निम्नलिखित दो खिलाड़ी शून्य योग खेल को प्रमुखता सिद्धांत द्वारा 2×2 खेल में समानीत कीजिए और इस प्रकार खेल को हल कीजिए।

बताइए निम्नलिखित कथनों में से कौन-से कथन सत्य हैं और कौन-से असत्य। अपन उत्तर की पुष्टि एक संक्षिप्त उपपत्ति या प्रत्युदाहरण द्वारा कीजिए।

a) परिमित संख्या में अवमुख समुच्चयों का सर्वनिष्ठ अवमुख नहीं होता है।

b) यदि 2×2 आव्यूह खेल  का मान 4 हो, तो p≥4.

c) यदि किसी 3×3 नियतन समस्या में लागत आव्यूह की प्रत्येक प्रविष्टि में 10 जोड़ा जाए, तो परिवर्तित लागत आव्यूह के लिए इष्टतम नियतन की कुल लागत में 10 की वृद्धि हो जाएगी।

d) किसी अधिकतमीकरण रैखिक प्रोग्रामन (LP) निदर्श में, जब सभी मान c_j, z_j, 20. हों, तो एकधा विधि सम्पन्न हो जाती है।

e) एक परिवहन समस्या में अपभ्रष्ट हल से बचने के लिए काल्पनिक (dummy) स्रोत या गंतव्य जोड़ा जाता है।

Consider a uniform square plate of length x ya and mass m. Obtain the moment of inertia along  z-axis

Using appropriate labelled diagram, obtain the kinetic energy of a free symmetrical top of mass M.

Use the Hamilton-Jacobi method to find Hamilton's principal function W for a particle in the three dimensional isotropic oscillator well with a potential  Hence obtain the corresponding momentum  and the corresponding action variables 

For a particle moving in central force field, the Hamiltonian of the system is given by

 Assuming the motion to be elliptical, obtain action variables J_r and J_o.

Consider a damped harmonic oscillator whose Lagrangian is given by:

(i) Write the equation of motion and hence obtain the corresponding Hamiltonian.

(ii) Using Hamilton-Jacobi equation associated with the Hamiltonian, solve the equations of motion.

State Liouville's theorem and write the mathematical expression. Show that if pdepends on q.p through the Hamiltonian 

Using symplectic condition for canonical transformation, show that the transformation is canonical

Obtain the generating function for the transformation.

What is a canonical transformation? Why do we use canonical transformation?

Give an example where a canonical transformation becomes useful.

(ii) The Hamiltonian of a harmonic oscillator if given by 

Solve the problem of the harmonic oscillator using canonical transformation with the generator 

Consider a Lagrangian L where q is cyclic. Show that the momentum conjugate to q is conserved under the translation q→q+a.

Consider a particle of mass m moving in a plane attracted to the origin due to the potential k/r:

(i) Choose an appropriate coordinate

(ii) Write down the Lagrangian, the momenta conjugate to your choice of coordinates, the Hamiltonian and the action for the system.

(ii) Show that the action is invariant under translation of time. What conser vation law does this yield?

What is Legendre transformation of a function? Express the Legendre transformation of a Lagrangian to obtain the Hamilton's function H.

(ii) The Hamiltonian of a simple harmonic oscillator is given as:

Show that the velocity vector is tangential to the curve defined by the Hamiltonian. Draw a labeled diagram of the velocity vector for the phase trajectory for energy E.

Find all values of k for which the vectors:

are linearly independent.

Solve the following LPP graphically:

Maximize:

                 z= 10x₁ + 10x₂

subject to the constraints:

4x₁ + 3x₂ ≤ 12

6x₁ +18x₂ ≤ 36

X₁,X₂ ≥ 0.

Solve the following game graphically:

Write the LPP formulation of the following assignment problem:

Solve the following assignment problem for profit maximization:

Write the LPP formulation of the following transportation problem:

Solve the following LP problem using simplex method:

Maximize z = 6x₁ + 4x₂

Subject to 2x₁ + 3x₂ ≤ 30

                 3x₁ + 2x₂ ≤ 24

                  x₁ + x₂ ≥ 3

                  x₁ , x₂ ≥ 0