Sect. 2.5: Advantages of a Variational Principle Formulation

2. Sect. 2.5: Advantages of a Variational Principle Formulation • HP  δ∫Ldt = 0 (limits t1 < t < t2). An example of a variational principle. • Most useful when a coordinate system-independent Lagrangian L = T - V can be set up. • HP: “Elegant”. Contains all of mechanics of holonomic systems in which forces are derivable from potentials. • HP: Involves only physical quantities (T, V) which can be generally defined without reference to a specific set of generalized coords.  A formulation of mechanics which is independent of the choice of coordinate system!

3. HP  δ∫Ldt = 0(limits t1 < t < t2). • From this, we can see (again) that the Lagrangian Lis arbitrary to within the derivative(dF/dt) of an arbitrary functionF = F(q,t). • If we form L´ = L + (dF/dt) & do the integral, ∫L´dt, we get ∫Ldt + F(q,t2) - F(q,t1). By the definition of δ, the variation at t1 & t2 is zero δ∫L´dt will not depend on the end points. • Another advantage to HP: Can extend Lagrangian formalism to systems outside of classical dynamics: • Elastic continuum field theory • Electromagnetic field theory • QM theory of elementary particles • Circuit theory!

4. Lagrange Applied to Circuit Theory • System: LR Circuit (Fig.) Battery, voltage V, in series with inductor L & resistor R (which will give dissipation). Dynamical variable = charge q. PE = V = qV KE = T = (½)L(q)2 Lagrangian: L = T - V Dissipation Function: (last chapter!) ₣= (½)R(q)2 = (½)R(I)2 Lagrange’s Eqtn (with dissipation): (d/dt)[(L/q)] - (L/q) + (₣/q) = 0 switch 

5. Lagrange Applied to RL circuit • Lagrange’s Eqtn (with dissipation): (d/dt)[(L/q)] - (L/q) + (₣/q) = 0  V = Lq + Rq I = q = (dq/dt)  V = LI + RI Solution, for switch closed at t = 0 is: I = (V/R)[1 - e(-Rt/L)] Steady state (t  ): I = I0 = (V/R)

6. Mechanical Analogue to RL circuit • Mechanical analogue: Sphere, radius a, (effective) mass m´, falling in a const density viscous fluid, viscosity η under gravity. m´  m - mf , m  actual mass, mf  mass of displaced fluid (buoyant force acting upward: Archimedes’ principle) • V = m´gy, T = (½)m´v2, L = T - V (v = y) Dissipation Function: ₣= 3πηav2 Comes from Stokes’ Law of frictional drag force: Ff = 6πηav and (Ch. 1 result that) Ff = - v₣ Lagrange’s Eqtn (with dissipation): (d/dt)[(L/y)] - (L/y) + (₣/y) = 0

7. V = m´gy, T = (½)m´v2, L = T - V (v = y) Dissipation Function: ₣= 3πηav2 Comes from Stokes’ Law frictional drag force: Ff = 6πηav and (Ch. 1 result that) Ff = - v₣ Lagrange’s Eqtn (with dissipation): (d/dt)[(L/y)] - (L/y) + (₣/y) = 0  m´g = m´y + 6πηay Solution, for v = y starting from rest at t = 0: v = v0 [1 - e(-t/τ)]. τ m´ (6πηa)-1Time it takes sphere to reach e-1 of its terminal speedv0. Steady state (t  ):v = v0 = (m´g)(6πηa)-1 = gτ = terminal speed.

8. Lagrange Applied to Circuit Theory • System: LC Circuit (Fig.) Inductor L & capacitor C in series. Dynamical variable = charge q. Capacitor acts a PE source: PE = (½)q2C-1, KE = T = (½)L(q)2 Lagrangian: L = T - V (No dissipation!) Lagrange’s Eqtn: (d/dt)[(L/q)] - (L/q) = 0  Lq + qC-1 = 0 Solution (for q = q0at t = 0): q = q0 cos(ω0t), ω0= (LC)-(½) ω0natural or resonant frequency of circuit

9. Mechanical Analogue to LC Circuit • Mechanical analogue: Simple harmonic oscillator (no damping) mass m, spring constant k. • V = (½)kx2, T = (½)mv2, L = T - V (v = x) Lagrange’s Eqtn: (d/dt)[(L/x)] - (L/x) = 0  mx + kx = 0 Solution (for x = x0at t = 0): x = x0 cos(ω0t), ω0 = (k/m)½ ω0 natural or resonant frequency of circuit

10. Circuit theory examples give analogies: Inductance L plays an analogous role in electrical circuits that massm plays in mechanical systems (an inertial term).  ResistanceRplays an analogous role in electrical circuits that viscosityηplays in mechanical systems (a frictional or drag term).  CapacitanceC (actually C-1) plays an analogous role in electrical circuits that a Hooke’s “Law” type spring constantk plays in mechanical systems (a “stiffness” or tensile strength term).

11. With these analogies, consider the system of coupled electrical circuits (fig): Mjk = mutual inductances! • Immediately, can write Lagrangian: L = (½)∑jLj(qj)2 + (½)∑j,k(j)Mjkqjqk - (½) ∑j(1/Cj)(qj)2 + ∑jEj(t)qj Dissipation function:₣ = (½)∑jRj(qj)2

12. Lagrangian: L = (½)∑jLj(qj)2 + (½)∑j,k(j)Mjkqjqk - (½)∑j(1/Cj)(qj)2 + ∑jEj(t)qj Dissipation function:₣ = (½)∑jRj(qj)2Lagrange’s Eqtns: (d/dt)[(L/qj)] - (L/qj) + (₣/qj) = 0  Eqtns of motion (the same as coupled, driven, damped harmonic oscillators!) Lj(d2qj/dt2) + ∑k(j)Mjk(d2qk/dt2) + Rj(dqj/dt) +(1/Cj)qj = Ej(t)

13. Describe 2 different physical systems by Lagrangians of the same mathematical form (circuits & harmonic oscillators):  ALL results & techniques devised for studying & solving one system can be taken over directly & used to study & solve the other.  Sophisticated studies of electrical circuits & techniques for solving them have been very well developed. All such techniques can be taken over directly & used to study analogous mechanical (oscillator) systems. These have wide applicability to acoustical systems. Also true in reverse.

14. HP & resulting Lagrange formalism can be generalized to apply to subfields of physics outside mechanics. • Similar variational principles exist in other subfields: Yielding Maxwell’s Eqtns (E&M) the Schrödinger Eqtn Quantum Electrodynamics Quantum Chromodynamics, …..etc.