Advanced Classical Mechanics (NS-350B) - Lecture 3. Alessandro Grelli Institute for Subatomic Physics

Transcription

1 Advanced Classical Mechanics (NS-350B) - Lecture 3 Alessandro Grelli Institute for Subatomic Physics

2 Outline Part I Variational problems Euler-Lagrange equation Fermat principle Brachistochrone problem Minimum surface problem Geodesic Dido problem Part II } See Extending to n-variables Lagrange Multipliers Introduction to Lagrange formulation of mechanics derivations on the blackboard 1

3 Announcements The solutions of the werkcolleges 1 and 2 are in blackboard under course content 2

4 Calculus of Variations Deals with problems where functionals appear A functional is kind of a function where the independent variable is itself a function or a curve (look at your calculus book for the math. definition!!) Historically: shortest path (Fermat), brachistochrome, isoperimetric problems The field has drawn the attention of a remarkable range of scientists, beginning with Newton and Leibniz, then initiated as a subject in its own right by the Bernoulli brothers Jakob and Johann. The first major developments appeared in the work of Euler, Lagrange, and Laplace. 3

5 Calculus of Variations Single variable calculus Functions take extreme values on bounded domains If f is differentiable, needed condition for an extreme is: Calculus of Variations Determine the function y(x) such that: is an extremus. (x is the independent variabile and the integration limit are fixed) Needed condition: y(ε,x) = y(0,x) + εη(x) 4

6 A bit of calculations small change on y(x): setting it to 0 integrating by parts: Now δy =0 for y=a and y=b and therefore the right term is 0 5

7 Euler-Lagrange equation The previus equation must be satisfied for all the δy and therefore the argument in brackets need to be 0 Euler-Lagrange equation Special case 1: Special case 2: F does not depend on x 6

8 Functional of several variables If f is functional of several variables: F = F{y1(x),y1 (x),y2(x),y2 (x),.;x} then we can write: i = 1,2,.,n See derivations on the blackboard 7

9 Function subject to constraints Lagrange multipliers, also called Lagrangian multipliers can be used to find the extrema of a multivariate function f(x1,x2,...,xn) subject to the constraint g(x1,x2,...,xn)=0, where f and g are functions with continuous first partial derivatives See derivation on the blackboard General case: Lagrange multiplier 8

10 Preview of Lagrangian mechanics In Lagrangian mechanics, the independent variable is the time t. The dependent coordinates are the one that define the configuration of the system (q1,q2,.,qn) - generalized coordinates The number n is equal to the number of degrees of freedom of the system The final goal of lagrangian mechanics is to find how our general coordinates vary with time (q1(t),..,qn(t)) The motion of the system can be described by Lagrange (Euler- Lagrange) equations. Usually much easier to write down and use than Newton second law. The Lagrangian L and depends on n coord. and n time derivatives. 9

11 Additional material

12 Functional derivative If the (first) variation of a functional I: can be written as: then the functional derivative is defined as: Functional derivative A functional derivative can be interpreted as follows: imagine a small and localized perturbation of y(x) near point x, δi/δy(x) is the change of I divided by the change of area under the curve y(x) 10