Ch. 2: Variational Principles & Lagranges Eqtns Sect. 2.1: Hamiltons Principle Our derivation of Lagranges Eqtns from DAlemberts Principle: Used Virtual Work - A Differential Principle. (A LOCAL principle). Here: An alternate derivation from Hamiltons Principle: An Integral (or Variational) Principle (A GLOBAL principle). More general than DAlemberts Principle. Based on techniques from the Calculus of Variations. Brief discussion of derivation & of Calculus of Variations. More details: See the text! System: n generalized coordinates q1,q2,q3,..qn.
At time t1: These all have some value. At a later time t2: 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 qi as n coordinate axes. At time t1: Configuration of System is represented by a point in this space. At a later time t2: 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. 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(qi,qi,t): Qi - (U/qi) + (d/dt)[(U/qi)] If U depends only on qi (& not on qi & t), U = V & the system is conservative. Monogenic systems, Hamiltons Principle: The motion of the system (in configuration space) from time t1 to time t2 is such that the line integral (the action or action integral) I = L dt (limits t1 < t < t2) 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 qi & t) Hamiltons Principle (HP) I = L dt (limits t1 < t < t2, 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 t1 to t2, it will ACTUALLY travel along path for which I is
an extremum (usually a minimum). I = L dt (limits t1 < t < t2, L = T - V ) In the terminology & notation from the calculus of variations: HP the motion is such that the variation of I (fixed t1 & t2) is zero: L dt = 0 (limits t1 < t < t2) (1)
Arbitrary variation (calculus of variations). 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 Lagranges Eqtns. That is, we can derive (1) from Lagranges Eqtns. However this text & (most texts) do it the other way around & derive Lagranges Eqtns from (1). Advantage: Valid in any system of generalized coords.!! More on HP (from Marions book) History, philosophy, & general discussion, which is worth briefly mentioning (not in Goldstein!). Historically, to overcome some practical difficulties of Newtons
mechanics (e.g. needing all forces & not knowing the forces of constraint) Alternate procedures were developed Hamiltons Principle Lagrangian Dynamics Hamiltonian Dynamics Also Others! All such procedures obtain results 100% equivalent to Newtons 2nd Law: F = dp/dt Alternate procedures are NOT new theories! But
reformulations of Newtonian Mechanics in different math language. Hamiltons Principle (HP): Applicable outside particle mechanics! For example to fields in E&M. HP: Based on experiment! 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 Newtons Laws!
HP: Given as a (single, simple) postulate. HP & Lagrange Eqtns apply (as weve seen) to non-conservative systems. 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: Heros 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. i = r Says nothing about the Law of Refraction! Minimal Principles: Fermat, 1657: Optics: Fermats 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: i = r
Law of Refraction: Snells Law Maupertuis, 1747: Mechanics: Maupertuiss 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 Hamiltons 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: [T - V]dt = Ldt = 0 (limits t1 < t < t2) Sect. 2.2: Variational Calculus Techniques Could spend a semester on this. Really (should be) a math course! Brief pure math discussion! Marions 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! 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 x]dx (fixed limits x1 < x < x2) 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]dx x] which depends on the functional form of the dependent variable y(x). A function of a function. Basic problem restated: Given f[y(x),y(x);x]dx x], find (for fixed x1, x2) the function(s) y(x) which minimize (or maximize) J f[y(x),y(x);x]dx x]dx (limits x1 < x < x2) 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 J = f[y(x),y(x);x]dx 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! Solution to basic problem : The text proves (p 37 & 38. More details, see Marion, Ch. 6) that to J f[y(x),y(x);x]dx x]dx or minimize (or maximize) (limits x1 < x < x2)
J = f[y(x),y(x);x]dx x]dx =0 The functional f must satisfy: (f/y) - (d[f/y]/dx) = 0 Euler, 1744. Applied to mechanics Euler - Lagrange Equation Various pure math applications, p 39-43 Read on your own! Eulers Equation
Sect. 2.3 Derivation of Lagrange Eqtns from HP 1st, 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 J = 0. Assumed one dependent variable y(x). In mechanics, we often have problems with many dependent variables: y1(x), y2(x), y3(x), In general, have a functional like: f = f[y1(x),y1(x),y2(x),y2(x), ;x]dx x] yi(x) dyi(x)/dx Abbreviate as f = f[yi(x),yi(x);x]dx x], i = 1,2, ,n
Functional: f = f[yi(x),yi(x);x]dx x], i = 1,2, ,n Calculus of variations problem: Simultaneously find the n paths yi(x), i = 1,2, ,n, which minimize (or maximize) the integral: J f[yi(x),yi(x);x]dx x]dx (i = 1,2, ,n, fixed limits x1 < x < x2) Or for which J = 0 Follow the derivation for one independent variable & get: (f/yi) - (d[f/yi]dx) = 0 Eulers Equations (Several dependent variables)
(i = 1,2, ,n) Summary: Forcing J f[yi(x),yi(x);x]dx x]dx (i = 1,2, ,n, fixed limits x1 < x < x2) To have an extremum (or forcing J = f[yi(x),yi(x);x]dx x]dx = 0) requires f to satisfy: (f/yi) - (d[f/yi]dx) = 0 (i = 1,2, ,n) Eulers Equations
HP The system motion is such that I = L dt is an extremum (fixed t1 & t2) The variation of this integral I is zero: L dt = 0 (limits t1 < t < t2) HP Identical to abstract calculus of variations problem of with replacements: J L dt;x]dx J L dt x t ;x]dx yi(x) qi(t) yi(x) dqi(t)/dt = qi(t) f[yi(x),yi(x);x]dx x] L(qi,qi;x]dx t)
The Lagrangian L satisfies Eulers eqtns replacements! Combining HP with Eulers eqtns gives: (d/dt)[(L/qj)] - (L/qj) = 0 (j = 1,2,3, n) with these Summary: HP gives Lagranges Eqtns: (d/dt)[(L/qj)] - (L/qj) = 0 (j = 1,2,3, n) Stated another way, Lagranges Eqtns ARE Eulers eqtns in the
special case where the abstract functional f is the Lagrangian L! They are sometimes called the Euler-Lagrange Eqtns.
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