Ch. 2: Variational Principles & Lagrange’s Eqtns Sect. 2.1: Hamilton’s Principle Our derivation of Lagrange’s Eqtns from D’Alembert’s Principle: Used.

Presentation on theme: "Ch. 2: Variational Principles & Lagrange’s Eqtns Sect. 2.1: Hamilton’s Principle Our derivation of Lagrange’s Eqtns from D’Alembert’s Principle: Used."— Presentation transcript:

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2 Ch. 2: Variational Principles & Lagrange’s Eqtns Sect. 2.1: Hamilton’s Principle Our derivation of Lagrange’s Eqtns from D’Alembert’s Principle: Used Virtual Work - A Differential Principle. (A LOCAL principle). Here: An alternate derivation from Hamilton’s Principle: An Integral (or Variational) Principle (A GLOBAL principle). More general than D’Alembert’s Principle. –Based on techniques from the Calculus of Variations. –Brief discussion of derivation & of Calculus of Variations. More details: See the text!

3 System: n generalized coordinates q 1,q 2,q 3,..q n. –At time t 1 : These all have some value. –At a later time t 2 : They have changed according to the eqtns of motion & all have some other value. System Configuration: A point in n-dimensional space (“Configuration Space”), with q i as n coordinate “axes”. –At time t 1 : Configuration of System is represented by a point in this space. –At a later time t 2 : Configuration of System has changed & that point has moved (according to eqtns of motion) in this space. –Time dependence of System Configuration: The point representing this in Configuration Space traces out a path.

4 Monogenic Systems  All Generalized Forces (except constraint forces) are derivable from a Generalized Scalar Potential that may be a function of generalized coordinates, generalized velocities, & time: U(q i,q i,t): Q i  - (  U/  q i ) + (d/dt)[(  U/  q i )] –If U depends only on q i (& not on q i & t), U = V & the system is conservative.

5 Monogenic systems, Hamilton’s Principle: The motion of the system (in configuration space) from time t 1 to time t 2 is such that the line integral (the action or action integral) I = ∫L dt (limits t 1 < t < t 2 ) has a stationary value for the actual path of motion. L  T - V = Lagrangian of the system L = T - U, (if the potential depends on q i & t)

6 Hamilton’s Principle (HP) I = ∫L dt (limits t 1 < t < t 2, L = T - V ) Stationary value  I is an extremum (maximum or minimum, almost always a minimum). In other words: Out of all possible paths by which the system point could travel in configuration space from t 1 to t 2, it will ACTUALLY travel along path for which I is an extremum (usually a minimum).

7 I = ∫L dt (limits t 1 < t < t 2, L = T - V ) In the terminology & notation from the calculus of variations : HP  the motion is such that the variation of I (fixed t 1 & t 2 ) is zero: δ∫L dt = 0 (limits t 1 < t < t 2 ) (1) δ  Arbitrary variation (calculus of variations). δ plays a role in the calculus of variations that the derivative plays in calculus. Holonomic constraints  (1) is both a necessary & a sufficient condition for Lagrange’s Eqtns. –That is, we can derive (1) from Lagrange’s Eqtns. –However this text & (most texts) do it the other way around & derive Lagrange’s Eqtns from (1). –Advantage: Valid in any system of generalized coords.!!

8 More on HP (from Marion’s book) History, philosophy, & general discussion, which is worth briefly mentioning (not in Goldstein!). Historically, to overcome some practical difficulties of Newton’s mechanics (e.g. needing all forces & not knowing the forces of constraint)  Alternate procedures were developed Hamilton’s Principle  Lagrangian Dynamics  Hamiltonian Dynamics  Also Others!

9 All such procedures obtain results 100% equivalent to Newton’s 2 nd Law: F = dp/dt  Alternate procedures are NOT new theories! But reformulations of Newtonian Mechanics in different math language. Hamilton’s Principle (HP): Applicable outside particle mechanics! For example to fields in E&M. HP: Based on experiment!

10 HP: Philosophical Discussion HP:  No new physical theories, just new formulations of old theories HP: Can be used to unify several theories: Mechanics, E&M, Optics, … HP: Very elegant & far reaching! HP: “More fundamental” than Newton’s Laws! HP: Given as a (single, simple) postulate. HP & Lagrange Eqtns apply (as we’ve seen) to non-conservative systems.

11 HP: One of many “Minimal” Principles: (Or variational principles) –Assume Nature always minimizes certain quantities when a physical process takes place –Common in the history of physics History: List of (some) other minimal principles: –Hero, 200 BC: Optics: Hero’s Principle of Least Distance: A light ray traveling from one point to another by reflection from a plane mirror, always takes shortest path. By geometric construction:  Law of Reflection. θ i = θ r Says nothing about the Law of Refraction!

12 “Minimal” Principles: –Fermat, 1657: Optics: Fermat’s Principle of Least Time: A light ray travels in a medium from one point to another by a path that takes the least time.  Law of Reflection: θ i = θ r  Law of Refraction: “Snell’s Law” –Maupertuis, 1747: Mechanics: Maupertuis’s Principle of Least Action: Dynamical motion takes place with minimum action: Action  (Distance)  (Momentum) = (Energy)  (Time) Based on Theological Grounds!!! (???) Lagrange: Put on firm math foundation. Principle of Least Action  HP

13 Hamilton’s Principle (As originally stated 1834-35) –Of all possible paths along which a dynamical system may move from one point to another, in a given time interval (consistent with the constraints), the actual path followed is one which minimizes the time integral of the difference in the KE & the PE. That is, the one which makes the variation of the following integral vanish: δ∫[T - V]dt = δ∫Ldt = 0 (limits t 1 < t < t 2 )

14 Sect. 2.2: Variational Calculus Techniques Could spend a semester on this. Really (should be) a math course! –Brief pure math discussion! –Marion’s book on undergrad mechanics, devotes an entire chapter (Ch. 6) –Useful & interesting. Read details (Sect. 2.2) on your own. –Will summarize most important results. No proofs, only results!

15 Consider the following problem in the xy plane: The Basic Calculus of Variations Problem: Determine the function y(x) for which the integral J  ∫f[y(x),y(x);x]dx (fixed limits x 1 < x < x 2 ) is an extremum (max or min) y(x)  dy/dx (Note: The text calls this y(x)!) –Semicolon in f separates independent variable x from dependent variable y(x) & its derivative y(x) –f  A GIVEN functional. Functional  Quantity f[y(x),y(x);x] which depends on the functional form of the dependent variable y(x). “A function of a function”.

16 Basic problem restated: Given f[y(x),y(x);x], find (for fixed x 1, x 2 ) the function(s) y(x) which minimize (or maximize) J  ∫f[y(x),y(x);x]dx (limits x 1 < x < x 2 )  Vary y(x) until an extremum (max or min; usually min!) of J is found. Stated another way, vary y(x) so that the variation of J is zero or δJ = δ∫f[y(x),y(x);x]dx =0 Suppose the function y = y(x) gives J a min value:  Every “neighboring function”, no matter how close to y(x), must make J increase!

17 Solution to basic problem : The text proves (p 37 & 38. More details, see Marion, Ch. 6) that to minimize (or maximize) J  ∫f[y(x),y(x);x]dx (limits x 1 < x < x 2 ) or δJ = δ∫f[y(x),y(x);x]dx =0  The functional f must satisfy: (  f/  y) - (d[  f/  y]/dx) = 0  Euler’s Equation –Euler, 1744. Applied to mechanics  Euler - Lagrange Equation –Various pure math applications, p 39-43 –Read on your own!

18 Sect. 2.3 Derivation of Lagrange Eqtns from HP 1 st, extension of calculus of variations results to Functions with Several Dependent Variables Derived Euler Eqtn = Solution to problem of finding path such that J = ∫f dx is an extremum or δJ = 0. Assumed one dependent variable y(x). In mechanics, we often have problems with many dependent variables: y 1 (x), y 2 (x), y 3 (x), … In general, have a functional like: f = f[y 1 (x),y 1 (x),y 2 (x),y 2 (x), …;x] y i (x)  dy i (x)/dx Abbreviate as f = f[y i (x),y i (x);x], i = 1,2, …,n

19 Functional: f = f[y i (x),y i (x);x], i = 1,2, …,n Calculus of variations problem: Simultaneously find the “n paths” y i (x), i = 1,2, …,n, which minimize (or maximize) the integral: J  ∫f[y i (x),y i (x);x]dx (i = 1,2, …,n, fixed limits x 1 < x < x 2 ) Or for which δJ = 0 Follow the derivation for one independent variable & get: (  f/  y i ) - (d[  f/  y i ]dx) = 0 (i = 1,2, …,n)  Euler’s Equations (Several dependent variables)

20 Summary: Forcing J  ∫f[y i (x),y i (x);x]dx (i = 1,2, …,n, fixed limits x 1 < x < x 2 ) To have an extremum (or forcing δJ = δ∫f[y i (x),y i (x);x]dx = 0) requires f to satisfy: (  f/  y i ) - (d[  f/  y i ]dx) = 0 (i = 1,2, …,n)  Euler’s Equations HP  The system motion is such that I = ∫L dt is an extremum (fixed t 1 & t 2 )  The variation of this integral I is zero: δ∫L dt = 0 (limits t 1 < t < t 2 )

21 HP  Identical to abstract calculus of variations problem of with replacements: J  ∫L dt; δJ  δ∫L dt x  t ; y i (x)  q i (t) y i (x)  dq i (t)/dt = q i (t) f[y i (x),y i (x);x]  L(q i,q i ;t)  The Lagrangian L satisfies Euler’s eqtns with these replacements !  Combining HP with Euler’s eqtns gives: (d/dt)[(  L/  q j )] - (  L/  q j ) = 0 (j = 1,2,3, … n)

22 Summary: HP gives Lagrange’s Eqtns: (d/dt)[(  L/  q j )] - (  L/  q j ) = 0 (j = 1,2,3, … n) –Stated another way, Lagrange’s Eqtns ARE Euler’s eqtns in the special case where the abstract functional f is the Lagrangian L!  They are sometimes called the Euler-Lagrange Eqtns.