Mechanics Hamiltonian Equation of Motion 2, Lecture Notes - Physics, Study notes of Mechanics

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Mechanics Physics 151 Lecture 19 Hamiltonian Equations of Motion (Chapter 8) What We Did Last Time Constructed Hamiltonian formalism Equivalent to Lagrangian formalism Simpler, but twice as many, equations Hamiltonian is conserved (unless explicitly t-dependent) Equals to total energy (unless it isn’t) ( , , ) ( , , )i iH q p t q p L q q t= − i i H q p ∂ = ∂ ii H p q ∂ = − ∂ H L t t ∂ ∂ = − ∂ ∂ Hamilton’s Principle Rewrite action integral using Hamiltonian Now δ denotes variation in the phase space i.e. both qi(t) and pi(t) are varied independently Calculation of δI goes exactly the same way Just consider 2n variables qi and pi instead of qi only ( )2 2 1 1 ( , , ) t t i it t I Ldt p q H q p t dtδ δ δ≡ = −∫ ∫ ( , ) ( ) ( )i i iq t q t tα αη≡ + ( , ) ( ) ( )i i ip t p t tα αζ≡ + ( , , , , ) ( , , )i if q q p t p qp H q p t= − This can be omitted Hamilton’s Principle δI = 0 is equivalent to Well, that was easy… One subtlety remains – the end points 0 i i f d f dq dt q ⎛ ⎞∂ ∂ − =⎜ ⎟∂⎝ ⎠ 0 i i f d f dp dt p ⎛ ⎞∂ ∂ − =⎜ ⎟∂⎝ ⎠ and ( , , , ) ( , , )i if q q p t p q H q p t= − 0i i H p dq ∂ + = 0i i Hq dp ∂ − = Hamilton’s equations End Point Constraints Variation in Lagrangian formalism required Derivation of Hamilton’s equations requires Or does it? What we really need is But f does not depend on ! So we need only 1 2( ) ( ) 0q t q tδ δ= = The end points are fixed in the config space 11 22 ( ) ( ) 0( ) ( )q t q t p t p tδ δδ δ == == More restrictive 2 2 1 1 ( ) ( ) 0 t t t t f ft t q p η ζ ⎡ ⎤ ⎡ ⎤∂ ∂ + =⎢ ⎥ ⎢ ⎥∂ ∂⎣ ⎦ ⎣ ⎦ ( , , , ) ( , , )i if q q p t p q H q p t= − p This is 0 This doesn’t have to be 0 1 2( ) ( ) 0q t q tδ δ= = as before Canonical Transformation Lagrangian Hamiltonian formalism meant moving from the configuration space to the phase space Lagrangian formalism is invariant under coordinate transformations such as Is Hamiltonian formalism invariant under similar but more general transformations? 1( , , )nq q… 1 1( , , , , , )n nq q p p… … 1( , , , )i i nQ Q q q t= … 1 1 1 1 ( , , , , , , ) ( , , , , , , ) i i n n i i n n Q Q q q p p t P P q q p p t = = … … … … Canonical Transformation Well, no. It’s much too general, but… There is a subset of such transformations – canonical transformations – that work We will find the rules Hamiltonian formalism is more forgiving Goal: Find the transformation that makes the problem easiest to solve We may make coordinates cyclic, as we did by using polar coordinates in the central-force problem 1 1( , , , , , , )i i n nQ Q q q p p t= … … 1 1( , , , , , , )i i n nP P q q p p t= … … Mechanics Physics 151 Lecture 19 Hamiltonian Equations of Motion (Chapter 8) What We Did Last Time Constructed Hamiltonian formalism Equivalent to Lagrangian formalism Simpler, but twice as many, equations Hamiltonian is conserved (unless explicitly t-dependent) Equals to total energy (unless it isn’t) ( , , ) ( , , )i iH q p t q p L q q t= − i i H q p ∂ = ∂ ii H p q ∂ = − ∂ H L t t ∂ ∂ = − ∂ ∂ Hamilton’s Principle Rewrite action integral using Hamiltonian Now δ denotes variation in the phase space i.e. both qi(t) and pi(t) are varied independently Calculation of δI goes exactly the same way Just consider 2n variables qi and pi instead of qi only ( )2 2 1 1 ( , , ) t t i it t I Ldt p q H q p t dtδ δ δ≡ = −∫ ∫ ( , ) ( ) ( )i i iq t q t tα αη≡ + ( , ) ( ) ( )i i ip t p t tα αζ≡ + ( , , , , ) ( , , )i if q q p t p qp H q p t= − This can be omitted Hamilton’s Principle δI = 0 is equivalent to Well, that was easy… One subtlety remains – the end points 0 i i f d f dq dt q ⎛ ⎞∂ ∂ − =⎜ ⎟∂⎝ ⎠ 0 i i f d f dp dt p ⎛ ⎞∂ ∂ − =⎜ ⎟∂⎝ ⎠ and ( , , , ) ( , , )i if q q p t p q H q p t= − 0i i H p dq ∂ + = 0i i Hq dp ∂ − = Hamilton’s equations End Point Constraints Variation in Lagrangian formalism required Derivation of Hamilton’s equations requires Or does it? What we really need is But f does not depend on ! So we need only 1 2( ) ( ) 0q t q tδ δ= = The end points are fixed in the config space 11 22 ( ) ( ) 0( ) ( )q t q t p t p tδ δδ δ == == More restrictive 2 2 1 1 ( ) ( ) 0 t t t t f ft t q p η ζ ⎡ ⎤ ⎡ ⎤∂ ∂ + =⎢ ⎥ ⎢ ⎥∂ ∂⎣ ⎦ ⎣ ⎦ ( , , , ) ( , , )i if q q p t p q H q p t= − p This is 0 This doesn’t have to be 0 1 2( ) ( ) 0q t q tδ δ= = as before Canonical Transformation Lagrangian Hamiltonian formalism meant moving from the configuration space to the phase space Lagrangian formalism is invariant under coordinate transformations such as Is Hamiltonian formalism invariant under similar but more general transformations? 1( , , )nq q… 1 1( , , , , , )n nq q p p… … 1( , , , )i i nQ Q q q t= … 1 1 1 1 ( , , , , , , ) ( , , , , , , ) i i n n i i n n Q Q q q p p t P P q q p p t = = … … … … Canonical Transformation Well, no. It’s much too general, but… There is a subset of such transformations – canonical transformations – that work We will find the rules Hamiltonian formalism is more forgiving Goal: Find the transformation that makes the problem easiest to solve We may make coordinates cyclic, as we did by using polar coordinates in the central-force problem 1 1( , , , , , , )i i n nQ Q q q p p t= … … 1 1( , , , , , , )i i n nP P q q p p t= … … Principle of Least Action Principle of Least Action is a confusing term Action changed its meaning historically I would rather call Hamilton’s principle as “the principle of least action” Least is not strictly true – Extreme would be the right word Let’s follow Goldstein’s usage for today “Principle of least action” is expressed as 2 1 0 t i it p q dt∆ =∫ “action” ∆-variation What are they? ∆-Variation In the configuration space End points of integration move by 1st-order approximation Same for t2 We will later impose ( )q t 1t 2t 1 1t t+ ∆ 2 2t t+ ∆ ( ) ( )q t q tδ+1 1 1 1( ) ( , ) ( )i i iq t q t t q tα∆ = + ∆ − 1 1 1 1( ) ( ) ( )i i iq t q t t q tδ∆ = ∆ + 1 2( ) ( ) 0i iq t q t∆ = ∆ = These points will be the same ∆-Variation Considering only 1st-order terms of ∆t and η(t) Last 2 terms are δ-variation with free end points Assume Lagrange’s equation 2 2 2 2 1 1 1 1 ( ) (0) t t t t t t t t Ldt L dt L dtα +∆ +∆ ∆ = −∫ ∫ ∫ 2 2 2 1 1 1 2 2 1 1( ) ( ) ( ) (0) t t t t t t Ldt L t t L t t L dt L dtα∆ = ∆ − ∆ + −∫ ∫ ∫ 2 2 2 1 1 1 t t t i it t i i i t L d L LLdt q dt q dq dt q q δ δ δ ⎧ ⎫⎛ ⎞ ⎡ ⎤∂ ∂ ∂⎪ ⎪= − +⎨ ⎬⎜ ⎟ ⎢ ⎥∂ ∂⎪ ⎪⎝ ⎠ ⎣ ⎦⎩ ⎭ ∫ ∫ [ ]2 2 11 t t i i tt Ldt L t p qδ∆ = ∆ +∫ [ ]2 2 11 t t i i tt Ldt p qδ δ=∫ ∆-Variation Using ( )q t 1t 2t 1 1t t+ ∆ 2 2t t+ ∆ ( ) ( )q t q tδ+ 1 1 1 1( ) ( ) ( )i i iq t q t t q tδ∆ = ∆ + 2 2 2 2( ) ( ) ( )i i iq t q t t q tδ∆ = ∆ + [ ]2 2 11 t t i i tt Ldt L t p qδ∆ = ∆ +∫ [ ] [ ] 2 2 11 2 1 t t i i i i tt t i i t Ldt L t p q t p q p q H t ∆ = ∆ − ∆ + ∆ = ∆ − ∆ ∫ This is a very general form Now we examine a more restrictive case Principle of Least Action Now, what is ? Let’s consider a simple example A particle under conservative potential Principle of least action is equivalent to The particle “tries” to move from point 1 to point 2 with minimum kinetic energy × time As slowly as possible, yet spending as little time as possible 2 1 i i p q dt∫ 2 ( ) 2 mL V= −v x 2 2i ip q m T= =v 2 1 0 t t Tdt∆ =∫ Principle of Ideal Commuting? Principle of Least Action For a free particle, kinetic energy T is constant This also means that a free particle takes the shortest path = straight line between point 1 and point 2 Similar to Fermat’s principle in optics Light travels the fastest path between two points 2 1 2 1( ) 0 t t Tdt T t t∆ = ∆ − =∫ Principle of Least Time Summary Hamilton’s Principle in the Hamiltonian formalism Derivation was simple Additional end-point constraints Not strictly needed, but adds flexibility to the definition of the action integral Principle of Least Action ∆-derivative allows change of time For simple systems, equivalent to 1 2 1 2( ) ( ) ( ) ( ) 0q t q t p t p tδ δ δ δ= = = = ( )2 1 ( , , ) 0 t i it I p q H q p t dtδ δ≡ − =∫ 2 1 0 t i it p q dt∆ =∫ 2 1 0 t t Tdt∆ =∫