EULER-LAGRANGE EQUATIONS EUGENE LERMAN Contents 1. Clssicl system of N prticles in R 3 1 2. Vritionl formultion 2 3. Constrine systems n Alembert principle. 4 4. Legenre trnsform 6 1. Clssicl system of N prticles in R 3. Consier mechnicl system consisting of N prticles in R 3 subject to some forces. Let x i R 3 enote the position vector of the ith prticle. Then ll possible positions of the system re escribe by N-tuples (x 1,..., x N ) (R 3 ) N. The spce (R 3 ) N is n exmple of configurtion spce. The time evolution of the system is escribe by curve (x 1 (t),..., x N (t)) in (R 3 ) N n is governe by Newton s secon lw: Here m i 2 x i t 2 = F i(x 1,..., x N, ẋ 1,..., ẋ N, t) F i enotes the force on ith prticle (which epens on the positions n velocities of ll N prticles n on time), x i = xi t, n m i enotes the mss of the ith prticle. We now re-lble the vribles. Let q 3i, q 3i+1, q 3i+2 be respectively the first, the secon n the thir coorinte of the vector x i, i = 1,..., N: (q 3i, q 3i+1, q 3i+2 ) = x i. The configurtion spce of our system is then R n, where now n = 3N. The equtions of motion tke the form (1.1) m α 2 q α t 2 = F α(q 1,..., q n, q 1,..., q n, t), 1 α n. We now suppose tht the forces re time-inepenent n conservtive/ Tht is, we ssume tht there exists function V : R n R ( potentil) such tht F α (q 1,..., q n, q 1,..., q n, t) = F α (q 1,..., q n ) = V (q 1,..., q n ). Exmple 1.1. For exmple if N prticles interct by grvittionl ttrction, then the potentil is V (x 1,..., x N ) = γ m i m j x i x j, i j where γ is universl constnt. typeset Februry 26, 2013. 1
Uner the ssumption of time inepenent conservtive forces the system of equtions (1.1) tkes the form (1.2) m α 2 q α t 2 (1.3) = V (q 1,..., q n ), 1 α n. We rewrite eqution (1.2) s first orer ODE by oubling the number of vribles; this is stnr trick. Cll the new vribles, the velocities, v α : { v m α α q α t t = v α = V (q 1,..., q n ) A solution (q(t), v(t)) of the bove system of equtions is curve in the tngent bunle T R n with t q(t) = v(t). The tngent bunle T R n is n exmple of phse spce. It is stnr to introuce function L(q, v) = 1 m α vα 2 V (q) 2 α on the phse spce T R n. The function is clle the Lgrngin of the system. It is the ifference of the kinetic n the potentil energies. We will see shortly tht we cn re-write (1.3) s ( ) L (1.4) L = 0, 1 α n t The system of equtions (1.4), which we will see re equivlent to Newton s lw of motion, is n exmple of the Euler-Lgrnge equtions. We now check tht equtions (1.3) n (1.4) re, inee, the sme. Since n we get Hence ( 1 2 m β vβ) 2 = m α v α β ( 1 mβ vβ 2 V ) = V = F α 2 0 = t ( L) L = t (m αv α ) + V. v α m α = V. t So fr introucing the Lgrngin i not give us nything new. We now show tht it oes inee llow us to look t Newton s lw from nother point of view, n tht the new point of view hs interesting consequences. 2. Vritionl formultion Let L : T R n R be Lgrngin (i.e., smooth function). Let q (0), q (1) be two points in R n. Consier the collection P = P(q (0), q (1) ) of ll possible twice continuously ifferentible (C 2 ) pths γ : [, b] R n with γ() = q (0), γ(b) = q (1). Tht is, set The Lgrngin L efines mp clle n ction. We re now in position to stte: P := {γ : [, b] R n γ() = q (0), γ(b) = q (1) }. A L : P R, A L (γ) := L(γ(t), γ(t)) t, 2
Hmilton s principle: physicl trjectories between two points q (0), q (1) of the system governe by the Lgrngin L re criticl points of the ction functionl A L : P(q (0), q (1) ) R. Proposition 2.1. Hmilton s principle implies Euler-Lgrnge equtions n hence Newton s lw of motion. Proof. The proof is well-known. Let γ : [, b] R n be criticl point (pth, trjectory) of n ction functionl A L ; let γ(t, ɛ) be fmily of pths epening on ɛ R such tht γ(t, 0) = γ(t) n such tht for ll ɛ we hve γ(, ɛ) = γ() n γ(b, ɛ) = γ(b) (i.e., we fix the en points). Let y(t) = ɛ ɛ=0 γ(t, ɛ). Note tht y() = 0 n y(b) = 0 since the en points re fixe. Also ɛ ɛ=0 ( t γ(t, ɛ)) = t ( ɛ ɛ=0 γ(t, ɛ)) = ẏ(t). Conversely, given curve y : [, b] R n with y() = y(b) = 0, we cn fin fmily of pths γ(t, ɛ) with fixe en points such tht γ(t, 0) = γ(t) n ɛ ɛ=0 ( γ(t, ɛ)) = ẏ(t). t For exmple, we my tke Now (2.1) γ(t, ɛ) = γ(t) + ɛy(t). 0 = ɛ A L (γ(t, ɛ)) = ɛ=0 ɛ L(γ(t, ɛ), γ(t, ɛ)) t ɛ=0 = = α = α = α ɛ 0 L t = α We now recll without proof: { ( L y α + L ẏ α ) t L y α t + L y α b ( L t ( L ))y α t ( L ɛ ɛ=0 + L ɛ ɛ=0 ) t t ( L ) y α t} (integrtion by prts) (since y α () = y α (b) = 0) for ll α). Lemm 2.2. If f : [, b] R is continuously ifferentible function with the property tht for ny continuously ifferentible function y : [, b] R with y() = y(b) = 0, we hve then f hs to be ienticlly zero. f(t)y(t) t = 0, We conclue tht for ny inex α we must hve L t ( L ) = 0. Tht is to sy, Hmilton s principle implies the Euler-Lgrnge equtions. Note tht we hve prove tht given Lgrngin there is vector fiel on T R n whose integrl curves re the criticl curves of the corresponing ction. 3
3. Constrine systems n Alembert principle. We strt by listing exmples of constrine systems (ll constrints re time-inepenent, tht is, scleronomic). Exmple 3.1 (Sphericl penulum). The system consists of mssive prticle in R 3 connecte by very light ro of length l to universl joint. The configurtion spce of this system is sphere S 2 of rius l. The phse spce is T S 2. This is n exmple of holonomic constrint. Exmple 3.2 (Free rigi boy). The system consists of N point msses in R 3 mintining fixe istnces between ech other: x i x j = const ij. We will see lter tht the configurtion spce is E(3), the Euclien group of istnce preserving trnsformtions of R 3. It is not hr to show tht E(3) consists of rottions n trnsltions. In fct we cn represent E(3) s certin collection of mtrices: ( ) A v E(3) { R 42 A T A = I, et A = ±1, v R 3 }. 0 1 The group E(3) is 6-imensionl mnifol, hence the phse spce for free rigi boy is T E(3). The free rigi boy is lso n exmple of system with holonomic constrint. Exmple 3.3 (A quter rolling on rough plne in n upright position (without slipping)). The configurtion spce is (R 2 S 1 ) S 1, where the elements of R 2 keep trck of the point of contct of the qurter with the rough plne, the points in the first S 1 keeps trck of the orienttion of the plne of the qurter n points of the secon S 1 keep trck of the orienttion of the esign on the qurter. The phse spce of the system is smller thn T (R 2 S 1 S 1 ) becuse the point of contct of the qurter with the plne hs to be sttionry. This is n exmple of non-holonomic constrints, since the constrints on position o not etermine the constrints on velocity: the roll-no-slip conition is extr. This les us to efinition. Definition 3.4. Time-inepenent constrints re holonomic if the constrint on possible velocities re etermine by the constrints on the configurtions of the system. In other wors if the constrints confine the configurtions of the system to submnifol M of R n n the corresponing phse spce is T M, then the constrints re holonomic. We will stuy only holonomic systems with n e ssumption: constrint forces o no work. We re now in position to formulte: Alembert s principle: If constrint forces o no work, then the true physicl trjectory of the system re extremls of the ction functionl of the free system restricte to the pths lying in the constrint submnifol. This principle is very powerful: we no longer nee to know nything bout the constrining forces except for the fct tht they limit the possible configurtions to constrint submnifol. We now investigte the equtions of motion tht Alembert s principle prouces. Let M R n be submnifol n let L : T R n T M R be Lgrngin for n unconstrine( free ) system. By Alembert s principle our system evolves long pth γ : [, b] M 4
such tht γ is criticl for A L : {σ : [, b] M σ() = q (0), σ(b) = q (1) } R A L (σ) = L(σ, σ) t Suppose the en points q (0) n q (1) lie in some coorinte ptch on M. Let (q 1,..., q n ) be the coorintes on the ptch n let (q 1,..., q n, v 1,..., v n ) be the corresponing coorintes on the corresponing ptch in T M. The sme rgument s before (cf. Proposition 2.1) gives us Euler-Lgrnge equtions t ( L ) L = 0! Note tht these equtions represent vector fiel on coorinte ptch in the tngent bunle T M. Exmple 3.5 (Plnr penulum). The system consists of hevy prticle in R 3 connecte by very light ro of length l to fixe point. Unlike the sphericl penulum the ro is only llowe to pivot in fixe verticl plne. The configurtion spce M is the circle S 1 R 2 of rius l. The Lgrngin of the free system is where L(x, v) = 1 2 m(v2 1 + v 2 2) mgx 2, m is the mss of the prticle, x 1, x 2 re coorintes on R 2, x 1, x 2, v 1, v 2 re the corresponing coorintes on T R 2 n g is the grvittionl ccelertion (9.8 m/s 2 ). We compute the equtions of motion for the constrint system s follows Consier the embeing S 1 R 2, ϕ (l sin ϕ, l cos ϕ). The corresponing embeing T S 1 T R 2 is given by (ϕ, v ϕ ) (l sin ϕ, l cos ϕ, l cos ϕ v ϕ, l sin ϕ v ϕ ). Therefore the constrint Lgrngin is given by L(ϕ, v ϕ ) = 1 2 m(l2 cos 2 ϕv 2 ϕ + l 2 sin 2 ϕv 2 ϕ) + mgl cos ϕ = 1 2 ml2 v 2 ϕ + mgl cos ϕ. The corresponing Euler-Lgrnge eqution is i.e., t ( L ) L v ϕ ϕ = 0, Therefore is the eqution of motion. ml 2 v ϕ t + mgl sin ϕ = 0. ϕ = g l sin ϕ 5
4. Legenre trnsform We strt with brief igression which will serve s toy exmple of Legenre trnsform. Let be smooth function on vector spce V n let L : V R v 1,... v n : V R be coorintes on V. For exmple, {v i } n i=1 coul be liner functionls on V tht form bsis of the ul spce V. For ech v V consier the mtrix ( 2 L v i v j (v)) of secon orer prtils. It cn be interprete s qurtic form 2 L(v) on V s follows: for u, w V with coorintes (u 1,..., u n ) n (w 1,..., w n ) respectively we set 2 L(v)(u, w) = 2 ( ( L (v) u i w j = u T 2 ) ) L (v) w. v i,j i v j v i v j The qurtic form 2 L(v) lso hs coorinte-free efinition. By the chin rule, for ny u, w V, 2 L(v)(u, w) = 2 L(v + su + tw) st. (0,0) We note tht the mtrix ( 2 L v iv j (v)) is invertible if n only if the qurtic form 2 L(v) is nonegenerte. This ens igression n we go bck to stuying Lgrngins on phse spces. Recll tht given Lgrngin L : T M R n two points m 1, m 2 M, the corresponing ction is efine by A L : { C 1 pths connecting m 1 to m 2 } R A L (σ) = L(σ(t), σ(t)) t. Suppose tht the points m 1, m 2 lie in coorinte ptch U with coorintes x 1,..., x n : U R. Let x 1,..., x n, v 1,..., v n be the corresponing coorintes on T U T M. We sw tht pth γ : [, b] U, γ() = m 1, γ(b) = m 2, is criticl for the ction A L if n only if the Euler-Lgrnge equtions hol. Now t ( L v i (γ(t), γ(t)) ) t ( L (γ, γ)) = v i j Hence the Euler-Lgrnge equtions re γ j = L v i v j x i j j L x i (γ, γ) = 0, 1 i n ( 2 L γ j + 2 L γ j ). x j v i v j v i x j v i γ j 1 i n. We now mke n importnt ssumption: L is regulr Lgrngin. Tht is. we ssume tht the mtrix ( 2 L v i v j (x, v)) is invertible for ll (x, v) T U. Equivlently we ssume tht for ll x U the form 2 L TxM (v) 6
is nonegenerte for ll v T x M. Then there exists n inverse mtrix ( (M ki ) = (M ki (x, v)) = (x, v) v i v j It is efine by i M ki = δ kj, v i v j where, s usul, δ kj is the Kronecker elt function. Uner this ssumption M ki γ j = M ki ( L γ j ), v i,j i v j x }{{} i i x j j v i δ kj hence (4.1) γ k = i M ki ( L x i j x j v i γ j ) Exercise 4.1. Let g be Riemnnin metric on M n let L(x, v) = 1 2g(x)(v, v). Check tht this Lgrngin is regulr. Wht oes (4.1) look like for this L? (4.2) We cn rewrite (4.1) s first orer system in 2n vribles. x j = v j (ẍ k =) v k = i M ki ( L x i j ) 1 x j v i v j ) Note tht in physics literture the coorintes x i s re usully clle q i s n the corresponing coorintes v i s re usully clle q i s. The confusing point here is tht the ot bove q i oes not stn for nything; q i is simply nme of coorinte. Eqution (4.2) mens tht we hve vector fiel X L on T U: X L (x, v) = j v j x j + k,i M ki ( L x i j x j v i v j ) v k The vector fiel X L is clle the Euler-Lgrnge vector fiel. One cn show tht Proposition 4.1. X L is well-efine vector fiel on the tngent bunle T M, i.e. it trnsforms correctly uner the chnge of vribles. 7