The Lagrangian of a system is given by L({q_{i}, v_{i}}),
where {qi} are linearly independent generalized coordinates and {v_{i} =
dq_{i}/dt} are the generalized velocities.

d/dt(∂L/∂v_{i}) - ∂L/∂q_{i} = 0, ∂L/∂v_{i} = p_{i}.

A symmetry is a coordinate transformation that does not change the form of the
Lagrangian.

Consider a continuous coordinate transformation, which is a transformation that
can be built from infinitesimal transformations of the form {q_{i} -->
q_{i}' = q_{i }+ δq_{i}}, with δq_{i} = f({q_{i}})ε,

where ε --> 0. Show that if the Lagrangian is invariant under this
transformation (δL = 0), then the quantity Q = ∑_{ }p_{i} f({q_{i}})
is a constant of motion, i.e. a conserved quantity.

Solution:

- Concepts:

Lagrangian mechanics - Reasoning:

δL({q_{i}, v_{i}}) = ∑[(∂L/∂v_{i}) δv_{i}- ∂L/∂q_{i}) δq_{i}] - Details of the calculation:

δL = ∑[p_{i}δv_{i}- ∂L/∂q_{i}) δq_{i}].

From Lagrange's equations ∂L/∂q_{i}= dp_{i}/dt.

δL = ∑[p_{i}δ(dq_{i}/dt) – (dp_{i}/dt)) δq_{i}] = ∑[p_{i}d(δq_{i})/dt – (dp_{i}/dt)) δq_{i}] = d/dt∑ [p_{i}δq_{i}]

Note: δ(dq_{i}/dt) = d(δq_{i})/dt

δL = 0 --> ∑ [p_{i}δq_{i}] = constant.

Let {q_{i}} and {p_{i}} be the generalized coordinates and momenta of a system and
let A, B, and C be arbitrary functions of the q's and p's.

The Poisson Bracket (PB) of A and C is defined as

{A, C} = ∑_{i} [(∂A/∂q_{i}) (∂C/∂p_{i}) - (∂A/∂p_{i})
(∂C/∂q_{i})].

Properties of the PB that can be easily verified are

{A, C} = -{C, A}, {kA, C} = k{A, C} for any constant k.

{(A+B), C} = {A, C} + {B, C}.

{(AB), C} = A{B, C} + B{A, C}

(a) Evaluate {q_{i}, q_{j}}, {p_{i}, p_{j}},
and {q_{i}, p_{j}} for arbitrary i, j.

(b) Evaluate {q_{i}^{n}, p_{i}}. (Hint: use
mathematical induction)

(c) Evaluate {F, p} for an arbitrary smooth function F of the generalized
coordinate q.

(Any smooth function can be arbitrarily well approximated by a polynomial.)

Solution:

- Concepts:

Poisson Bracket - Reasoning:

We are asked to evaluate various Poisson Brackets. - Details of the calculation:

(a) {q_{i}, q_{j}} = 0, {p_{i}, p_{j}} = 0, {q_{i}, p_{j}} = δ_{ij}.

(b) {q_{i}^{2}, p_{i}} = 2 q_{i}{q_{i}, p_{i}}= 2 q_{i }{q_{i}^{n}, p_{i}} = n q_{i}^{n-1}, for n = 2.

Assume {q_{i}^{n-1}, p_{i}} = (n-1) q_{i}^{n-2}.

Then {q_{i}^{n}, p_{i}} = q_{i}{q_{i}^{n-1}, p_{i}} + q_{i}^{n-1}{q_{i}, p_{i}} = (n-1) q_{i}^{n-1}+ q_{i}^{n-1}= n q_{i}^{n-1}.

{q_{i}^{n}, p_{i}} = n q_{i}^{n-1}therefore holds for all n.

(c) F(q) = ∑_{n}A_{n}q^{n}. {F, p} = ∑_{n}A_{n}{q^{n}, p} = ∑_{n}A_{n}n q_{i}^{n-1}= dF/dq.

{F, p} = dF/dq.

Let {q_{i}} and {p_{i}} be the generalized coordinates and momenta of a system and
let F, and G be arbitrary functions of the q's and p's.

The Poisson Bracket (PB) of F and G is defined as

{F, G} = ∑_{i} [(∂F/∂q_{i}) (∂G/∂p_{i}) - (∂F/∂p_{i})
(∂G/∂q_{i})].

(a) Show that dq_{j}/dt = {q_{j}, H}, dp_{j}/dt
= {p_{j}, H} is another way of writing Hamilton's equations of motion.

(b) Write dF/dt in terms of a PB.

(c) Consider the three Cartesian components of the angular momentum **L**.

We have {Li, Lj} = ∑_{k} ε_{ijk} L_{k}.

Assume the Hamiltonian of a system is given by H = ωL_{z}. Use the
Poisson Bracket formulation to work out the equations of motion for the vector
**L**.

Solution:

- Concepts:

Poisson Bracket formulation of classical mechanics - Reasoning:

We are asked to show how the PB formalism can be used to find the evolution of a physical quantity. - Details of the calculation:

(a) {q_{j}, H} = ∑_{i}[(∂q_{j}/∂q_{i}) (∂H/∂p_{i}) - (∂q_{j}/∂p_{i}) (∂H/∂q_{i})] = ∂H/∂p_{j }dq_{j}/dt = ∂H/∂p_{j }{p_{j}, H} = ∑_{i}[(∂q_{j}/∂q_{i}) (∂H/∂p_{i}) - (∂q_{j}/∂p_{i}) (∂H/∂q_{i})] = -∂H/∂q_{j }dp_{j}/dt = -∂H/∂q_{j }(b) dF/dt = ∑_{i}[(∂F/∂q_{i}) (dq_{i/}dt) + (∂F/∂p_{i}) (dp_{i/}dt)]

(dq_{i/}dt) = (∂H/∂p_{i}), (dp_{i/}dt) = -(∂H/∂q_{i})

dF/dt = ∑_{i}((∂F/∂q_{i}) (∂H/∂p_{i}) - (∂F/∂p_{i}) (∂H/∂q_{i})) = {F, H}

(c) dL_{i}/dt = {L_{i}, H} = ω{L_{i}, L_{z}}

dL_{z}/dt = 0.

dL_{x}/dt = -ωL_{y}.

dL_{y}/dt = ωL_{x}.

L_{x}= A cos(ωt + φ)

L_{y}= A sin(ωt + φ)

The z-component of**L**does not change. The component of**L**perpendicular to the z-axis in the xy-plane rotate ccw about the origin with constant angular velocity**ω**= ω**k**.

Assume every point in phase space is shifted by an amount δq_{i} = {q_{i},G},
δp_{i} = {p_{i},G}, where G is an arbitrary function of the
coordinates and conjugate momenta.

G is called the generator of the transformation δq_{i}, δp_{i}.

Show that the function of the coordinates and conjugate momenta F is changed by
an amount δF = {F,G} by the transformation.

Solution:

- Concepts:

Poisson Bracket formulation of classical mechanics - Reasoning:

We are asked to explore the connection between symmrtry and conservation laws. - Details of the calculation:

δF = (∂F/∂q_{i}) δq_{i}+ (∂F/∂p_{i}) δp_{i}(summation convention)

δF = (∂F/∂q_{i}) {q_{i},G}, + (∂F/∂p_{i}) {p_{i},G} = {F, G} (show by writing out all the terms in the PB)

When the transformation generated by G does not change the Lagrangian or Hamiltonian, then δH = {H, G} = 0.

Assume G is such a symmetry transformation. Find dG/dt.

dG/dt = (∂G/∂q_{i}) (∂q_{i}/∂t) + (∂G/∂p_{i}) (∂p_{i}/∂t) = (∂G/∂q_{i}) (∂H/∂p_{i}) - (∂G/∂p_{i}) (∂H/∂q_{i}) = {G,H}

dG/dt = 0. If G generates a symmetry transformation, the G is a constant of motion.

Consider the motion of a particle in a potential U(**r**) in a rotating
frame. Let **Ω** be the angular velocity of the rotating frame with
respect to an inertial frame and let **Ω** be constant.

(a) Express
the Lagrangian of the particle in terms of **r** and **v** of the particle
in the rotating frame.

(b) Obtain the equations of motion in the
rotating frame.

(c) Obtain the Hamiltonian of the particle in the
rotating frame.

Solution:

- Concepts:

Lagrangian and Haniltonian mechanics, rotating frames - Reasoning:

We are asked to write down the lagrangian and the Hamiltonian in the rotating frame and to obtain the equations of motion in the rotating frame. - Details of the calculation:

(a) L = T - U. In the inertial frame L = ½m**v**_{i}^{2}- U(**r**).

The relationship between the velocity**v**_{i}in the inertial frame and the velocity**v**in a frame rotating with constant angular velocity**Ω**is**v**_{i}=**v**+**Ω**×**r**.

In terms of**r**and**v**in the rotationg frame L = ½m(**v**+**Ω**×**r**)^{2}- U(**r**), or

L = ½m**v**^{2}+ ½m(**Ω**×**r**)^{2}+ m**v∙**(**Ω**×**r**) - U(**r**),

(b) ∂L/∂**v**= m**v**+ m(**Ω**×**r**)

(d/dt)∂L/∂**v**= md**v**/dt + m(**Ω**× v)

∂L/∂r = (∂/∂r)(½m(**Ω**×**r**)^{2}+ m**v∙**(**Ω**×**r**) - U(**r**))

d(**Ω**×**r**)^{2}= 2(**Ω**×**r**)**∙**(**Ω**× d**r**) = 2[(**Ω**×**r**) ×**Ω**]**∙**d**r**(∂/∂r)(**Ω**×**r**)^{2}= 2[(**Ω**×**r**) ×**Ω**] = -2[(**Ω**× (**Ω**×**r**)]

(∂/∂r)(**v∙**(**Ω**×**r**)) = (∂/∂r)[(**v**×**Ω**)**∙r**] = (**v**×**Ω**) = -(**Ω**×**v**)

The equation of motion is (d/dt)∂L/∂**v**= ∂L/∂r.

md**v**/dt + m(**Ω**× v) = -m[(**Ω**× (**Ω**×**r**)] - m(**Ω**×**v**) - ∂U(**r**)/∂r

md**v**/dt = - ∂U(**r**)/∂r - m[(**Ω**× (**Ω**×**r**)] - 2m(**Ω**×**v**)

is the equation of motion in the rotating frame.

-2m(**Ω**×**v**) = Coriolis force

-m**Ω**× (**Ω**×**r**) = centrifugal force

(c)**p**= ∂L/∂**v**= m**v**+ m(**Ω**×**r**),**v**= (**p**- m(**Ω**×**r**))/m

H =**p∙v**- L

H(**p**,**r**) =**p∙**(**p**- m(**Ω**×**r**))/m - ½(**p**- m(**Ω**×**r**))^{2}/m - ½m(**Ω**×**r**)^{2}- (**p**- m(**Ω**×**r**))**∙**(**Ω**×**r**) + U(r)

= p^{2}/m -**p∙**(**Ω**×**r**) - p^{2}/(2m) - ½m(**Ω**×**r**))^{2}+**p∙**(**Ω**×**r**)

- ½m(**Ω**×**r**)^{2}-**p∙**(**Ω**×**r**) + m(**Ω**×**r**))^{2}+ U(r)

= p^{2}/(2m) -**p∙**(**Ω**×**r**) + ½m(**Ω**×**r**))^{2}- ½m(**Ω**×**r**)^{2}+ U(r)

H(**p**,**r**) = (**p**- m(**Ω**×**r**))^{2}/(2m) - ½m(**Ω**×**r**)^{2}+ U(r)

is the Hamiltonian of the particle in the rotating frame.

The Lagrangian of a system of N degrees of freedom is

What is the Hamiltonian for a symmetric mass matrix M_{ij} = M_{ji}?

Solution:

- Concepts:

The Lagrangian and the Hamiltonian - Reasoning:

Given the Lagrangian, we are asked to find the Hamiltonian of a system. - Details of the calculation:

Therefore:

We then have:

In matrix notation: