Theoretical Mechanics - PHY6200 Chapter 6 Introduction to the calculus of variations Prof. Claude A Pruneau, Physics and Astronomy Department Wayne State University

Introduction Many mechanics problem more easily analyzed/solved by means of calculus of variation, Lagrange Eqs., etc. We will: Consider some general principles Omit formal existence proofs As an example, consider Fermat’s principle.

Fermat’s Principle Light travels between two points along the path that takes the least amount of time. air water L

Snell’s Law By construction (geometry): Total travel time: Extremum for : Simplification/Substitution : Snell’s Law :

Statement of the problem In classical mechanics, the problem amounts to finding a trajectory between an initial and a final point (boundary conditions). Express the problem to solve as a minimization problem. Find a functional that once minimized will yield an optimal trajectory which corresponds to the physical trajectory (according to Newton’s laws of motion). 2 1

Basic Formulation (1 dimension) Determine a function, y(x), such that the following integral is an extremum. f(y,y’;x) is a function considered as “given” Limits of integration are fixed. y(x), the trajectory is to be varied until an extremum is found for J. y’(x)=dy(x)/dx x : independent variable 2 1

Basic Formulation (cont’d) If the optimal trajectory y(x) produces a minimal value for J, then any neighboring trajectory y  (x) corresponds to a large value of J. Hence by minimizing J, one finds the optimal trajectory. 2 1

Parametric Representation The function y(x) can be considered with a parametric representation y  (x) = y( ,x) Where the variable  denotes a parameter that describes functions close to y(x) but that differ by an “amount” proportional to . By definition,  =0 corresponds to the optimal trajectory y(x). y  (x) = y(0,x)=y(x)

Parametric Representation (cont’d) Consider the specific representation y( ,x)=y(0,x)+  (x) where  (x) is some “arbitrary” function of x, with continuous 1 st derivative, and vanishing at the boundaries  (x 1 ) =  (x 2 ) = 0

Parametric Representation (cont’d) The functional J is written This integral has an extremum if This must be true for all functions  (x) Note: This is a necessary but not sufficient condition.

Example 1 Consider the function f(y’;x)=(dy/dx) 2. Where y(x)=x. Consider  (x)=sin(x) Find J(  ) between x=0, and x=2 . Show that the stationary value (extremum) of J occurs for  =0.

Example 1 - Solution Let: y( ,x) = x +  sin(x) Note that by choice/construction, we have  (0) = 0  (2  ) = 0 Proceed to calculate…

Example 1 - Solution (cont’d)

Derivation of Euler’s Equation Calculate the derivative of J For fixed integration limits, one can change the order of operations

By construction: Thus:

Integrate 2nd term by parts 0 The function  (x) being totally arbitrary. We get Provided the integrand itself vanishes. So…

Euler’s Equation This equation yields a solution for y(x) and y’(x) which produces an extremum for J.

Example 2 - Brachistochrone Consider a particle moving in a constant force field, starting at rest from some point (x 1,y 1 ) and ending at (x 2,y 2 ). Find the path (trajectory) that allows the particle to accomplish the motion/transit in the least amount of time.

Example 2 - Solution (1) Constant force, neglect friction, implies a conservative system. T+U=constant. Let U(x=0)=0, and T(0)+U(0)=0. T= 0.5 mv 2 U=-Fx=-mgx x y (x 1,y 1 ) (x 2,y 2 )

Example 2 - Solution (2) Time required for transit:

Example 2 - Solution (3) Use Euler’s Equation Thus…

Example 2 - Brachistochrone - Answer Solution passes through (0,0) (0,2  a) y (0,0) (0,  a) x (2a,0)  cycloid

2 nd Form of Euler’s Equation For a function “f” that does not explicitly depend on “x”. 1) 2)

2 nd Form of Euler’s Equation (cont’d) We find: 2nd Form of Euler’s Equation is thus If f does not depend on “x”, there is a conserved quantity

Function with Several Dependent Variables Consider a more general case where f is a functional of several dependent variables. E.g. Motion of one particle in 3 dimensions, or many particles…

Several Dependent Variables… Repeat our previous reasoning… y i ( ,x)=y i (0,x)+  i (x)

Several Dependent Variables… So we have Functions  i are all independent. Each term of the sum must be null for the above derivative to vanish at  =0. Euler’s Equations for many dependent variables

Euler Equations with Auxiliary Conditions In many problems, additional constrains come into play. Consider e.g. motion constrained to a spherical shell, or some other type of curved surface. Then obviously the path (motion) must be on the surface, and thereby satisfy the equation of the surface, which can be generally be written: g{y i ;x}=0 One therefore introduce constraint equations.

Euler Equations with Auxiliary Conditions Consider a case: Also include constraints: The variations are thus no longer independent.and

No terms in x appear since We write: or: The constraints equation becomes:

The left hand side involves only derivatives of f and g with respect to y and y’, whereas the right hand side involves only derivatives of f and g with respect to z and z’. Because y and z are both functions of x, the two sides may be set equal to some function of x, which we note: - (x). The complete solution requires one finds three functions: y(x), z(x), and (x). Fortunately, three relations may be used: the above two equations and the constraints Eq.

The function (x) is known as Lagrange underdetermined multiplier. In general, if there are many external constraints, one has: m equations m+n unknowns n equations Note that the constraints can also be written: Often more “useful” in problem solutions….

The  Notation To simplify writing in calculus of variations, one uses a shorthand notation. Consider:

The  Notation Condition of extremum Note: Swap  and integral: So…

The  Notation Do keep in mind that the  notation is only a shorthand expression for a more detailed and precise quantity. You can visualize the varied path  y as a virtual displacement from the actual path consistent with all forces and constraints. It is to be distinguished from dy (differential displacement) by the condition that dt=0 (fixed time)

Geodesic on a sphere