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This 6 page Class Notes was uploaded by Dr. Simeon Wiza on Wednesday September 9, 2015. The Class Notes belongs to PHYS 505 at University of Washington taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/192447/phys-505-university-of-washington in Physics 2 at University of Washington.
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Date Created: 09/09/15
Lecture 5 The Methods of Lagrange I See Chapter 3 in FampW the text by Lanczos and Chapter 12 in Mathews amp Walker Mathematical Methods of Physics We have seen that solving Newton s laws can become quite complicated depending on the choice of reference frame and that often the underlying symmetries are not obvious Luckily various techniques have been developed that allow us to generalize the analysis of mechanics These techniques serve not only to facilitate both changes of reference frame and the studies of symmetries but they also transcend classical mechanics and provide a basis for the study of quantum mechanics The basic idea is that instead of thinking simply in terms of finding the trajectory of a particle or system of particles by solving certain differential equations we focus instead on finding the desired trajectory as the extremal path in phase space with respect to variations in a quantity called the action This formalism will release us from the need to focus on specific reference frames e g Cartesian coordinates in an inertial frame in order to get started on any given problem It will also allow us to systematically include the effects of constraints Finally we can imagine including quantum effects by simply enlarging the region studied in phase space configuration space 17 momentum space g phase space I743 or 027 Instead ofa unique trajectory the classical solution we include a larger space of trajectories a subspace of the full phase space whose transverse dimension with respect to the classical trajectory is characterized by h In this lecture and the next we will introduce and develop these new techniques An important concept for the studies of extremal solutions is that of a functional Up to now we have been largely discussing ordinary functions e g y fx which yield a value for y that depends only on the specific point x not the path followed to reach x If instead the quantity of interest depends on the path followed to reach a given point then the quantity is a functional of that path ie of the function that defines that path This concept typically arises from definitions in terms of integrals if 2Tfxdx 51 The value of this expression will in general depend on the details of the function f ie the path followed in the integral unless f x dx is a perfect differential f x dx dF x I F x2 F x1 In this latter case I depends only on the values of F at the endpoints of the integral Another label for this distinction is Physics 505 Lecture 5 1 Autumn 2005 provided by the term holonomic Holonomic variables functions are independent of the underlying paths while nonholonomic variables depend on the paths In the applications to classical mechanics we will be interested in nonlinear functionals of both the path and its derivative 1qu l y y39 xdx 52 Our challenge will be to determine the path y f x that yields an extremum of I typically with the endpoints of the integral fixed and perhaps with other constraints on the possible functions The corresponding mathematics is called the calculus of variations see Section 17 in FampW and the text by Lanczos The concept of finding the extrema of an ordinary function of several variables is familiarly linked to the stationary points of the function where all of the partial derivatives vanish j l to m 6fx1xm 6x J 0 53 The limit In gt 00 brings us to the idea of a functional derivative the derivative of a functional with respect to the functions that constitute its domain of definition Consider the space of all allowed functions y f x with the same values at the endpoints x1 and x2 Define a virtual variation as arising from taking the difference between two such nearby functions and their slopes 5y 5fx 5 12x f1x at a xed value ofx Due to this latter feature ofthe definition the operation of taking the functional variation commutes with the ordinary derivative 5y 5 dydx d 5 y dx The corresponding variation in the integral is given by quotZacD 6CD Physics 505 Lecture 5 2 Autumn 2005 Using the fact that the variation operation and the derivative operation commute we can perform an integration by parts to yield 6CD quot2 quot2 61 of 6CD 51 VSy j 5ydx 6y 1 1 6y dx 6y quot2 6CD d 6CD 55 2 I r j 1 6y dx 6y where the second step arises from the fact that by our definition of fixed no variation endpoints 5y x1 5yx2 0 Thinking of the righthandside as a Riemann sum over variations at each x value we can identify the corresponding local functional derivative the final expression is also sometimes labeled as MIDBy am kmm 51 5lt1gtac1gt d 6 56 5yx gg3 ay We can now generalize the stationary point result for the extremum of an ordinary function Eq 53 to apply to the extremum of a functional Thus 5 0 for arbitrary 5y X at each value of x means that agiacp0 6y dx 6y39 57 at each x This is called the EulerLagrange equation The essential feature is that it is equivalent to the original variational problem when boundary conditions are included In certain situations we can integrate the EulerLagrange equation once trivially For example consider the case when d is independent of y Then it follows that Physics 505 Lecture 5 3 Autumn 2005 d 3CD 3CD r 0 gt r constant 58 dx 6y 3y As a trivial example consider the problem of determining the shortest distance between two points in 2 dimensions The corresponding functional and Euler Lagrange equation are 12 zjdszj ldxz 0322 2 1y392dx l 1 x1 CDyy39x 41 y 520 51 6y 6y39 Jim 59 52i5 i y39 0 6y dx ay39 dx ly392 It follows that the quantity y391 y392 is a constant which is possible if and only if y39 itself is a constant If y 1y 2 C then y39 a CVl c2 and yaxb 510 the expected straight line A second case with simple properties is when d has no explicit x dependence Again we can reduce the EL equation to a firstorder equation In this case we have old 61 61 61 y 7 yquot dx ax ay ay39 52yraqy 511 5y 3y39 39 Physics 505 Lecture 5 4 Autumn 2005 We can use this expression in the EL equation after multiplying through by y39 and adding and subtracting yquot 513 ay These manipulations yield r vi 5CD y 3y MAW Dyagw yac1gt1ac1gt0 5y 3f 3 dx 5quot iqy39620 512 dx 3y39 V 3CD gt CD y constant 5y This approach can be applied to the founding problem of the calculation of variations analyzed by Bernoulli the brachistochrone problem Greek for shortest time see exercise 310 in FampW If a particle is released from rest in a uniform gravitational field and travels along a frictionless wire that connects two fixed points in the xy plane where the ydirection is the local down direction what shape for the wire yields the minimum time of descent In this case we want to find the extremal time 2ds 2 dx2ay2 M 1 2 Tlflflfd lt51 Xi with respect to variations in the function yx We can use the conservation of energy to express the velocity in terms of the path y 0 0 the point of release is the origin 1 Emv2 mgygtvx1 2gyx 514 This yields Physics 505 Lecture 5 5 Autumn 2005
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