Sect. 1.4: D’Alembert’s Principle & Lagrange’s Eqtns Virtual (infinitesimal) displacement  Change in the system configuration as result of an arbitrary infinitesimal change of coordinates δri, consistent with the forces & constraints imposed on the system at a given time t. “Virtual” distinguishes it from an actual displacement dri, occurring in small time interval dt (during which forces & constraints may change)

Consider the system at equilibrium: The total force on each particle is Fi = 0. Virtual work done by Fi in displacement δri: δWi = Fiδri = 0. Sum over i:  δW = ∑iFiδri = 0. Decompose Fi into applied force Fi(a) & constraint force fi: Fi = Fi(a) + fi  δW = ∑i (Fi(a) + fi )δri  δW(a) + δW(c) = 0 Special case (often true, see text discussion): Systems for which the net virtual work due to constraint forces is zero: ∑ifiδri  δW(c) = 0

Principle of Virtual Work  Condition for system equilibrium: Virtual work due to APPLIED forces vanishes: δW(a) = ∑iFi(a)δri = 0 (1)  Principle of Virtual Work Note: In general coefficients of δri , Fi(a)  0 even though ∑iFi(a)δri = 0 because δri are not independent, but connected by constraints. In order to have coefficients of δri = 0, must transform Principle of Virtual Work into a form involving virtual displacements of generalized coordinates q , which are independent. (1) is good since it does not involve constraint forces fi . But so far, only statics. Want to treat dynamics!

D’Alembert’s Principle Dynamics: Start with Newton’s 2nd Law for particle i: Fi = (dpi/dt) Or: Fi - (dpi/dt) = 0  Can view system particles as in “equilibrium” under a force = actual force + “reversed effective force” = -(dp/dt) Virtual work done is δW = ∑i[Fi - (dpi/dt)]δri = 0 Again decompose Fi: Fi = Fi(a) + fi  δW = ∑i[Fi(a) - (dpi/dt) + fi ]δri = 0 Again restrict consideration to special case: Systems for which the net virtual work due to constraint forces is zero: ∑i fiδri  δW(c) = 0

 δW = ∑i[Fi - (dpi/dt)]δri = 0 (2)  D’Alembert’s Principle Dropped the superscript (a)! Transform (2) to an expression involving virtual displacements of q (which, for holonomic constraints, are indep of each other). Then, by linear independence, the coefficients of the δq = 0

δW = ∑i[Fi - (dpi/dt)]δri = 0 (2) Much manipulation follows! Only highlights here! Transformation eqtns: ri = ri(q1,q2,q3,.,t) (i = 1,2,3,…n) Chain rule of differentiation (velocities): vi  (dri/dt) = ∑k(ri/qk)(dqk/dt) + (ri/t) (a) Virtual displacements δri are connected to virtual displacements δq: δri = ∑j (ri/qj)δqj (b)

Generalized Forces 1st term of (2) (Combined with (b)): ∑i Fi δri = ∑i,j Fi (ri/qj)δqj  ∑jQjδqj (c) Define Generalized Force (corresponding to Generalized Coordinate qj): Qj  ∑iFi(ri/qj) Generalized Coordinates qj need not have units of length!  Corresponding Generalized Forces Qj need not have units of force! For example: If qj is an angle, corresponding Qj will be a torque!

2nd term of (2) (using (b) again): ∑i(dpi/dt)δri = ∑i[mi (d2ri/dt2)δri ] = ∑i,j[mi (d2ri/dt2)(ri/qj)δqj] (d) Manipulate with (d): ∑i[mi (d2ri/dt2)(ri/qj)] = ∑i[d{mi(dri/dt)(ri/qj)}/dt] – ∑i[mi(dri/dt)d{(ri/qj)}/dt] Also: d{(ri/qj)}/dt = {dri/dt}/qj  (vi/qj) Use (a): (vi/qj) = ∑k(2ri/qjqk)(dqk/dt) + (2ri/qjt) From (a): (vi/qj) = (ri/qj) So: ∑i[mi (d2ri/dt2)(ri/qj)] = ∑i[d{mivi(vi/qj)}/dt] - ∑i[mivi(vi/qj)]

More manipulation  (2) is: ∑i[Fi-(dpi/dt)]δri = 0 ∑j{d[(∑i (½)mi(vi)2)/qj]/dt - (∑i(½)mi(vi)2)/qj - Qj}δqj = 0 System kinetic energy is: T  (½)∑imi(vi)2  D’Alembert’s Principle becomes ∑j{(d[T/qj]/dt) - (T/qj) - Qj}δqj = 0 (3) Note: If qj are Cartesian coords, (T/qj) = 0  In generalized coords, (T/qj) comes from the curvature of the qj. (Example: Polar coords, (T/θ) becomes the centripetal acceleration). So far, no restriction on constraints except that they do no work under virtual displacement. qj are any set. Special case: Holonomic Constraints  It’s possible to find sets of qj for which each δqj is independent.  Each term in (3) is separately 0!

(d[T/qj]/dt) - (T/qj) = Qj (4) (j = 1,2,3, … n) Holonomic constraints  D’Alembert’s Principle: (d[T/qj]/dt) - (T/qj) = Qj (4) (j = 1,2,3, … n) Special case: A Potential Exists  Fi = - iV Needn’t be conservative! V could be a function of t!  Generalized forces have the form Qj  ∑i Fi(ri/qj) = - ∑i iV(ri/qj)  - (V/qj) Put this in (4): (d[T/qj]/dt) - ([T-V]/qj) = 0 So far, V doesn’t depend on the velocities qj  (d/dt)[(T-V)/qj] - (T-V)/qj = 0 (4´)

Lagrange’s Equations Define: The Lagrangian L of the system: L  T - V  Can write D’Alembert’s Principle as: (d/dt)[(L/qj)] - (L/qj) = 0 (5) (j = 1,2,3, … n) (5)  Lagrange’s Equations

(d/dt)[(L/qj)] - (L/qj) = 0 (j = 1,2,3, … n) Lagrange’s Eqtns Lagrangian: L  T - V Lagrange’s Eqtns: (d/dt)[(L/qj)] - (L/qj) = 0 (j = 1,2,3, … n) Note: L is not unique, but is arbitrary to within the addition of a derivative (dF/dt). F = F(q,t) is any differentiable function of q’s & t. That is, if we define a new Lagrangian L´ L´= L + (dF/dt) It is easy to show that L´satisfies the same Lagrange’s Eqtns (above).