拉格朗日与哈密顿力学代写 物理学代写 作业代写

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拉格朗日与哈密顿力学代写 Answer two questions. If you answer more than two questions, credit will only be given for the best two answers.

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Answer two questions. If you answer more than two questions, credit will only be given for the best two answers.

1. 拉格朗日与哈密顿力学代写

Two rods joined together at point O at a fixed angle α = 60° rotate around an axis passing through O that is perpendicular to the plane defined by the rods with constant angular velocity Ω. Two beads of mass m are constrained to move along the rods and are connected by a spring of unit natural length and spring constant κ.

(a) Find a Lagrangian for the system, where the generalised coordinates u and v measure distances from each bead to the point 0 . In your answer you should clearly identify the physical origin of each term.  

(b) Describe a modification to the system which would reduce it to one degree of freedom.

For the ongoing calculation, the mass of the beads is set to be that of unit mass, the spring constant to unit spring constant.

(c) Define the generalised momenta p_u and p_v and thus deduce the Hamiltonian H. Explain why in the present case H is not equal to the total energy T +V .

(e) The above potential V^eff(u, v) can be expanded around the equilibrium position {u₀, v₀}. By undertaking this expansion derive the normal modes and frequencies of small oscillations about equilibrium. Use your solution to explain why this equilibrium is stable.

2. 拉格朗日与哈密顿力学代写

The laws of geometrical optics can be derived from Fermat’s principle: the actual path between two points taken by a beam of light is the one which is traversed in the least time.

The speed of light in a material is inversely proportional to the refractive index, c = c₀/n(r), so that the length ds of the path travelled in an interval of time dt satisfies n(r)ds = c₀dt (c₀ is the light speed in vacuum).

Consider light propagation in this material in a plane perpendicular to z.

(a) Sketch how the speed of light varies as a function of r(x, y).

(b) Argue (with both mathemtical and written justification) that in this case Fermat’s principle for finding the optimal path between points r₁(θ₁) and r₂(θ₂)reduces to a minimization of the following functional:

(c) It is known that finding the optimal path for any function f(r, r’ ; θ)is equivalent to solving the Euler – Lagrange equation

Argue by analogy with Lagrangian mechanics (stating explicitly which parameters are in correspondence) or show by a direct calculation that the Euler – Lagrange equation has the first integral (a quantity conserved along the optimal path), a, given by

(e) It is known that the first integral takes the value r(0) = a along a certain light trajectory. Find r as an explicit function of θ for this trajectory.

3. 拉格朗日与哈密顿力学代写

(a) Describe a test for a transformation of a Hamiltonian system to be canonical. Why are we interested in canonical transformations? Your audience is a student that has studied up to week 7 (i.e. the Lagrangian part) of this module.

Approximately 100 words (not including equations, or any references from outside of the lecture notes).  

(b) Derive Hamilton’s equations for this system.

(c) Show that I = q2+p2 is a conserved quantity by differentiating with respect to time. How can we deduce that I is a conserved quantity by inspection of the Hamiltonian?

(d) Use the results of parts (a) and (b) to deduce the most general motion of the system, writing q(t) and p(t) in terms of two constants.

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