Problem 1. (Likharev-EGP) Stretchable pendulum

Consider a stretchable pendulum (i.e. a mass hung on an elastic cord that exerts a forceF = −κ(` − `0) where κ and `0 are positive constants), confined to the x − y plane of thepage.

Essential Graduate Physics CM: Classical Mechanics

Chapter 2 Page 11 of 14

is conserved. This is just a particular (2D) case of the angular momentum conservation – see Eq. (1.24).

Indeed, for the 2D motion within the [x, y] plane, the angular momentum vector,

zmymxmzyx

zyx

���

nnnprL u{ , (2.51)

has only one nonvanishing component, perpendicular to the motion plane:

).()( xmyymxLz �� � (2.52)

Differentiating the well-known relations between the polar and Cartesian coordinates,

,sin,cos MM ryrx (2.53)

over time, and plugging the result into Eq. (52), we see that .2

MM pmrLz �

Thus the Lagrangian formalism provides a powerful way of searching for non-evident integrals

of motion. On the other hand, if such conserved quantity is evident or known a priori, it is helpful for

the selection of the most appropriate generalized coordinates, giving the simplest Lagrange equations.

For example, in the last problem, if we have known in advance that pM had to be conserved, this could

provide a motivation for using angle M as one of generalized coordinates.

2.5. Exercise problems

In each of Problems 2.1-2.10:

(i) introduce a set of convenient generalized coordinate(s) qj of the system,

(ii) write down Lagrangian L as a function of ,j jq q� , and (if appropriate) time,

(iii) write down the Lagrangian equation(s) of motion,

(iv) calculate the Hamiltonian function H; find out whether it is conserved,

(v) calculate energy E; is E = H?; is energy conserved?

(vi) any other evident integrals of motion?

2.1. Double pendulum – see Fig. on the right. Consider only the motion

confined to a vertical plane containing the suspension point.

2.2. Stretchable pendulum (i.e. a mass hung on an elastic cord that exerts force F

= -N(l - l0), where N and l0 are positive constants), confined to a vertical plane:

m g l

m' g

m l

l'

Essential Graduate Physics CM: Classical Mechanics

Chapter 2 Page 11 of 14

is conserved. This is just a particular (2D) case of the angular momentum conservation – see Eq. (1.24).

Indeed, for the 2D motion within the [x, y] plane, the angular momentum vector,

zmymxmzyx

zyx

���

nnnprL u{ , (2.51)

has only one nonvanishing component, perpendicular to the motion plane:

).()( xmyymxLz �� � (2.52)

Differentiating the well-known relations between the polar and Cartesian coordinates,

,sin,cos MM ryrx (2.53)

over time, and plugging the result into Eq. (52), we see that .2

MM pmrLz �

Thus the Lagrangian formalism provides a powerful way of searching for non-evident integrals

of motion. On the other hand, if such conserved quantity is evident or known a priori, it is helpful for

the selection of the most appropriate generalized coordinates, giving the simplest Lagrange equations.

For example, in the last problem, if we have known in advance that pM had to be conserved, this could

provide a motivation for using angle M as one of generalized coordinates.

2.5. Exercise problems

In each of Problems 2.1-2.10:

(i) introduce a set of convenient generalized coordinate(s) qj of the system,

(ii) write down Lagrangian L as a function of ,j jq q� , and (if appropriate) time,

(iii) write down the Lagrangian equation(s) of motion,

(iv) calculate the Hamiltonian function H; find out whether it is conserved,

(v) calculate energy E; is E = H?; is energy conserved?

(vi) any other evident integrals of motion?

2.1. Double pendulum – see Fig. on the right. Consider only the motion

confined to a vertical plane containing the suspension point.

2.2. Stretchable pendulum (i.e. a mass hung on an elastic cord that exerts force F

= -N(l - l0), where N and l0 are positive constants), confined to a vertical plane:

m g l

m' g

m l

l'

Essential Graduate Physics CM: Classical Mechanics

Chapter 2 Page 11 of 14

is conserved. This is just a particular (2D) case of the angular momentum conservation – see Eq. (1.24).

Indeed, for the 2D motion within the [x, y] plane, the angular momentum vector,

zmymxmzyx

zyx

���

nnnprL u{ , (2.51)

has only one nonvanishing component, perpendicular to the motion plane:

).()( xmyymxLz �� � (2.52)

Differentiating the well-known relations between the polar and Cartesian coordinates,

,sin,cos MM ryrx (2.53)

over time, and plugging the result into Eq. (52), we see that .2

MM pmrLz �

Thus the Lagrangian formalism provides a powerful way of searching for non-evident integrals

of motion. On the other hand, if such conserved quantity is evident or known a priori, it is helpful for

the selection of the most appropriate generalized coordinates, giving the simplest Lagrange equations.

For example, in the last problem, if we have known in advance that pM had to be conserved, this could

provide a motivation for using angle M as one of generalized coordinates.

2.5. Exercise problems

In each of Problems 2.1-2.10:

(i) introduce a set of convenient generalized coordinate(s) qj of the system,

(ii) write down Lagrangian L as a function of ,j jq q� , and (if appropriate) time,

(iii) write down the Lagrangian equation(s) of motion,

(iv) calculate the Hamiltonian function H; find out whether it is conserved,

(v) calculate energy E; is E = H?; is energy conserved?

(vi) any other evident integrals of motion?

2.1. Double pendulum – see Fig. on the right. Consider only the motion

confined to a vertical plane containing the suspension point.

2.2. Stretchable pendulum (i.e. a mass hung on an elastic cord that exerts force F

= -N(l - l0), where N and l0 are positive constants), confined to a vertical plane:

m g l

m' g

m l

l'

Essential Graduate Physics CM: Classical Mechanics

Chapter 2 Page 11 of 14

is conserved. This is just a particular (2D) case of the angular momentum conservation – see Eq. (1.24).

Indeed, for the 2D motion within the [x, y] plane, the angular momentum vector,

zmymxmzyx

zyx

���

nnnprL u{ , (2.51)

has only one nonvanishing component, perpendicular to the motion plane:

).()( xmyymxLz �� � (2.52)

Differentiating the well-known relations between the polar and Cartesian coordinates,

,sin,cos MM ryrx (2.53)

over time, and plugging the result into Eq. (52), we see that .2

MM pmrLz �

Thus the Lagrangian formalism provides a powerful way of searching for non-evident integrals

of motion. On the other hand, if such conserved quantity is evident or known a priori, it is helpful for

the selection of the most appropriate generalized coordinates, giving the simplest Lagrange equations.

For example, in the last problem, if we have known in advance that pM had to be conserved, this could

provide a motivation for using angle M as one of generalized coordinates.

2.5. Exercise problems

In each of Problems 2.1-2.10:

(i) introduce a set of convenient generalized coordinate(s) qj of the system,

(ii) write down Lagrangian L as a function of ,j jq q� , and (if appropriate) time,

(iii) write down the Lagrangian equation(s) of motion,

(iv) calculate the Hamiltonian function H; find out whether it is conserved,

(v) calculate energy E; is E = H?; is energy conserved?

(vi) any other evident integrals of motion?

2.1. Double pendulum – see Fig. on the right. Consider only the motion

confined to a vertical plane containing the suspension point.

2.2. Stretchable pendulum (i.e. a mass hung on an elastic cord that exerts force F

= -N(l - l0), where N and l0 are positive constants), confined to a vertical plane:

m g l

m' g

m l

l'

Essential Graduate Physics CM: Classical Mechanics

Chapter 2 Page 11 of 14

is conserved. This is just a particular (2D) case of the angular momentum conservation – see Eq. (1.24).

Indeed, for the 2D motion within the [x, y] plane, the angular momentum vector,

zmymxmzyx

zyx

���

nnnprL u{ , (2.51)

has only one nonvanishing component, perpendicular to the motion plane:

).()( xmyymxLz �� � (2.52)

Differentiating the well-known relations between the polar and Cartesian coordinates,

,sin,cos MM ryrx (2.53)

over time, and plugging the result into Eq. (52), we see that .2

MM pmrLz �

Thus the Lagrangian formalism provides a powerful way of searching for non-evident integrals

of motion. On the other hand, if such conserved quantity is evident or known a priori, it is helpful for

the selection of the most appropriate generalized coordinates, giving the simplest Lagrange equations.

For example, in the last problem, if we have known in advance that pM had to be conserved, this could

provide a motivation for using angle M as one of generalized coordinates.

2.5. Exercise problems

In each of Problems 2.1-2.10:

(i) introduce a set of convenient generalized coordinate(s) qj of the system,

(ii) write down Lagrangian L as a function of ,j jq q� , and (if appropriate) time,

(iii) write down the Lagrangian equation(s) of motion,

(iv) calculate the Hamiltonian function H; find out whether it is conserved,

(v) calculate energy E; is E = H?; is energy conserved?

(vi) any other evident integrals of motion?

2.1. Double pendulum – see Fig. on the right. Consider only the motion

confined to a vertical plane containing the suspension point.

2.2. Stretchable pendulum (i.e. a mass hung on an elastic cord that exerts force F

= -N(l - l0), where N and l0 are positive constants), confined to a vertical plane:

m g l

m' g

m l

l'

Essential Graduate Physics CM: Classical Mechanics

Chapter 2 Page 11 of 14

is conserved. This is just a particular (2D) case of the angular momentum conservation – see Eq. (1.24).

Indeed, for the 2D motion within the [x, y] plane, the angular momentum vector,

zmymxmzyx

zyx

���

nnnprL u{ , (2.51)

has only one nonvanishing component, perpendicular to the motion plane:

).()( xmyymxLz �� � (2.52)

Differentiating the well-known relations between the polar and Cartesian coordinates,

,sin,cos MM ryrx (2.53)

over time, and plugging the result into Eq. (52), we see that .2

MM pmrLz �

Thus the Lagrangian formalism provides a powerful way of searching for non-evident integrals

of motion. On the other hand, if such conserved quantity is evident or known a priori, it is helpful for

the selection of the most appropriate generalized coordinates, giving the simplest Lagrange equations.

For example, in the last problem, if we have known in advance that pM had to be conserved, this could

provide a motivation for using angle M as one of generalized coordinates.

2.5. Exercise problems

In each of Problems 2.1-2.10:

(i) introduce a set of convenient generalized coordinate(s) qj of the system,

(ii) write down Lagrangian L as a function of ,j jq q� , and (if appropriate) time,

(iii) write down the Lagrangian equation(s) of motion,

(iv) calculate the Hamiltonian function H; find out whether it is conserved,

(v) calculate energy E; is E = H?; is energy conserved?

(vi) any other evident integrals of motion?

2.1. Double pendulum – see Fig. on the right. Consider only the motion

confined to a vertical plane containing the suspension point.

2.2. Stretchable pendulum (i.e. a mass hung on an elastic cord that exerts force F

= -N(l - l0), where N and l0 are positive constants), confined to a vertical plane:

m g l

m' g

m l

l'

Essential Graduate Physics CM: Classical Mechanics

Chapter 2 Page 11 of 14

is conserved. This is just a particular (2D) case of the angular momentum conservation – see Eq. (1.24).

Indeed, for the 2D motion within the [x, y] plane, the angular momentum vector,

zmymxmzyx

zyx

���

nnnprL u{ , (2.51)

has only one nonvanishing component, perpendicular to the motion plane:

).()( xmyymxLz �� � (2.52)

Differentiating the well-known relations between the polar and Cartesian coordinates,

,sin,cos MM ryrx (2.53)

over time, and plugging the result into Eq. (52), we see that .2

MM pmrLz �

Thus the Lagrangian formalism provides a powerful way of searching for non-evident integrals

of motion. On the other hand, if such conserved quantity is evident or known a priori, it is helpful for

the selection of the most appropriate generalized coordinates, giving the simplest Lagrange equations.

For example, in the last problem, if we have known in advance that pM had to be conserved, this could

provide a motivation for using angle M as one of generalized coordinates.

2.5. Exercise problems

In each of Problems 2.1-2.10:

(i) introduce a set of convenient generalized coordinate(s) qj of the system,

(ii) write down Lagrangian L as a function of ,j jq q� , and (if appropriate) time,

(iii) write down the Lagrangian equation(s) of motion,

(iv) calculate the Hamiltonian function H; find out whether it is conserved,

(v) calculate energy E; is E = H?; is energy conserved?

(vi) any other evident integrals of motion?

2.1. Double pendulum – see Fig. on the right. Consider only the motion

confined to a vertical plane containing the suspension point.

2.2. Stretchable pendulum (i.e. a mass hung on an elastic cord that exerts force F

= -N(l - l0), where N and l0 are positive constants), confined to a vertical plane:

m g l

m' g

m l

l'

Essential Graduate Physics CM: Classical Mechanics

Chapter 2 Page 11 of 14

is conserved. This is just a particular (2D) case of the angular momentum conservation – see Eq. (1.24).

Indeed, for the 2D motion within the [x, y] plane, the angular momentum vector,

zmymxmzyx

zyx

���

nnnprL u{ , (2.51)

has only one nonvanishing component, perpendicular to the motion plane:

).()( xmyymxLz �� � (2.52)

Differentiating the well-known relations between the polar and Cartesian coordinates,

,sin,cos MM ryrx (2.53)

over time, and plugging the result into Eq. (52), we see that .2

MM pmrLz �

Thus the Lagrangian formalism provides a powerful way of searching for non-evident integrals

of motion. On the other hand, if such conserved quantity is evident or known a priori, it is helpful for

the selection of the most appropriate generalized coordinates, giving the simplest Lagrange equations.

For example, in the last problem, if we have known in advance that pM had to be conserved, this could

provide a motivation for using angle M as one of generalized coordinates.

2.5. Exercise problems

In each of Problems 2.1-2.10:

(i) introduce a set of convenient generalized coordinate(s) qj of the system,

(ii) write down Lagrangian L as a function of ,j jq q� , and (if appropriate) time,

(iii) write down the Lagrangian equation(s) of motion,

(iv) calculate the Hamiltonian function H; find out whether it is conserved,

(v) calculate energy E; is E = H?; is energy conserved?

(vi) any other evident integrals of motion?

2.1. Double pendulum – see Fig. on the right. Consider only the motion

confined to a vertical plane containing the suspension point.

2.2. Stretchable pendulum (i.e. a mass hung on an elastic cord that exerts force F

= -N(l - l0), where N and l0 are positive constants), confined to a vertical plane:

m g l

m' g

m l

l'

(a) Introduce a set of convenient generalized coordinate(s) q of the system,

(b) Write down Lagrangian L as a function of q and (if appropriate) time,

(c) Write down the Lagrangian equation(s) of motion,

(d) Calculate the Hamiltonian H(q, p) ; find out whether it is conserved.

(e) Are there any other evident integrals of motion?

1

Problem 2. Particle in an electro-magnetic field

A non-relativistic particle of charge q in a electro-magnetic field is described by the La-grangian (try to remember this!)

L =1

2mx2 − qφ+ q

v

c·A (1)

where φ(t,x(t)) is the scalar potential, and A(t,x(t)) is the vector potential of electricityand magnetisim. The electric and mangic fields are related to φ and A through

E(t,x) =−∇φ− 1

c∂tA (2)

B(t,x) =∇×A (3)

(a) Show that the Euler-Lagrange equations give the expected EOM for a particle experi-encing the force law: F = q(E + v

c×B).

(b) Compute the canonical momementa. Does

dp

dt= F (4)

with p the (canonical) momentum hold? Explain.

(c) Determine the Hamiltonian H(q, p) and Hamiltonian function h(q, q).

2

Problem 3. (MIT/OCW) Spring system on a plane

✓ ◆ ✓ ◆

3 Physics 8.09, Classical Physics III, Fall 2014

4. Spring System on a Plane [10 points]

A massless spring has an unstretched length b and spring constant k, and is used toconnect two particles of mass m1 and m2. The system rests on a frictionless tableand may oscillate and rotate.

(a) What is the Lagrangian? Write it with two-dimensional cartesian coordinates. (b) Setup a suitable set of generalized coordinates to better account for the symme-

tries of this system. What are the Lagrangian and equations of motion in these variables?

(c) Identify as many conserved generalized momenta as you can that are associated to cyclic coordinates in the Lagrangian from part b). If you think you are missing some, try to improve your answer to b). Show that there is a solution that rotates but does not oscillate, and discuss what happens to this solution for an increased rate of rotation. (A closed form solution is not necessary. A graphical solution will su�ce.)

5. Jerky Mechanics [6 points]

Consider an extension of classical mechanics where the equation of motion involves...

a triple time derivative, x = f(x, x,˙ x, t¨ ). Lets use Hamilton’s principle to derive the corresponding Euler-Lagrange equations. Start with a Lagrangian of the form L(qi, qi, qi, t) for n generalized coordinates qi, and make use of Hamilton’s principle for paths qi(t) that have zero variation of both qi and qi at the end points. Show that

d2 @L d @L @L � + = 0 dt2 @qi dt @qi @qi

for each i = 1, . . . , n.

6.* Extra Problem: Equivalent Lagrangians [not for credit] Some of you have seen this problem already in Classical Mechanics II. If it does not sound familiar then I suggest you try it. (It will not be graded but solutions will be provided.)

Let L(q, q, t) be the Lagrangian for a particle with coordinate q, which satisfies the Euler-Lagrange equations. Show that the Lagrangian

dF (q, t)L0 = L +

dt

also satisfies the Euler-Lagrange equations where F is an arbitrary di↵erentiable function.

3

Problem 4. (Goldstein/MIT OCW) Jerky Mechanics

✓ ◆ ✓ ◆

3 Physics 8.09, Classical Physics III, Fall 2014

4. Spring System on a Plane [10 points]

A massless spring has an unstretched length b and spring constant k, and is used toconnect two particles of mass m1 and m2. The system rests on a frictionless tableand may oscillate and rotate.

(a) What is the Lagrangian? Write it with two-dimensional cartesian coordinates. (b) Setup a suitable set of generalized coordinates to better account for the symme-

tries of this system. What are the Lagrangian and equations of motion in these variables?

(c) Identify as many conserved generalized momenta as you can that are associated to cyclic coordinates in the Lagrangian from part b). If you think you are missing some, try to improve your answer to b). Show that there is a solution that rotates but does not oscillate, and discuss what happens to this solution for an increased rate of rotation. (A closed form solution is not necessary. A graphical solution will su�ce.)

5. Jerky Mechanics [6 points]

Consider an extension of classical mechanics where the equation of motion involves...

a triple time derivative, x = f(x, x,˙ x, t¨ ). Lets use Hamilton’s principle to derive the corresponding Euler-Lagrange equations. Start with a Lagrangian of the form L(qi, qi, qi, t) for n generalized coordinates qi, and make use of Hamilton’s principle for paths qi(t) that have zero variation of both qi and qi at the end points. Show that

d2 @L d @L @L � + = 0 dt2 @qi dt @qi @qi

for each i = 1, . . . , n.

6.* Extra Problem: Equivalent Lagrangians [not for credit] Some of you have seen this problem already in Classical Mechanics II. If it does not sound familiar then I suggest you try it. (It will not be graded but solutions will be provided.)

Let L(q, q, t) be the Lagrangian for a particle with coordinate q, which satisfies the Euler-Lagrange equations. Show that the Lagrangian

dF (q, t)L0 = L +

dt

also satisfies the Euler-Lagrange equations where F is an arbitrary di↵erentiable function.

Problem 5. (MIT/OCW) Equivalent Lagrangians

Let L(q, q, t) be the Lagrangian for a particle with coordinate q, which satises the Euler-Lagrange equations. Show that the Lagrangian

L′ = L+dF (q, t)

dt(5)

also produces the Euler-Lagrange equations where F is an arbitrary dierentiable function.Give an algebraic proof and a proof based on the action. Using the previous problem andthe current problem, what is the equation of motion resulting from

L = −1

2mqq − 1

2ω20q

2 (6)

and what is it related to? Explain.

4