Mechanics Hamilton Principle, Lecture Notes - Physics, Study notes of Mechanics

Download Mechanics Hamilton Principle, Lecture Notes - Physics and more Study notes Mechanics in PDF only on Docsity! Mechanics Physics 151 Lecture 4 Hamilton’s Principle (Chapter 2) Administravia ! Problem Set #1 due ! Solutions will be posted on the web after this lecture ! Problem Set #2 is here ! Due next Thursday ! Next lecture (Tuesday) will be given by Srinivas and Abdol-Reza ! I will be attending a workshop at Stanford Mechanics Physics 151 Lecture 4 Hamilton’s Principle (Chapter 2) Administravia ! Problem Set #1 due ! Solutions will be posted on the web after this lecture ! Problem Set #2 is here ! Due next Thursday ! Next lecture (Tuesday) will be given by Srinivas and Abdol-Reza ! I will be attending a workshop at Stanford Configuration Space ! Generalized coordinates q1,...,qn fully describe the system’s configuration at any moment ! Imagine an n-dimensional space ! Each point in this space (q1,...,qn) corresponds to one configuration of the system ! Time evolution of the system " A curve in the configuration space configuration space real space configuration space Action Integral ! A system is moving as ! Lagrangian is ! Action I depends on the entire path from t1 to t2 ! Choice of coordinates qj does not matter ! Action is invariant under coordinate transformation ( ) 1...j jq q t j n= = ( , , ) ( ( ), ( ), ) ( )L q q t L q t q t t L t= =! ! integrate 2 1 t t I Ldt= ∫ Action, or action integral Hamilton’s Principle ! This is equivalent to Lagrange’s Equations ! We will prove this ! Three equivalent formulations ! Newton’s Eqn depends explicitly on x-y-z coordinates ! Lagrange’s Eqn is same for any generalized coordinates ! Hamilton’s Principle refers to no coordinates ! Everything is in the action integral The action integral of a physical system is stationary for the actual path We will also define “stationary” Hamilton’s Principle is more fundamental probably... Hamilton " Lagrange ! Consider 1 generalized coordinate q ! Add δq(t) to q(t), then make δq(t) " 0 ! Do this by ! α is a parameter " 0 ! η(t) is an arbitrary well-behaving function ! Let’s define ( ) ( )q t tδ αη= Continuous, non-singular, continuous η' and η'' 2 1 ( ) ( ( , ), ( , ), ) t t I L q t q t t dtα α α≡ ∫ ! ( , ) ( ) ( )q t q t tα αη= + 1 2( ) ( ) 0t tη η= = configuration space 1t 2t ( )q t ( ) ( )q t q tδ+ NB: this also depends on η(t) Calculus of Variations ! Let’s define ! If the action is stationary 2 1 ( ) ( ( , ), ( , ), ) t t I L q t q t t dtα α α= ∫ ! 0 ( ) 0dI d α α α =   =    2 1 ( ) t t dI L dq L dq dt d q d q d α α α α  ∂ ∂= + ∂ ∂ ∫ ! ! Some work! 2 1 t t L d L dq dt q dt q dα  ∂ ∂= − ∂ ∂ ∫ ! ( )tη= ( , ) ( ) ( )q t q t tα αη= + Arbitrary function for any η(t) NB: this also depends on η(t) Lagrange’s Equation ! Fundamental lemma ! We got 2 1 ( ) ( ) 0 for any ( ) x x M x x dx xη η=∫ 1 2( ) 0 for M x x x x= < < 2 1 ( ) 0 t t L d L t dt q dt q η ∂ ∂− = ∂ ∂ ∫ ! 0L d L q dt q ∂ ∂− = ∂ ∂ ! Done! Hamilton’s Principle ! Action I describes the entire motion of the system ! It is sufficient to derive the equations of motion ! Action I does not depend on the choice of the coordinates ! Lagrange formalism is coordinate invariant ! Adding dF/dt to L would add F(t2) – F(t1) to I ! It wouldn’t affect δI  Variations are 0 at t1 and t2 ! Arbitrarity of L is obvious 2 1 ( , , ) 0 t t I L q q t dtδ δ= =∫ ! ( , )dF q tL L dt ′ = + Calculus of Variation ! Technique has wider applications ! In general for ! Examples in Goldstein Section 2.2 ! Most famous: the brachistochrone problem 0Jδ = dyy dx ′ ≡2 1 ( ( ), ( ), ) x x J f y x y x x dx′= ∫ 0f d f y dx y  ∂ ∂− = ′∂ ∂  Fastest path via gravity Conservation Laws ! We’ve seen (in Lectures 1&2) conservation of linear, angular momenta and energy in Newtonian mechanics ! How do they work with Lagrange’s equations? ! Should better be the same… ! We’ll find a few differences and assumptions ! They are, in fact, limitations we ignored so far Generalized Momentum ! Generalized momentum may not look like linear momentum ! Dimension may vary, if qj is not a space coordinate ! pjqj always has the dimension of action (= work × time) ! Form may vary if V depends on velocity ! Example: a particle in EM field j j Lp q ∂≡ ∂ ! 21 2 L mv q qφ= − + ⋅A v x xp mx qA= +! Extra term due to velocity- dependent potential Symmetry ! Linear momentum p = (px, py, pz) is conjugate of (x, y, z) coordinates ! Conserved if Lagrangian does not depend explicitly on position ! I.e. if Lagrangian is invariant under space translation ! Such a system is called symmetric under space translation ! Symmetry of a system = Invariance of Lagrangian " Conservation of conjugate momentum ! Let’s study an example of angular momentum ( , , ) ( , , )x y z x x y y z z→ + ∆ + ∆ + ∆ Angular Momentum ! Consider a multi-particle system ! Suppose q1 turns the whole system around ! Example: φ in ! Assume V does not depend on ! Conjugate momentum is 1( ,..., , )i i nq q t=r r ( , , ) ( cos , sin , )i i i i i i ix y z r r zφ φ= =r iz ir dφ ( )i φr ( )i dφ φ+r φ! L Tpφ φ φ ∂ ∂≡ = ∂ ∂! ! i i = ⋅ = ⋅∑n L n Lbit ofwork Axis of rotation Total angular momentum n Conservation Laws ! Following statements are equivalent: ! System is symmetric wrt a generalized coordinate ! The coordinate is cyclic (does not appear in Lagrangian) ! The conjugate generalized momentum is conserved ! The associated generalized force is zero TorqueForceForce AngularLinearMomentum Angle around an axisDistance along an axisCoordinate RotationSpatial translationSymmetry Summary ! Derived Lagrange’s Eqn from Hamilton’s Principle ! Calculus of variation ! Discussed conservation laws ! Generalized (conjugate) momentum ! Symmetry of the system " Invariance of the Lagrangian " Conservation of momentum ! We are almost done with the basic concepts ! Finish up next Tuesday with energy conservation ! Some applications are in order " Central force problem j j Lp q ∂≡ ∂ !