Analytical Mechanics: Lagrangian and Hamiltonian Formalism

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MATH-F-204 Analytical mechanics: Lagrangian and Hamiltonian formalism

Glenn Barnich Physique Théorique et Mathématique Université libre de Bruxelles and International Solvay Institutes Campus Plaine C.P. 231, B-1050 Bruxelles, Belgique Bureau 2.O6. 217, E-mail: [email protected], Tel: 02 650 58 01.

The aim of the course is an introduction to Lagrangian and Hamiltonian mechanics. The fundamental equations as well as standard applications of Newtonian mechanics are derived from variational principles. From a mathematical perspective, the course and exercises (i) constitute an application of differential and integral calculus (primitives, inverse and implicit function theorem, graphs and functions); (ii) provide examples and explicit solutions of first and second order systems of differential equations; (iii) give an implicit introduction to differential manifolds and to realiza- tions of Lie groups and algebras (change of coordinates, parametrization of a rotation, Galilean group, canonical transformations, one-parameter subgroups, infinitesimal transformations). From a physical viewpoint, a good grasp of the material of the course is indispensable for a thorough understanding of the structure of quantum mechanics, both in its operatorial and path integral formulations. Furthermore, the discussion of Noether’s theorem and Galilean invariance is done in a way that allows for a direct generalization to field theories with Poincaré or conformal invariance. 2 Contents

1 Extremisation, constraints and Lagrange multipliers 5 1.1 Unconstrained extremisation ...... 5 1.2 Constraints and regularity conditions ...... 5 1.3 Constrained extremisation ...... 7

2 Constrained systems and d’Alembert principle 9 2.1 Holonomic constraints ...... 9 2.2 d’Alembert’s theorem ...... 10 2.3 Non-holonomic constraints ...... 10 2.4 d’Alembert’s principle ...... 13

3 Hamilton and least action principles 15

4 Lagrangian mechanics 17

5 Applications of the Lagrangian formalism 19

6 Covariance of the Lagrangian formalism 21 6.1 Change of dependent variables ...... 21 6.2 Change of the independent variable ...... 22 6.3 Combined change of variables ...... 23

7 Noether’s theorem and Galilean invariance 25 7.1 Noether’s ﬁrst theorem ...... 25 7.2 Applications ...... 26 7.2.1 Explicit time independence and conservation of energy ...... 26 7.2.2 Cyclic variables and conservation of canonical momentum ...... 26 7.2.3 One-parameter groups of transformations ...... 26 7.2.4 One-parameter groups of transformations involving time ...... 28 7.2.5 Galilean boosts ...... 30

8 Hamiltonian formalism 31

9 Canonical transformations 33

10 Dynamics in non-inertial frames 35

11 Charged particle in electromagnetic ﬁeld 37

References 37

3 4 CONTENTS Chapter 1

Extremisation, constraints and Lagrange multipliers

The purpose of this chapter is to re-derive in the ﬁnite-dimensional case standard results on constrained extremisation so as to motivate and clarify analogous results needed in the functional case. It is based on chapters II.1, II.2, II.5 of [8]. The discussion on constraints and regularity conditions is taken from chapter 1 of [11].

1.1 Unconstrained extremisation

The problem consists of ﬁnding the extrema (=stationary points) of a function F (x1, . . . , xn) of n variables xa, a = 1, . . . , n in the interior of a domain. We write the variation of such a function under variations δxa of the variables xa as ∂F δF = δxa . (1.1) ∂xa ∂F ∂F Here and throughout, we use the summation convention over repeated indices, δxa = Pn δxa . ∂xa a=1 ∂xa b Since for a collection of functions fa(x ),

a a faδx = 0, ∀δx =⇒ fa = 0, (1.2) it follows that the extrema x¯a are determined by the vanishing of the gradient of F ,

∂F δF = 0 ⇐⇒ | a a = 0 . (1.3) ∂xa x =¯x

Recall further that the nature (maximum, minimum, saddle point) of each extremum is determined by the eigenvalues of the matrix ∂2F | . (1.4) ∂xa∂xb x¯

1.2 Constraints and regularity conditions

Very often, one is interested in a problem of extremisation, where the variables xa are subjected to constraints, that is to say where r independent relations

Gm(x) = 0, m = 1, . . . r, (1.5)

5 6 CHAPTER 1. EXTREMISATION, CONSTRAINTS AND LAGRANGE MULTIPLIERS

∂G are supposed to hold. Standard regularity on the functions G are that the rank of the matrix m is m ∂xa r (locally, in a neighborhood of Gm = 0). This means that one may choose locally a new system of a α coordinates such that Gm are r of the new coordinates, x ←→ q ,Gm, with α = 1, . . . n − r. Lemma 1. A function f that vanishes when the constraints hold can be written as a combination of the functions deﬁning the constraints:

m f|Gm=0 = 0 ⇐⇒ f = g Gm , (1.6) for some coefﬁcients gm(x). That the condition is sufﬁcient (⇐=) is obvious. That it is necessary (=⇒) is shown in the new α coordinate system, where f(q ,Gm = 0) = 0 and thus

1 1 Z d Z ∂f f(qα,G ) = dτ f(qα, τG ) = G dτ (qα, τG ), (1.7) m dτ m m ∂G m 0 0 m

m R 1 ∂f α so that g = dτ (q , τGm). 0 ∂Gm In the following, we will use the notation f ≈ 0 for a function that vanishes when the constraints hold, m so that the lemma becomes f ≈ 0 ⇐⇒ f = g Gm. Lemma 2. Suppose that the constraints hold and that one restricts oneself to variations δxa tangent to the constraint surface, ∂G G = 0, m δxa ≈ 0. (1.8) m ∂xa It follows that ∂G f δxa ≈ 0, ∀δxa satisfying (1.8) ⇐⇒ f ≈ µm m , (1.9) a a ∂xa for some µm. NB: In this lemma, the use of ≈ means that these equalities in (1.8) and (1.9) are understood as holding when Gm = 0. That the condition is sufﬁcient (⇐=) is again direct when contracting the expression for fa in the RHS of (1.9) with δxa and using the second of (1.8). That the condition is necessary (=⇒) follows by ﬁrst substracting from the LHS of (1.9) the combination ∂G −λm m δxa ≈ 0, with arbitrary λm, which gives ∂xa ∂G (f − λm m )δxa ≈ 0. (1.10) a ∂xa Without loss of generality (by renaming if necessary some of the variables), we can assume that

∂G | m |= 6 0, for a = n − r + 1, . . . , n. (1.11) ∂xa This implies that one may locally solve the constraints in terms of the r last coordinates,

∆ ∆ α Gm = 0 ⇐⇒ x = X (x ), ∆ = n − r + 1, . . . , n, α = 1, . . . , n − r. (1.12)

In this case, one refers to the x∆ as dependent and to the xα as independent variables. This also implies that, locally, there exists an invertible matrix M m∆ such that

m∆ ∂Gm0 m ∂Gm mΓ Γ M = δ 0 , M = δ . (1.13) ∂x∆ m ∂x∆ ∆ 1.3. CONSTRAINED EXTREMISATION 7

In terms of the various types of variables, (1.10) may be decomposed as

∂G ∂G (f − λm m )δx∆ + (f − λm m )δxα ≈ 0. (1.14) ∆ ∂x∆ α ∂xα Since the λm are arbitrary, we are free to ﬁx

∂G λm = M mΓf := µm ⇐⇒ f = µm m , (1.15) Γ ∆ ∂x∆ so that the ﬁrst term (1.14) vanishes. We then remain with

∂G (f − µm m )δxα ≈ 0. (1.16) α ∂xα Since the variables xα are unconstrained and the variations δxα independent, this implies as in (1.2) that

∂G f ≈ µm m . (1.17) α ∂xα

1.3 Constrained extremisation

a Extremisation of a function F (x ) under r constraints Gm = 0 can then be done in two equivalent ways:

Method 1: The ﬁrst is to solve the constraints as in (1.12) and to extremise the function

F G(xα) = F (xα,X∆(xβ)) . (1.18)

In terms of the unconstrained variables xα, we are back to an unconstrained extremisation problem, whose extrema are determined by ∂F G = 0 . (1.19) ∂xα In order to relate to the previous discussion and the second method to be explained below, note that, ∆ ∆ α by using Lemma 1, Gm = 0 ⇐⇒ x − X (x ) = 0 implies

∆ ∆ α ∆m x − X (x ) = g Gm, (1.20) for g∆m(xa) invertible. Differentiation with respect to xΓ yields

∂g∆m ∂G δ∆ = G + g∆m m . (1.21) Γ ∂xΓ m ∂xΓ ∂G Hence, if the constraints are satisﬁed, i.e., if G = 0, g∆m is the inverse matrix to m , g∆m ≈ M ∆m. m ∂x∆ On the one hand, ∂F G ∂F ∂F ∂XΓ = | ∆ ∆ + | ∆ ∆ . (1.22) ∂xα ∂xα x =X ∂xΓ x =X ∂xα On the other hand, ∂XΓ ∂(XΓ − xΓ) ∂gΓm ∂G = = − G − gΓm m , (1.23) ∂xα ∂xα ∂xα m ∂xα ∆ ∆ so that, if Gm = 0, or equivalently, if one substitutes x by X , (1.19) may be written as ∂F ∂G ∂F ≈ µm m , µm ≈ g∆m. (1.24) ∂xα ∂xα ∂x∆ 8 CHAPTER 1. EXTREMISATION, CONSTRAINTS AND LAGRANGE MULTIPLIERS

Furthermore, if Gm = 0, one may also write

∂F ∂F ∂F ∂G ∂G = δΓ ≈ gΓm m ≈ µm m . (1.25) ∂x∆ ∂xΓ ∆ ∂xΓ ∂x∆ ∂x∆ by using (1.21) and the deﬁnition of µm in (1.24).

Method 2: Instead of restricting oneself to the space of independent variables xa, one extends the space of variables xa by additional variables λm, called Lagrange multipliers, and one considers the unconstrained extremisation of the function,

λ a m a m λ F (x , λ ) = F (x ) − λ Gm, δF = 0 . (1.26)

By unconstrained extremisation, we understand that the variables xa, λm are considered as unconstrained at the outset, with independent variations δxa, δλm. That this gives the same extrema can be seen as follows: ∂F ∂G 0 = δF λ = ( − λm m )δxa − δλmG , (1.27) ∂xa ∂xa m implies ∂F ∂G = λm m ,G = 0. (1.28) ∂xa ∂xa m ∂G A variation of the second equation then implies that m δxa = 0, while contraction of the ﬁrst equation ∂xa with δxa gives us (1.10) of the proof of lemma 2. We can then continue as in the proof of this lemma to ∂F ∂G conclude that ≈ µm m for some µm, and thus that both methods deﬁne the same extrema. ∂xa ∂xa Chapter 2

Constrained systems and d’Alembert principle

2.1 Holonomic constraints

3 α Consider N particles moving in space R , with coordinates are denoted by xi , i = 1,..., N , α = 1, 2, 3. Alternatively, we can label them by xa, with a = 1,..., 3N . We assume that they are submitted to r constraints of the form a Gm(x , t) = 0. (2.1) Such constraints that only involve the positions but not the velocitites of the particles are called holonomic. ∂G As before, we assume regularity conditions: the rank of matrix m is maximal for all values of t. When ∂xa the constraints do not depend explicitly on time, they are called scleronomic, if they do, they are called rheonomic. During a motion, the particles are assumed to remain on the surface deﬁned by the constraints. If this motion dxα takes place during an inﬁnitesimally small time interval dt, this means that

∂G ∂G dG = dxa m + dt m = 0. (2.2) m ∂xa ∂t In other words, the constraints are constraints on the trajectories of the particles and gives rise by differentiation with respect to time to constraints on the velocities,

∂G ∂G x˙ a m + m = 0, (2.3) ∂xa ∂t

dxa d2xa where, for a trajectory xa(t), the notation x˙ a = and x¨a = is used. dt dt2 a a Consider two trajectories x (t) and xe (t). What we will be interested in below is (instantaneous) a a a virtual displacements i.e., the difference of these trajectories ∆x (t) = xe (t) − x (t) at ﬁxed times t. It follows that

dxa(t) xa(t + ) + xa(t) xa(t + ) − xa(t + ) − (xa(t) − xa(t)) ∆ = ∆ lim = lim e e dt →0 →0 dxa(t) dxa(t) d = e − = ∆xa(t). (2.4) dt dt dt More speciﬁcally we will be interested in inﬁnitesimal displacements δxa of this type that are compatible with the constraints, ∂G δxa m ≈ 0. (2.5) ∂xa

9 10 CHAPTER 2. CONSTRAINED SYSTEMS AND D’ALEMBERT PRINCIPLE

Lemma 3. The elementary work δ/W of the reaction forces due to the constraints vanishes for all in- ﬁnitesimal virtual displacements that are compatible with the constraints if and only if the reaction forces are perpendicular to the constraints,

∂G δ/W = R δxa ≈ 0 ⇐⇒ R ≈ µm m , (2.6) a a ∂xa for some µm(xa, t).

Indeed, except for the interpretation in terms of work and forces as well as the geometric meaning of a linear combination of the gradient of constraints, this follows directly from the previous lemma. The notation δ/W means that this elementary work is not necessarily the variation of a function. In what follows, we limit ourselves (almost) exclusively to such constraints, i.e., to constraints whose reaction forces do not do any work during a virtual displacement. In particular this means that there can be no friction due to these constraints. Such constraints are called ideal.

2.2 d’Alembert’s theorem

The dynamics of the N particles is governed by Newton’s equations,

α α α m(i)x¨i = Fi + Ri , (2.7)

α where the parenthesis around the index i means that there is no summation, and where Fi are the com- α ponents of the applied forces acting on the ith particles, while Ri are the reaction forces due to the constraints, assumed to be ideal. The latter keep the particles conﬁned to the constraint surface.

Theorem 1 (d’Alembert). Newton’s equation for a system of point particles subjected to ideal constraints are equivalent to the condition that the elementary work of the sum of the applied forces and of the forces α of inertia (−m(i)x¨i ) vanishes for all inﬁnitesimal virtual displacements compatible with the constraints,

α α α (Fi − m(i)x¨i )δxi ≈ 0 . (2.8)

The theorem states that the Newton’s equations (2.7), together with the fact that the reaction forces i m ∂Gm associated to ideal constraints are of the form Rα ≈ µ α on account of equation (2.6) of Lemma 3 are ∂xi equivalent to (2.8) when the constraints (2.1) hold, and when limiting oneself to compatible infinitesimal α virtual displacements, (2.5). That the condition is suffisant (=⇒) follows by contracting (2.7) with δxi and using (2.6) together with (2.5) to conclude that (2.8) holds. That the condition is necessary (⇐=) α α m ∂Gm follows because (2.1), (2.5) together with (2.8) imply, as in lemma 3, that Fi − m(i)xï ≈ −µ α , for ∂xi some µm, which are Newton’s equation (2.7), together with (2.6).

2.3 Non-holonomic constraints

Non-holonomic constraints are constraints on the velocities of point particles as in (2.3), but that do not originate from holonomic constraints, i.e., from constraints on positions, through differentiation with respect to time. 2.3. NON-HOLONOMIC CONSTRAINTS 11

Figure 2.1: Source: Goldstein [5], chapter 1.3

As an example, consider a vertical disk or coin rolling without sliding on a horizontal plane, z = 0 in standard Cartesian coordinates. Let (xC , yC , zC ) the coodinates of the center of the disk. Only xC , yC are relevant since zC = a, with a the radius of the disk. Let φ be the angle of rotation of the disk starting from a ﬁxed initial direction and θ the angle made by ~1x and the axes of rotation of the disk. Since the velocity ~v of the center of the disk is normal to this axes, and lies in a plane of constant z, it follows that the angle ~ π between 1x and ~v is θ − 2 , so that π π x = v cos(θ − ) = v sin θ, y = v sin(θ − ) = −v cos θ. (2.9) C 2 C 2

Futhermore, the velocity of a point on the edge of the disk is aφ˙. This is the module of the velocity of the point of contact of the disk with the plane, and since there is no sliding this is also the velocity of the center of the disk, v = aφ˙. By injecting into the previous relations, we thus get the following relations between velocitites for this motion ˙ x˙ C − a sin θφ = 0, ˙ (2.10) y˙C + a cos θφ = 0, and the question is whether these constraints on velocities come from constraints on positions by differentiation with respect to time. More generally, consider r constraints that are at most linear in velocities x˙ a,

m a m Ka x˙ + K0 = 0, (2.11)

m b m b a where Ka (x , t),K0 (x , t) are functions that depend on the coordinates x , a = 1, . . . , n (which may α or may not simply be the positions xi in Cartesian space of the N particles), and possibly an explicit dependence on time t. The question then is whether the relation on the velocities (2.11), or any equivalent relation on the velocities m m0 a m0 Λ m0 (Ka x˙ + K0 ) = 0, (2.12)

m b with Λ m0 (x , t) an invertible matrix, can be written as

m m m m0 a m0 ∂G a ∂G Λ 0 (K x˙ + K ) = x˙ + . (2.13) m a 0 ∂xa ∂t 12 CHAPTER 2. CONSTRAINED SYSTEMS AND D’ALEMBERT PRINCIPLE for r functions Gm(xa, t). Since the velocities are arbitrary, this relation can be decomposed as

 m ∂G m m0 0  a = Λ m Ka , ∂xm (2.14) ∂G m m0 = Λ 0 K ,  ∂t m 0 In order to answer this question, it is useful to introduce xA = (t, xa), A = 0, . . . , n, so that the above n + 1 equations can be written in a uniﬁed way as

m ∂G m m0 = Λ 0 K . (2.15) ∂xA m A Necessary conditions1 for the existence of such functions Gm is that mixed second derivatives of the right hand sides vanish,

2 2 ∂ Gm ∂ Gm ∂ m m0 ∂ m m0 = =⇒ (Λ 0 K ) = (Λ 0 K ). (2.16) ∂xA∂xB ∂xB∂xA ∂xA m B ∂xB m A m This is however not the condition that we need because it still contains the unknown functions Λ m0 . m What we need is a condition on the functions KA alone. A standard technical assumption on these functions, that we aasume to hold is that they are linearly independet in the following sense: if there exist B m some functions λm(x ) such that λmKA = 0 then necessarily λm = 0. The condition we need is then a consequence of the following theorem.

m Theorem 2 (Frobenius). For linearly independent functions KA , the set of conditions

∂ m 1 r ( KA )KA ...KA ] = 0, (2.17) ∂x[A1 2 3 r+2

m B m B is necessary and sufﬁcient for the existence of functions Λ m0 (x ) (invertible) and G (x ) such that

m ∂G m m = Λ 0 K . (2.18) ∂xA m A

In equation (2.17), the square bracket denotes complete antisymmetrization of the indices A1,...Ar+2, divided by (r + 2)!, the number of permutations of r + 2 objects. More explicitly,

∂ m 1 r 1 X |σ| ∂ m 1 r ( KA )KA ...KA ] = (1) ( KA )KA ...KA , (2.19) ∂x[A1 2 3 r+2 (r + 2)! ∂xAσ(1) σ(2) σ(3) σr+2 σ∈Pr+2 where Pr+2 denotes a permutations of r +2 elements and |σ| = 0 if the permutation is even, while |σ| = 1 if the permutation is odd. Let us prove that these conditions are indeed necessary (⇐=). We start from

m ∂ ∂G 1 r 0 = ( )KA ...KA ], (2.20) ∂x[A1 ∂xA2 3 r+2 ∂2Gm which holds because the second order derivatives are symmetric in A1,A2, so that, when com- ∂xA1 ∂xA2 pletely antisymmetrizing in the right hand side of equation (2.20), one ﬁnds always 0. Since we assume that (2.18) is true, we ﬁnd by substitution

∂ m m0 1 r 0 = ( (Λ m0 KA ))KA ...KA ] = ∂x[A1 2 3 r+2 ∂ m m0 1 r m ∂ m0 1 r ( Λ m0 )KA KA ...KA ] + Λ m0 ( KA )KA ...KA ] (2.21) ∂x[A1 2 3 r+2 ∂x[A1 2 3 r+2 1It can in fact be shown, as done later in the course, that these conditions are also locally sufﬁcient. 2.4. D’ALEMBERT’S PRINCIPLE 13

The ﬁrst term on the last line vanishes because the antisymmetry in A2,...Ar+2 implies the antisymmetry in m0, 1, . . . , r, but m0 takes a value between 1 and r so that one of the values arises twice and a completely m antisymmetric expression then vanishes. Equation (2.17) then follows because Λ m0 is invertible. The proof that the condition is sufﬁcient (=⇒) can be found in [3] or [10]. In the particular case of the disk, we have

A x = (t, xC , yC , φ, θ), (2.22) and we have two constraints, r = 2. More explicitly,

1 1 1 1 4 1 K0 = 0,K1 = 1,K2 = 0,K3 = −a sin θ = −a sin x ,K4 = 0, 2 2 2 1 4 2 (2.23) K0 = 0,K1 = 0,K2 = 1,K3 = a cos θ = a cos x ,K4 = 0. Let us show that the constraints are non-holonomic, i.e., that there is at least one of the conditions in (2.17) that is not satisﬁed. If we take m = 1, they become

∂ 1 1 2 ( KA )KA KA ] = 0. (2.24) ∂x[A1 2 3 4

Choosing A1 = 4,A2 = 3,A3 = 1,A4 = 2, one can check that only the ﬁrst term in the complete 1 4 antisymmetrization is non-vanishing, so that the result of the left hand side is given by 4! (−a cos x ) which does not vanish.

2.4 d’Alembert’s principle

In the proof of d’Alembert theorem, nothing crucially depended on the fact that the constraints were ∂Gm holonomic. It can thus be extended to non-holonomic ones (by the replacement → Km in the ∂xa a proof). Suppose then that there are r non holonomic constraints

m a m Ka x˙ + K0 = 0, (2.25) and consider inﬁnitesimal virtual displacements δxa that are compatible,

m a Ka δx = 0. (2.26) d’Alembert’s principle then states that Newton’s equation

a a a m(a)x¨ = F + R , (2.27) together with the condition that the forces of reaction are of the form

m Ra ≈ λmKa , (2.28) are equivalent to the condition that the sum of the applied forces and of the forces of inertia produce no work during any compatible inﬁnitesimal virtual displacement δxa,

a a a (F − m(a)x¨ )δx ≈ 0. (2.29)

Systems covered by d’Alembert’s theorem and principle are

• systems of point particles that move on polished surfaces that are ﬁxed or moving (holonomic constraints), 14 CHAPTER 2. CONSTRAINED SYSTEMS AND D’ALEMBERT PRINCIPLE

• motions of solids with one point having a prescribed trajectory

• sliding of solids on polished surfaces

• rolling without sliding of solids on rough surfaces

• ﬁxed links between solids

What is not covered by the theorem and principle are constraints that involve friction or constraints that are expressed by inequalities. Chapter 3

Hamilton and least action principles

a a Consider trajectories x = x (t) that pass through a fixed point at some initial time ti and another fixed a a a a point at a final time tf , x (ti) = xi , x (tf ) = xf .A natural trajectory is a trajectory that satisfies Newton’s equations together with the constraints. Virtual trajectories

15 16 CHAPTER 3. HAMILTON AND LEAST ACTION PRINCIPLES Chapter 4

Lagrangian mechanics

17 18 CHAPTER 4. LAGRANGIAN MECHANICS Chapter 5

Applications of the Lagrangian formalism

19 20 CHAPTER 5. APPLICATIONS OF THE LAGRANGIAN FORMALISM Chapter 6

Covariance of the Lagrangian formalism

6.1 Change of dependent variables

Consider a non-singular change of variables of the generalized Lagrange coordinates, α 0 ∂q qα = qα(qβ , t), det ( ) 6= 0, (6.1) ∂qβ0 with inverse transformation denoted by qβ0 = qβ0 (qα, t). The Lagrangian in the new coordinates is d L0(q0, q˙0, t) = L(q(q0, t), q(q0, t), t) := L| (6.2) dt d α β0 where the bar means that one has done the change of variables inside the function, with dt q (q , t) = ∂qα γ0 ∂qα 0 q˙ + . ∂qγ ∂t By direct computation one ﬁnds

0 0 0 α 2 α 2 α α δL ∂L d ∂L ∂L ∂q ∂L ∂ q 0 ∂ q d ∂L ∂q = − = | + | q˙γ + − | δqβ0 ∂qβ0 dt ∂q˙β0 ∂qα ∂qβ0 ∂q˙α ∂qβ0 ∂qγ0 ∂qβ0 ∂t dt ∂q˙α ∂qβ0 ∂qα h ∂L d ∂L i = 0 | − | . (6.3) ∂qβ ∂qα dt ∂q˙α ∂qα Indeed, when d in the last term of the first line hits the previous term in the first line is cancelled. As a consequence, dt ∂qβ0 ∂L out of the second and third term of the first line, one only remains with d that hits |. dt ∂q˙α Furthermore,

0 d 0 d2 0 where f(q, q,˙ q,¨ t)| = f q(q , t), dt q(q , t), dt2 q(q , t), t . Hence,

δL0 ∂qα δL 0 = 0 | . (6.5) δqβ ∂qβ δqα

This equation captures the covariance of Euler-Lagrange equations under changes of Lagrange coordinates. The meaning is as follows. Suppose that q¯α(t) is a natural trajectory, i.e., a solution to the Euler-Lagrange equations of motion,

δL d (¯qα(t), q¯α(t), t) = 0. (6.6) δqα dt

21 22 CHAPTER 6. COVARIANCE OF THE LAGRANGIAN FORMALISM

If one changes variables, qβ0 = qβ0 (qα, t), with inverse given in (6.1), it follows that q¯β0 (t) = qβ0 (¯qα(t), t) is a solution for the variational pinciple deﬁned by L0(q0, q˙0, t). Indeed, (6.5) is equivalent to

β0 0 2 ∂q δL 0 d 0 d 0 δL 0 q (q, t), q (q, t), q (q, t), t = (q, q,˙ q,¨ t). (6.7) ∂qα δqβ dt dt2 δqα

0 α δL β0 When evaluating at q¯ (t), the RHS vanishes. The result that 0 evaluated at q¯ (t) vanishes then follows δqβ 0 ∂qβ from invertibility of . ∂qα

6.2 Change of the independent variable

The action is a functional, it produces a number if one evaluates it on a trajectory,

Z tf d S[q] = dt L q(t), q(t), t ∈ R. (6.8) ti dt

Under an arbitrary reparametrisation of time, t = t(t0), with inverse denoted by t0 = t0(t), one ﬁnds the same number provided that

0 0 Z t =t (tf ) 0 0 dt 0 dt d 0 0 S[q] = dt 0 L q(t(t )), 0 q(t(t )), t(t ) . (6.9) 0 0 t =t (ti) dt dt dt

This suggests that the Euler-Lagrange equations are covariant under t = t(t0) and q0α(t0) = qα(t(t0)) provided that dq0 dt dt0 d L0 q0, , t0) = L q(t(t0)), q(t(t0)), t(t0) . (6.10) dt0 dt0 dt dt0 with 0α α dq dq dt α dt = | 0 =q ˙ | 0 . (6.11) dt0 dt t=t(t ) dt0 t=t(t ) dt0 δL Consider ﬁrst a simple example. If L = 1 mq˙2, = 0 ⇐⇒ q¨ = 0 ⇐⇒ q(t) = vt + c with √ 2 δq 0 0 0 2 v, c ∈ R. Let t, t > 0 and take√t = t ⇐⇒ t = t . √ dt0 0 0 0 0 It follows that dt = 2t = 2 t , and the solution becomes q (t ) = v t + c. The new Lagranian is

dt 1 dt0 dq0 √ dq0 L0 = m( )2 = m t0( )2. (6.12) dt0 2 dt dt0 dt0 The new equations of motion are

0 0 0 √ 0 0 √ 2 0 δL ∂L d ∂L d 0 dq 1 dq 0 d q = − 0 = − 2m t = −m√ − 2m t = 0, (6.13) δq0 ∂q0 0 dq 0 0 0 0 02 dt ∂ dt0 dt dt t dt dt which is equivalent to d2q0 1 dq0 = − . (6.14) dt02 2t0 dt0 This equation holds since dq0 1 v d2q0 1 v = √ , = − √ . (6.15) dt0 2 t0 dt02 4 ( t0)3 6.3. COMBINED CHANGE OF VARIABLES 23

The proof in the general case proceeds as follows.

0 δL0 ∂L0 d ∂L0 dt ∂L d dt¡ ∂L dt¡ dt h ∂L d ∂L i dt h δL i = − 0α = | − ¡ | ¡ = | − | = |, (6.16) δq0α ∂q0α 0 dq 0 ∂qα 0 0 ∂q˙α 0 ∂qα ∂q˙α 0 δqα dt ∂ dt0 dt dt ¡dt ¡dt dt dt dt where the last equality holds provided one can show that

d ∂L d ∂L | = |. (6.17) dt ∂q˙α dt ∂q˙α

δL0 δL As in the preceding section, = dt ( )| means that, if qα(t) is a natural trajectory for L(q, q,˙ t), δq0α dt0 δqα 0α 0 α 0 0 0 dq0 0 then so is q (t ) = q (t(t )) for L (q , dt0 , t ). In order to show (6.17), we have

d ∂L dt0 dq0 ∂2L dt0 dq0β ∂2L d2t0 dq0β dt0 d2q0β q0(t0), , t(t0) = | + | + ( )2 dt ∂q˙α dt dt0 ∂qβ∂q˙α dt dt0 ∂q˙β∂q˙α dt2 dt0 dt dt02 ∂2L dt¡0 dt¡ + | ¡ ¡ , (6.18) ∂t∂q˙α ¡dt ¡dt0 while d ∂L ∂2L ∂2L ∂2L | = |q˙β| + |q¨β| + |. (6.19) dt ∂q˙α ∂qβ∂q˙α ∂q˙β∂q˙α ∂t∂q˙α When taking into account (6.11), the ﬁrst terms on the RHS do agree, since so do the last terms, we only have to show that middle terms do as well, which is the case if

d2t0 dq0β dt0 d2q0β q¨β| = + ( )2 . (6.20) dt2 dt0 dt dt02 Differentiating (6.11) with respect to t0 gives

d2q0β d dt d2t dt = ( q˙β|) = q˙β + ( )2q¨β|, (6.21) dt02 dt0 dt0 dt02 dt0 which implies that dt0 d2q0β dt0 d2t q¨β| = ( )2 − ( )2 q˙β. (6.22) dt dt02 dt dt02 This gives the result when using (6.11) and

d dt0 dt d2t0 dt dt0 d2t d2t0 dt0 d2t 0 = ( ) = + ( )2 =⇒ = −( )3 . (6.23) dt dt dt0 dt2 dt0 dt dt02 dt2 dt dt02

6.3 Combined change of variables

When combining the results of the two previous sections, Euler-Lagrange equations are covariant under

qα = qα(qβ0 , t0) (6.24) t = t(t0) with inverse qα0 = qα0 (qβ, t) (6.25) t0 = t0(t) 24 CHAPTER 6. COVARIANCE OF THE LAGRANGIAN FORMALISM provided that

0 0 d 0 dt 0 dt d 0 L0 qβ (t0), qβ (t0), t0 = L qα(qβ (t0), t0), qα(qβ (t0), t0), t(t0) , (6.26) dt0 dt0 dt dt0

α 0 α d α β0 0 0 ∂q dqγ ∂q with q (q (t ), t ) = 0 + . Indeed, showing that dt0 ∂qγ dt0 ∂t0

δL0 dt δL ∂qα 0 = | 0 . (6.27) δqβ dt0 δqα ∂qβ can now be done in a straightforward way by combining the intermediate steps of the proofs of the previous two sections. Chapter 7

Noether’s theorem and Galilean invariance

7.1 Noether’s ﬁrst theorem

A first integral or conservation law is a function K(q, q,˙ t) that is constant on all natural trajectories, dK δL |q¯ = 0, with | = 0. (7.1) dt δqα q¯ An infinitesimal symmetry is a virtual variation that leaves the Lagrangian invariant up to a total deriva- tive1, dF (q, q,˙ t) δ qα = Qα(q, q,˙ t) such that δ L = . (7.2) Q Q dt Noether’s first theorem states that: Every infinitesimal symmetry gives rise to a conservation law. Indeed, ∂L ∂L ∂L dQα ∂L dF δ L = δ qα + δ q˙α = Qα + = , (7.3) Q Q ∂qα Q ∂q˙α ∂qα dt ∂q˙α dt where the second equality follows because virtual variations commute with the total derivative, while the last equality follows by the assumption that the variation is a symmetry. Using dQα ∂L d ∂L d ∂L = (Qα ) − Qα , (7.4) dt ∂q˙α dt ∂q˙α dt ∂q˙α the latter combines with the first term to produce the Euler-Lagrange derivatives, while the former combines with the total derivative on the RHS, δL d ∂L Qα = (F − Qα). (7.5) δqα dt ∂q˙α Since the LHS vanishes when evaluated at any natural trajectory, it follows that associated the conservation law is given by ∂L K = F − Qα . (7.6) ∂q˙α Remark: Neither the theorem nor the expression depends on the choice of representative for the df(q, t) action: if L0 = L + , it follows that δ is also an infinitesimal symmetry of L0, dt Q d d δL0 = δL + (δ f) = (F + δ f), (7.7) dt Q dt Q with associated conservation law ∂f ∂L ∂f 0 α α K = F + α Q − ( α + α )Q = K. (7.8) ∂q ∂q˙ ∂q 1This means that the variation leaves the action functional invariant if one considers only trajectories that vanish at the end-points and restricts f to satisfy f(0, t) = 0.

25 26 CHAPTER 7. NOETHER’S THEOREM AND GALILEAN INVARIANCE 7.2 Applications

7.2.1 Explicit time independence and conservation of energy ∂L Consider a Lagrangian that does not depend explicitly on time, L = L(q, q˙), = 0. It follows that ∂t dL ∂L ∂L δL d ∂L = q˙α + q¨α = q˙α + ( q˙α), (7.9) dt ∂qα ∂q˙α δqα dt ∂q˙α In other words, ∂L K = q˙α − L, (7.10) ∂q˙α is the conservation law associated to the inﬁnitesimal symmetry

α α δQq = −q˙ . (7.11) In particular, if 1 L = g (q) + A (q)q ˙α − V (q), (7.12) 2 αβ α the associated conservation law 1 1 K = g q˙αq˙β + A¨q˙¨α − g − A¨q˙¨α + V − = g q˙αq˙β + V = E (7.13) αβ ¨α 2 αβ ¨α 2 αβ corresponds to the mechanical energy.

7.2.2 Cyclic variables and conservation of canonical momentum Suppose that a given variable qα is “cyclic” which means that L does not depend on qα but may depend ∂L on the associated velocity q˙α. To fix ideas, suppose that the first variable q1 is cyclic, = 0. It follows ∂q1 α α that δQq = −δ1 is an infinitesimal symmetry, ∂L d(−δα) ∂L ∂L d(−1) ∂L δ L = −δα + 1 = − + = 0. (7.14) Q 1 ∂qα dt ∂q˙α ∂q1 dt ∂q˙1 Since F = 0, the associated conservation law is ∂L ∂L K = − (−δα) = , (7.15) ∂q˙α 1 ∂q˙1 which is the canonical momentum associated to q1.

7.2.3 One-parameter groups of transformations Suppose that the Lagrangian is invariant under a set of invertible transformations qα0 = qα0 (qβ, t; s) that depend continously on a parameters s, and reduce to the identity at s = 0, qα0 (qβ, t; 0) = qα. By invariance we mean that the transformed Lagragian in terms of the new variables is equal to the old Lagrangian in terms of the same variables, d L0(q0, q˙0, t) = L(q(q0, t; s), q(q0, t; s), t) = L(q0, q˙0, t), (7.16) dt where the ﬁrst equality merely expresses how the new Lagrangian in terms of the new coordinates is obtained in terms of the old one, while the second equality expresses the invariance of the Lagrangian. When using the inverse transformation, the latter is equivalent to d L(q0(q, t; s), q0(q, t; s), t) = L(q, q,˙ t). (7.17) dt 7.2. APPLICATIONS 27

In this case, the transformation 0 ∂qα δ qα = Rα(q, t) := | , (7.18) R ∂s s=0 defines an infinitesimal symmetry. α0 α α0 0 dq dR ∂ dq Indeed, qα = qα + sRα(q, t) + O(s2), while =q ˙α + s + O(s2). If δ q˙α = | , dt dt R ∂s dt s=0 dRα δ q˙α = , as it should for a virtual variation. In these terms, the invariance condition (7.17) becomes R dt dR L(q + sR + O(s2), q˙ + s + O(s2), t) = L(q, q,˙ t), ∀s. (7.19) dt In particular, for s = 0, this is an identity, while the terms linear in s on the LHS have to vanish. The latter ∂ can be obtained by taking at s = 0, and yield ∂s ∂L dRα ∂L Rα + = 0, (7.20) ∂qα dt ∂q˙α which can also be written as δRL = 0. According to Noether’s theorem, the associated conservation law is given by ∂L K = − δ qα. (7.21) ∂q˙α R Note that a cyclic variabe can be interpreted as a particular case. Indeed, the transformation that leaves α0 α α the Lagrangian invariant is given by q = q − sδ1 . Another particular case is a system of particles in three-dimensional Euclidean space, with a La- grangian of the type N 1 X L = T − V,T = m x˙ α x˙ ,V = V (r ). (7.22) 2 (i) (i) α(i) ij i=1 In this expression, the summation in the kinetic term over the different particles has been written out explicitly, while the summation over the 3 components of the position of a single particle is implicit β through the summation convention. Note also that we have defined xα(i) = δαβx(i), and that the potential V (rij) with i < j is assumed to depend only on the relative distances between the different particles,

q α α rij = (x(i) − x(j))(xα(i) − xα(j)). (7.23)

If one now considers translations in Euclidean space,

α0 α α xi = xi − a , (7.24) for constant aα (which correspond to three different parameters s), invariance of the Lagrangrian in the 0 dxα dxα form (7.17) is manifest since i = i and r0 = r . The associated inﬁnitesimal transformations are dt dt ij ij

0 ∂xα Rα = i | = δα, (7.25) γi ∂aγ a=0 γ with associated conservation laws given by the components of total linear momenta, X ∂L X P = = m x˙ . (7.26) γ ∂x˙ γ (i) γ(i) i (i) i For rotations, we have

α0 α β α γ T xi = A βxi ,A βδαγA δ = δβδ ⇐⇒ A A = I, (7.27) 28 CHAPTER 7. NOETHER’S THEOREM AND GALILEAN INVARIANCE with A a constant 3 × 3 orthogonal matrix. Again, the invariance of the Lagrangian is manifest since 0 0 dxα dxγ dxα dxγ orthogonality implies that (i) δ (i) = (i) δ (i) while distances are also left invariant. In order to dt αγ dt dt αγ dt α α α 2 α ﬁnd the parameters, we note that if A β = δβ +ω β +O(ω ), with ω β a “small” matrix, the orthogonality condition α α 2 γ γ 2 (δβ + ω β + O(ω ))δαγ(δδ + ω δ + O(ω )) = δβδ, (7.28) implies to ﬁrst order in ω that T ωδβ + ωβδ = 0 ⇐⇒ ω + ω = 0. (7.29) This implies that ω can be written in terms of three independent rotation parameters θγ as

α γ α ω β = θ γβ, (7.30) and that the associated inﬁnitesimal ﬁeld transformations are

0 ∂xα Rα = i | = α xβ, (7.31) γi ∂θγ θ=0 γβ i whith associated conservation laws given by the components of the total angular momentum,

X α β Jγ = γαβ m(i)x˙ (i)x(i). (7.32) i

We will show in the appendix that a rotation matrix that is connected to the identity can be written as a rotation around an axis ~ α p 1 2 2 2 3 3 θ = θ~n = θ ~eα, θ = (θ ) + (θ ) + (θ ) , (7.33) with ~e1 = ~1x, ~e2 = ~1y, ~e3 = ~1z, as

1 − cos θ sin θ Aα = δα cos θ + θαθ ( ) − α θγ . (7.34) β β β θ2 βγ θ As before, one then ﬁnds that ∂Aα β | = −α . (7.35) ∂θγ θ=0 βγ

7.2.4 One-parameter groups of transformations involving time Consider now a one-parameters group of invertible transformations that affects time as well,

α0 α0 β q = q (q , t; s), 0 0 =⇒ qα (t0) = qα (qβ(t(t0; s)), t(t0; s); s), (7.36) t0 = t0(t; s) with qα0 = qα0 (qβ, t; 0) = qα, t0(t; 0) = t. According to the equation (6.26) of the previous section, the transformed Lagrangian is

dq0(t0) dt dt0 d L0(q0(t0), , t0) = Lq(q0(t0), t0), q(q0(t0), t0), t(t0), (7.37) dt0 dt0 dt dt0 where we have temporarily suppressed the s dependence. Let us assume that the transformed Lagrangian is equal to the starting point Lagrangian (in terms of the new variables), up to a total derivative,

dq0(t0) d d L0(q0(t0), , t0) = L(q0(t0), q0(t0), t0) − G0(q0(t0), t0), (7.38) dt0 dt0 dt0 7.2. APPLICATIONS 29

dt0 with G0| = 0. When using the inverse transformation and multiplying by , the generalized invari- s=0 dt ance condition (7.37) can also be written as dt0 dt d d d Lq0(q, t), q0(q(t), t), t0(t) − G(q(t), t) = L(q(t), q(t), t), (7.39) dt dt0 dt dt dt with G(q(t), t) = G0(q0(q(t), t), t0(t)). In terms of the action, this generalized invariance condition means that as a functional of the new trajectories and the transformed end-points, the

t0=t0(t ) Z f d S[q0] = dtLq0(t), q0(t), t), (7.40) 0 0 dt t =t (ti) equals the action as a functional of the old trajectories, up to a boundary term. Indeed, by ﬁrst changing the name of the dummy integration variable, and then doing the change of variables from primed to unprimed ones, we get

0 0 Z t =t (tf ) Z t=tf 0 0 0 0 0 d 0 0 0 dt 0 dt d 0 0 S[q ] = dt L(q (t ), 0 q (t ), t ) = dt L q (q(t), t), 0 q (q(t), t), t (t) . (7.41) 0 0 dt dt dt dt t =t (ti) t=ti When using the generalized invariance condition (7.39), this gives

t Z f d h itf S[q0] = dtL(q(t), q(t), t) + G(q(t), t) . (7.42) dt t ti i Note that the boundary term does not contribute when computing the variation of trajectories with ﬁxed end-points. Expanding the transformed trajectories to ﬁrst order in s gives qα0 (t0) = qα(t) + sRα(qβ, t) + O(s2), ,G = sG (q, t) + O(s2), (7.43) t0 = t + sξ(t) + O(s2), 1 the associated equal-time virtual variation is

α 1 α0 α 1 α0 0 2 α δSq (t) = lim q (t) − q (t) = lim q (t − sξ(t) + O(s )) − q (t) s→0 s s→0 s 0 1 dqα α¨¨ α β 0 2 α¨¨ α α = lim ¨q (t) + sR (q (t), t) − s 0 (t )ξ(t) + O(s ) − ¨q (t) = R − q˙ ξ . (7.44) s→0 s dt Such a variation leaves the Lagrangian invariant up to a total derivative. Indeed, the generalized invaraince condition (7.37) can be written as d d (1 + sξ˙)L(q + sR, (1 − sξ˙) (q + sR), t + sξ − s G + O(s2) = L(q, q,˙ t). (7.45) dt dt 1 To order 0 in s, this is an identity, while the terms of order 1 yield

∂L ∂L d ∂L d ξL˙ (q, q,˙ t) + Rα + ( Rα − ξ˙q˙α) + (q,q,˙ t)ξ − G = 0, (7.46) ∂qα ∂q˙α dt ∂t dt 1 The ﬁrst term can be integrated by parts

d ∂L ∂L ∂L ξL˙ = (ξL) − ξ( q˙α + q¨α + ), (7.47) dt ∂qα ∂q˙α ∂t and, when combining like terms, we get ∂L ∂L d d d (Rα − q˙αξ) + (Rα − q˙αξ) = (G − ξL) ⇐⇒ δ L = (G − ξL). (7.48) ∂qα ∂q˙α dt dt 1 S dt 1 30 CHAPTER 7. NOETHER’S THEOREM AND GALILEAN INVARIANCE

The associated conservation law is

∂L K = G − ξL − δ qα . (7.49) 1 ∂q˙α S

∂L In particular, a Lagragian does not depend explicit on time, = 0 iff it possesses the symmetry ∂t α0 α 0 α α α q = q , t = t + s. In this case G = 0, ξ = 1, R = 0, δSq = −q˙ . The associated conservation law is

∂L E = q˙α − L, (7.50) ∂q˙α discussed previously, which now appears from the viewpoint of the symmetry of the system under constant time translations.

7.2.5 Galilean boosts Galilean transformations on a system of particles are described by

α0 α β α α 0 T xi = A βxi + v t − a , t = t + b, A A = I. (7.51) They consists of a combination of spatial translations, rotations, time-translations, described by the constant parameter b, and changes of observers with a constant velocity in a given direction, sometimes called “Galilean boosts” and described by the 3 parameters vα. We have already analyzed invariance of the particle Lagrangian (7.22) under spatial translations (parameters aα) and rotations (parameters ωα of the rotation matrix A) and the associated conservation laws Pα,Jα. For the time-translation, explicit time-indepence implies that the associated conservation law is P ∂L α E = i α x˙ (i) − L = T + V . So only the Galilean boosts remain to be analyzed explicitely. More ∂x˙ (i) 0 dt0 dxα (t0) dxβ(t) generally, for the whole set of Galilean transformations, we have = 1, i = Aα i + vα. dt dt0 β dt For the kinetic energy of the particle Lagrangian (7.22), this gives

α0 0 β γ dx (t ) 1 X dx(i) dx(i) T ( i ) = m (Aα + vα)(A + v ) dt0 2 (i) β dt αγ dt α i α β 1 X dx(i) dxα(i) X dx(i) 1 X = m + m v Aα + m vαv 2 (i) dt dt (i) α β dt 2 (i) α i i i dxα d X 1 = T ( i ) + ( m v (Aα xβ + tvα). (7.52) dt dt (i) α β (i) 2 i Since relative distances are invariant even if one translates the individual coordinates linearly in time (by the same amount), the potential is also invariant. Since X 1 G = m v (Aα xβ + tvα), (7.53) (i) α β (i) 2 i

∂G it follows that for Galilean boosts parametrized by vγ, Rα = δαt, ξ = 0, | = P m x , and γi γ ∂vγ a,b,θ,v=0 i (i) γ(i) the associated conservation laws are X Kγ = m(i)(xγ(i) − x˙ γ(i)t). (7.54) i Chapter 8

Hamiltonian formalism

31 32 CHAPTER 8. HAMILTONIAN FORMALISM Chapter 9

Canonical transformations

33 34 CHAPTER 9. CANONICAL TRANSFORMATIONS Chapter 10

Dynamics in non-inertial frames

35 36 CHAPTER 10. DYNAMICS IN NON-INERTIAL FRAMES Chapter 11

Charged particle in electromagnetic ﬁeld

37 38 CHAPTER 11. CHARGED PARTICLE IN ELECTROMAGNETIC FIELD Bibliography

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