Intro RCED/PLR5 More PE KE t PE KE t falling – constant force SHM For another, more formal, approach see the appendix on D’Alembert & Hamilton For many simple systems When averaged over a path This leads to the idea that (the action) evaluated along a path may take a minimum or stationary value

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LA Intro RCED/PLR6 Least Action by brute force x y O R A B C D Numerical integration… Fixed distance R and time t R Horizontal speed u 0 =R/t R Trajectory is of the form Note Vary v 0 - calculate For paths B, C, D A has v 0 < 0 (!) C is path with S stationary With

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LA Intro RCED/PLR7 Path integrals A B t 1 2 3 4 5 6 7 8 B A t L = T - V is the Lagrangian, a function of Minimise the action integral w.r.t. variations in the path? Calculus of variations? - No, use Euler’s geometrical approach. Approximate S as sum x i at start of each segment for segment

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LA Intro RCED/PLR8 Euler I Key observation – every section of action integral or sum must be stationary Consider two linearised sections of sum, with point at time t 2 moved to N M O N 1 2 3 x t

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LA Intro RCED/PLR9 Euler II

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LA Intro RCED/PLR10 From action to Newton

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LA Intro RCED/PLR11 Generalised coordinates

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LA Intro RCED/PLR12 Example – Newtonian gravity

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LA Intro RCED/PLR13 Sliding blocks B A x z Mass A M, mass B m Find the initial acceleration of A No friction

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LA Intro RCED/PLR14 Pendulum with oscillating support m

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LA Intro RCED/PLR15 Symmetry and conserved quantities The Lagrangian approach provides a useful alternative to a direct formulation using Newton’s equations However it also provides the framework in which fundamental questions about the nature of forces and interactions can studied. In particular the very close relationship between a symmetry of the Lagrangian and a conserved quantity By symmetry we mean an operation – eg coordinate rotation – that leaves the Lagrangian unchanged

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LA Intro RCED/PLR16 Noether’s theorem

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LA Intro RCED/PLR17 Examples Free particle and translational invariance

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LA Intro RCED/PLR18 Rotational Invariance

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LA Intro RCED/PLR19 Conservation of energy

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LA Intro RCED/PLR20 The Hamiltonian & Hamilton’s Eqs of Motion

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LA Intro RCED/PLR21

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LA Intro RCED/PLR22 Summary Lagrangian L=T-V plus Euler-Lagrange equations give a convenient way of generating equations of motion for complicated dynamical problems Noether’s theorem provides a mechanism for finding conserved quantities that follow from symmetry of the Lagrangian The Hamiltonian is an alternative formulation, useful in formal treatments, and with an analogue in Schrödinger’s equation of quantum mechanics. The method of ‘least action’ can be extended to many other areas of physics to learn more… try Classical Mechanics Short Option – S7 – next year

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LA Intro RCED/PLR23 Sources and further reading Any advanced text on classical mechanics –Kibble & Berkshire 5 th Ed, IC Press 2005 –Goldstein, Poole & Safko 3 rd Ed, AW 2002 –Fowles & Cassidy, Harcourt Brace 1993 –Cowan, RKP 1984 Feynman Lectures on Physics, Vol II 19-1 – Feynman, Leighton & Sands, AW 1964 For many interesting, almost evangelical, articles on the principles of least action http://www.eftaylor.com/leastaction.html (Prof Edwin Taylor (MIT) and collaborators)

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LA Intro RCED/PLR24 Appendix: D’Alembert, Hamilton & Least Action B A tBtB tAtA variation of path at fixed t Hamilton considers variation in paths between fixed points at A and B such that varied coordinates are evaluated at the same time, so t = 0. Note that all paths must have the same start and end points in space and time.

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LA Intro RCED/PLR25

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LA Intro RCED/PLR26 Problems r

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LA Intro RCED/PLR27 x z y r

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LA Intro RCED/PLR28 a x y b xBxB xAxA mAmA mBmB O

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