Understanding Classical Mechanics: Equations of Motion on Parabolic Surfaces
Explore the equations of motion for a particle on a parabolic surface influenced by gravity, highlighting stable circular motion, and transforming to polar coordinates. Delve into the Euler-Lagrange equations for a deeper understanding.
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PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 10: Continue reading Chapter 3 & 6 1. Constants of the motion 2. Conserved quantities 3. Legendre transformations 9/21/2018 PHY 711 Fall 2018 -- Lecture 10 1
9/21/2018 PHY 711 Fall 2018 -- Lecture 10 2
Public Talk Lecturer -- Mon. 9/24/2018 at 7 PM 9/21/2018 PHY 711 Fall 2018 -- Lecture 10 3
Physics Colloquium: Tue. 9/25/2018 at 4 PM Olin 101 Prof. Dava Newman Lecturer from MIT 9/21/2018 PHY 711 Fall 2018 -- Lecture 10 4
Summary of Lagrangian formalism (without constraints) independen For = L dt generalize t q coordinate d s ( : ) t q ( , ) q ) ( ( , ) t L L q t t d L = 0 q L d L = = Note that if then , 0 0 q dt q L (constant) = q 9/21/2018 PHY 711 Fall 2018 -- Lecture 10 5
Consider a particle of mass m moving frictionlessly on a parabola z=c(x2+y2) under the influence of gravity. Find the equations of motion, particularly showing stable circular motion. ( ) m ( ) ( ) 2 = + + + + 2 2 2 2 2 ( , , , ) L x y x y 4 x y c xx yy mgc x y 2 9/19/2018 PHY 711 Fall 2018 -- Lecture 9 6
( ) m ( ) ( ) 2 = + + + + 2 2 2 2 2 ( , , , ) L x y x y 4 x y c xx yy mgc x y 2 Transform to polar coor cos x r = dinates; = sin y r m ( ) ( , , , ) L r Euler-Lagrange equations L L d dt = + + 2 2 2 2 2 2 c r r 2 4 r r r mg cr 2 dmr dt = 2 0 0 = 0 2 Let (constant) mr z L r d dt L r = 0 9/19/2018 PHY 711 Fall 2018 -- Lecture 9 7
m ( ) ( , , , ) L r = + + 2 2 2 2 2 2 c r r 2 4 r r r mgcr 2 L r d dt L r = 0 ( ) d dt d dt ( ) + 1 4 + = 2 2 2 2 2 4 2 0 mr mr c r mgcr mr c r ( ) 2 ( ) + 1 4 + = 2 2 2 2 c r 2 4 0 mgcr mr c r mr z 3 mr Now consider the case where initially the particle is moving in a circle 2 g c = = 2 at height and 2 with 0. z mz mr g c r 0 0 0 0 z = + Consider small perturbation to the motion: r r r 0 9/19/2018 PHY 711 Fall 2018 -- Lecture 9 8
( ) 2 d dt ( ) + 1 4 + = 2 2 2 2 c r 2 4 0 mgcr mr c r mr z 3 mr Consider small perturbation to the motion: where initially the particle is moving in a circle = + r r r 0 2 g c = = 2 at height and 2 with 0. z m z mr gc r 0 0 0 0 z Keeping terms to linear orde r: ( ) 1 2 + = 2 2 c r 8 0 mgc r m r 0 8 gc = r r 1 2 + 2 2 c r 0 8 gc = + cos r A t + 2 2 1 2 c r 0 9/19/2018 PHY 711 Fall 2018 -- Lecture 9 9
Examples of constants of the motion: Example one : 1 - dimensiona potential l : ( ) = + + 2 2 2 ( ) L m x y z V z 1 2 d = 0 (constant) m x m x p x dt d = 0 (constant) m y m y p y dt d V = m z dt z 9/21/2018 PHY 711 Fall 2018 -- Lecture 10 10
Examples of constants of the motion: Example 2: Motion in a central potential = + ( ) 2 2 2 ( ) L m r r V r 1 2 dmr dt d mr dt = 2 2 0 (constant) mr p 2 p mr V r V r = = 2 mr 3 9/21/2018 PHY 711 Fall 2018 -- Lecture 10 11
Recall alternative form of Euler-Lagrange equations: Starting from : ( L ) = ( , ) ( , ) L L q t q t t d L = 0 dt q q dL L L L = + + Also note that : q q dt q q t d L L = + q dt q t d L L = L q dt q t 9/21/2018 PHY 711 Fall 2018 -- Lecture 10 12
Additional constant of the motion: L = If ; 0 t d L L = = the n : 0 L q dt q t L + = (constant) L q E q - Example = : 1 one dimensiona potential l : ( ) + 2 2 2 ( ) L m x y z V z 1 2 ( ( ) ) d + + = 2 2 2 2 2 2 ( ) 0 m x y z V z m x m y m z 1 2 dt ( For this case, we also have ( ) ) + + + = 2 2 2 ( ) (constant) x mx p m x y z V z E 1 2 and my p y 2 y p 2 x p = + + + 2 ( ) E mz V z 1 2 2 2 m m 9/21/2018 PHY 711 Fall 2018 -- Lecture 10 13
Additional constant of the motion -- continued: ; 0 If t L = d L L = = the n : 0 L q dt q t L = (constant) L q E q Example = 2 Motion : + central a in potential ( ) 2 2 2 ( ) L m r r V r 1 2 ( ( ) ) d + = 2 2 2 2 2 2 ( ) 0 m r r V r m r mr 1 2 dt ( ( ) ) + + = 2 2 2 ( ) (constant) 2 mr m r r V r so have E 1 2 For this case, we al p 2 p mr = + + 2 ( ) E mr V r 1 2 2 2 9/21/2018 PHY 711 Fall 2018 -- Lecture 10 14
Other examples q ( ) ( ) = + + + + 2 2 2 L m x y z B xy yx 1 2 0 2 c L z = = 0 (constant) mz p z L q = E q L q ( ) ( ) = + + + + 2 2 2 m x y z B xy yx 0 2 c q ( ) ( ) + + + 2 2 2 m x y z B xy yx 1 2 0 2 c 2 z p ( ) ( ) = + + = + + 2 2 2 2 2 m x y z m x y 1 2 1 2 2 m 9/21/2018 PHY 711 Fall 2018 -- Lecture 10 15
Other examples q c ( ) = + + 2 2 2 L m x y z B xy 1 2 0 L z L x = = 0 (constant) mz p z = = 0 (constant) mx p x L q = E q L q c ( ) = + + 2 2 2 m x y z B xy 0 q c ( ) + + + 2 2 2 m x y z B xy 1 2 0 2 2 p 2 zp m ( ) = + + = + + 2 2 2 2 m x y z my x 1 2 1 2 2 m 9/21/2018 PHY 711 Fall 2018 -- Lecture 10 16
Lagrangian picture independen For q L L generalize t q coordinate d s ( : ) t q ( , ) L ) = ( ( , ) t t t d L = 0 dt q q Second order differenti equations al for ( ) q t Switching variables Legendre transformation 9/21/2018 PHY 711 Fall 2018 -- Lecture 10 17
Mathematical transformations for continuous functions of several variables & Legendre transforms: Simple change of variables: ( , ) ( , ) ??? z x y x y z z z = + ( , ) z x y dz dx dy x y y x x x x = = + / ) )y ( , ) x y z dx dy dz y z y y ( ( z z But : x / y z x z 9/21/2018 PHY 711 Fall 2018 -- Lecture 10 18
Simple change of variables -- continued: z z = + ( , ) z x y dz dx dy x y y x x x = + ( , ) x y z dx dy dz y z y z ( ( ) ) / / z z y x 1 / x y x z = = x ( ) z x y z y y 9/21/2018 PHY 711 Fall 2018 -- Lecture 10 19
Simple change of variables -- continued: z z Example: = + ( , ) z x y dz dx dy x y 2 + = x y ( , ) z x y y e x x x ( ) 1/2 = ( , ) x y z ln z y = + ( , ) x y z dx dy dz y z y z ( ( ) ) / / z z y x ? x y = 1 / x z ? = x ( ) z x y z y y 1 z 1 2 + x y 1 e xe = = ( ) 1/2 2 + ( ) x y 2 1/2 xe 2 ln 2 z y + x y 2 2 ln z y 9/21/2018 PHY 711 Fall 2018 -- Lecture 10 20
Mathematical transformations for continuous functions of several variables & Legendre transforms continued: ) , ( y x z z = + z x y dz dx dy y x z z Let and u v x y y x Define new function w u w y = + ( , ) w u y dw du dy y u = = = + For dw , w xdu z + ux vdy dw dz udx xdu udx vdy udx xdu = w u w y z y = = = x v y u x 9/21/2018 PHY 711 Fall 2018 -- Lecture 10 21
For thermodynamic functions: = Internal energy : ( , ) U U S V = dU TdS PdV U U = + dU dS dV S V V S U U = = T P S : V V S = = + Enthalpy ( , ) H H S P U PV H H = + + = + = + dH dU PdV VdP TdS VdP dS dP S P P S H H = = T V S P P S 9/21/2018 PHY 711 Fall 2018 -- Lecture 10 22
9/21/2018 PHY 711 Fall 2018 -- Lecture 10 23
Lagrangian picture independen For L L generalize t q coordinate d s ( : ) t q ( , ) L ) = ( ( , ) t q t t d L = 0 dt q q Second order differenti equations al for ( ) q t Switching variables Legendre transformation ( t q H H = , ) ) Define : ( ( , ) p t t L = = where H q p L p q L L L = + dH q dp p d q dq d q dt q q t 9/21/2018 PHY 711 Fall 2018 -- Lecture 10 24
Hamiltonian picture continued ( , ) ) = ( ( , ) H H q t p t t L = = where H q p L p q L L L = + dH q dp p d q dq d q dt q q t H H H = + + dq dp dt q p t H L d L H L H = = = = q p p q dt q q t t 9/21/2018 PHY 711 Fall 2018 -- Lecture 10 25
