4. Lagrangians and Hamiltonians

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Setup

\(Clear["Global`*"]; 
Off[General::spell]; 
Off[General::spell1]; 
    \)

Needs["Calculus`VectorAnalysis`"]

<< "D:\Mma\Inits\ch4init.m"

Clear["Global`*"];

4.2 Problems

4.2.1  Lagrangian Problems

Problem 4:  Hoop Rolling on an Incline of Angle ϕ

Part (a)   Assume that x and θ are generalized coordinates and construct the Lagrangian.  Write the slippage constraint and use Lagrangian multipliers to obtain the equations of motion for the variables x and θ.

T  =  m/2 (x'[t]^2  +  r^2 θ'[t]^2)

V  =  m g (Len  -  x[t]) Sin[ϕ]

L  =  T - V

eq1  =  r θ[t]  -  x[t]  ==  0

\(constraintForce\  = 
      \ {Q[1] -> λ\ D[eq1[\([1]\)], x[t]]

        			\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , Q[2] -> λ\ D[eq1[\([1]\)], θ[t]]}\)

eqX  =  Lag[x, L, Q[1]] /. constraintForce

eqθ  =   Lag[θ, L, Q[2]] /. constraintForce

Part (b)   Solve the equarions of motion for x[t] and θ[t].

initial  = 
       {x[0] == x0, x'[0] == v0, θ[0] == x0/r,  
        θ'[0] == v0/r}

eq2  =  Join[{eqX, eqθ}, initial];

xtSol = DSolve[eq2, {x, θ}, t] // Flatten

λSol  =  Solve[eq1 //. xtSol, {λ}] // Flatten

xtSol2  =  xtSol //. λSol

Part (c)   Plot the results.

values = {x0 -> 0, v0 -> 0, g -> -0.1, ϕ -> π/4, r -> 2}

posx = x[t] Sin[ϕ]  +  r Sin[θ[t]]

posy = x[t] Cos[ϕ]  +  r Cos[θ[t]]

point = {posx, posy} /. xtSol2 //. values // N

\(\(ParametricPlot[point // Evaluate, {t, 0, 70}
, AspectRatio -> 1]; 
    \)\)

Problem 5:  Sphere Rolling on an Fixed Sphere

Clear["Global`*"]

Part (a)   Spheres have radii a and b.  Assume the generalized coordinates are {r[t], θ1[t], and θ2[t]} and construct the Lagrangian.  Write the  constraint that there is no slippage.  Use Lagrangian multipliers to obtain the equations of motion for the variables x and θ.

Take the origin at the center of the fixed sphere.  Let r be the distance to the center of the rolling sphere and I1 be its moment of inertia.

Tcm  =  m/2 (r'[t]^2  +  r[t]^2 θ1'[t]^2)

Trot  =  I1/2 (θ1'[t]  +  θ2'[t])^2

T  =  Tcm  +  Trot

V = m g r[t] Cos[θ1[t]]

L  =  T - V

\(constraints = {b\ θ1[t] - a\ θ2[t] == 0

        	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
        r[t] - a - b == 0}\)

\(conRule = {DSolve[constraints[\([1]\)], {θ2}, t]

          	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
          DSolve[constraints[\([2]\)], {r}, t]} // Flatten\)

coordinates = {r, θ1, θ2};

coordinatesT = coordinates // Map[Composition[#][t]&, #]&

\(Clear[Q]; 

    Q[i_] := 
	
      Sum[λ[j]*
          D[constraints[\([j, 1]\)]
			, coordinatesT[\([i]\)]\ ], {j, 1, 
          2}]\)

Array[Q, 3]

eqMotion = Table[Lag[coordinates[[i]], L, Q[i]], {i, 3}]

Part (b)   Show that the equation for θ1[t] can be reduced to a first integral.

\(sol1 = 
      \((Solve[eqMotion //. conRule, {\(\(θ1'\)'\)[t], λ[1], 
              λ[2]}]
	 // Flatten)\)\)

eqθ  =  
      (θ1''[t] //. sol1) - θ1''[t] == 0

SetOptions[Limit, Analytic -> True]

\(intθ\  = \ 
      \((Integrate[eqθ[\([1]\)]\ \(θ1'\)[t], {t, 0, t}]

            	 //. {θ1[0] -> 0, \(θ1'\)[0] -> 0} // Simplify)
        \)\)

solθ = 
      Solve[intθ == 0, {θ1'[t]}][[2]]

θ1'[t]^2 //. solθ // Simplify

Part (c)   Find the angle at which the sphere falls off.

eq1 = 0 == λ[2] //. sol1 //. solθ // Simplify

sol2  =  Solve[eq1, Cos[θ1[t]]] // Flatten

sol3  =  
      Solve[sol2 //. {Rule -> Equal}, θ1[t]][[2]]

(θ1[t]/Degree) //. sol3 //. {I1 -> (2/5) m a^2} // N

Problem 6:  Mass Falling Through a Hole in a Table

Clear["Global`*"]

Part (a)   Find the equations of motion for the variables r[t] and ϕ[t].

T1 = m1/2 (r'[t]^2  +  (r[t] ϕ'[t])^2)

V1 = 0;

T2 = m2/2 r'[t]^2

V2 = -m2 g (len - r[t])

L = (T1 + T2) - (V1 + V2)

Lag[r, L]

eqR = 
      ((Solve[Lag[r, L], r''[t]] // Flatten) //. {
            Rule -> Equal})

eqΦ = 
      ((Solve[Lag[ϕ, L], ϕ''[t]] // Flatten) //. {
            Rule -> Equal})

Part (b)   Find the two constants of motion that correspond to angular momentum and energy conservation.  Express the constants in therms if {r[0]=r0, r'[0]=0, ϕ[0]=0, ϕ'[0]=0}.

eq1 = D[L, ϕ'[t]] == m1 r0^2 dϕ0

eqΦ //. Solve[D[eq1, t], ϕ''[t]]

solΦ = DSolve[eq1, ϕ', t] // Flatten

energy = 
      (T1 + T2) + (V1 + V2) //. solΦ // Simplify

eqMotion = {eqR, eqΦ} /. {Equal -> Rule} // Flatten

D[energy, t] //. eqMotion //. solΦ // Simplify

initial = {r[0] -> r0,  r'[0] -> 0,  ϕ[0] -> 0,  
        ϕ'[0] -> dϕ0}

energy0 = energy //. {t -> 0} //. initial

rSol = Solve[energy == energy0, r'[t]][[2]]

Part (c)   Solve the limiting case where m1 describes a circle, r[t]=r0.

sol1 = 
      Solve[eqR //. {r''[t] -> 0, r[t] -> r0}, ϕ'[t]][
        [2]]

dsol1 = 
      DSolve[(sol1 /. {Rule -> Equal}), ϕ[t], t] // Flatten

Part (d)   If the circular orbit is displaced by δr[t], find the angular frequence ω with which δr[t] oscillates about a circular orbit.

radiusRule = {r -> (r0 + dr[#]&)}

eq2 = eqR //. solΦ //. radiusRule

\(rhs = 
      \((\(\((Series[eq2[\([1, 2]\)], {dr[t], 0, 1}] // Normal)\)

            	 //. {dϕ0 -> \(ϕ'\)[t]}\)
	 //. sol1)\)\)

displacement = dr''[t] == rhs

\(dsol9\  = \ 
      DSolve[{displacement, dr[0] == dr0, \(dr'\)[0] == 0}, {dr[t]}, t]

        	\ \ \  // Flatten\)

(Solve[(dr[t] /. dsol9) == dr0 Cos[ω t], ω] //. {
            ArcCos[Cos[x_]] :> x}) [[2]]