Q8
(a) Find the characteristics of the partial differential equation p² + q² = 2; p ≡ ∂z/∂x, q ≡ ∂z/∂y and determine the integral surface which passes through x = 0, z = y. (15 marks) (b) Find the constant p and error term for the quadrature formula ∫_{x_0}^{x_1} f(x)dx = (h/2)(f_0 + f_1) + ph²(f'_0 - f'_1) where x_0 + h = x_1, f_0 = f(x_0), f_1 = f(x_1) and prime (') represents derivative with respect to x. Hence deduce the composite rule for integrating ∫_a^b f(x)dx, a = x_0 < x_1 < ... < x_N = b (15 marks) (c) (i) A particle of mass m moves in a force field of potential V(r) = -k cosθ/r², k is constant Find the Hamiltonian and the Hamilton's equations in spherical polar coordinates (r, θ, φ). (10 marks) (ii) Consider the Lagrangian L = mẋẏ - mω₀²xy where m and ω₀ are constants. Find the Hamiltonian and Hamilton's equations of motion. Identify the system. (10 marks)
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(a) आंशिक अवकल समीकरण p² + q² = 2; p ≡ ∂z/∂x, q ≡ ∂z/∂y के अभिलक्षण (कैरेक्टरिस्टिक्स) ज्ञात कीजिए और x = 0, z = y से होकर जाने वाला समाकल पृष्ठ (सरफेस) प्राप्त कीजिए। (15 अंक) (b) क्षेत्रकलन-सूत्र ∫_{x_0}^{x_1} f(x)dx = (h/2)(f_0 + f_1) + ph²(f'_0 - f'_1) जहाँ x_0 + h = x_1, f_0 = f(x_0), f_1 = f(x_1) है और प्राइम ('), x के सापेक्ष अवकलज को निर्दिष्ट करता है, के लिए अचर p और त्रुटि-पद ज्ञात कीजिए। अतः समाकलन ∫_a^b f(x)dx, a = x_0 < x_1 < ... < x_N = b का मान ज्ञात करने के लिए संयुक्त नियम का निगमन कीजिए। (15 अंक) (c) (i) विभव V(r) = -k cosθ/r², k अचर है के एक बल क्षेत्र में द्रव्यमान m का एक कण गतिमान है। गोलीय ध्रुवीय निर्देशांकों (r, θ, φ) में हैमिल्टोनियन और हैमिल्टन का समीकरण ज्ञात कीजिए। (10 अंक) (ii) लैग्रांजी L = mẋẏ - mω₀²xy, जहाँ m और ω₀ अचर हैं, का विचार कीजिए। हैमिल्टोनियन और गति का हैमिल्टन समीकरण ज्ञात कीजिए। तंत्र (सिस्टम) की पहचान बताइए। (10 अंक)
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How this answer will be evaluated
Approach
Solve this multi-part problem by allocating approximately 30% time to part (a) on PDE characteristics, 30% to part (b) on numerical integration, 20% to part (c)(i) on Hamiltonian mechanics in spherical coordinates, and 20% to part (c)(ii) on the coupled oscillator system. Begin each part with clear statement of the governing equations, show systematic derivation with intermediate steps, and conclude with boxed final answers for each sub-part.
Key points expected
- For (a): Correctly identify Charpit's equations, solve dx/2p = dy/2q = dz/2(p²+q²) = dp/0 = dq/0, obtain p = a, q = √(2-a²), and find the complete integral z = ax + √(2-a²)y + b; apply initial condition x=0, z=y to determine the integral surface
- For (b): Use Taylor expansion of f(x) about x₀ and x₁ to match coefficients, determine p = 1/12, derive error term as -(h⁵/720)f⁽⁴⁾(ξ), and construct composite rule by summing over N subintervals with endpoint derivative corrections
- For (c)(i): Express kinetic energy T = ½m(ṙ² + r²θ̇² + r²sin²θ φ̇²), construct H = T + V with given potential, derive Hamilton's equations: ṙ = ∂H/∂pᵣ, θ̇ = ∂H/∂p_θ, etc., showing all six canonical equations
- For (c)(ii): Identify non-standard Lagrangian with coupled velocities, perform Legendre transform with pₓ = mẏ, pᵧ = mẺx, obtain H = pₓpᵧ/m + mω₀²xy, derive Hamilton's equations and identify as 2D isotropic oscillator with rotated coordinates
- Demonstrate dimensional consistency throughout: [p] = [q] = L⁰ for (a), [p] = T for (b) constant, [H] = ML²T⁻² for both mechanical parts
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly writes Charpit's auxiliary equations for (a), Taylor expansions with remainder for (b), spherical coordinate metric and canonical momenta definitions for (c)(i), and Legendre transform setup for (c)(ii); all initial conditions and constraints properly stated | Mostly correct setups with minor errors in Charpit equation form, Taylor expansion order, or momentum definitions; some missing initial conditions | Incorrect fundamental setup: wrong auxiliary equations, incorrect Taylor expansion approach, confused coordinate systems, or failure to identify canonical variables |
| Method choice | 20% | 10 | Selects optimal methods: Charpit's method for nonlinear PDE, undetermined coefficients with Taylor matching for quadrature, standard Hamiltonian formulation for spherical coordinates, and clever handling of coupled-velocity Lagrangian via matrix inversion or direct substitution | Acceptable methods chosen but with suboptimal approaches; e.g., using Lagrange-Charpit without simplification, or brute-force Taylor expansion without coefficient matching strategy | Inappropriate methods selected: attempting separation of variables for (a), using interpolation instead of Taylor expansion for (b), or failing to recognize need for Legendre transform in (c) |
| Computation accuracy | 20% | 10 | Flawless calculations: correct integration of Charpit equations yielding p=a, q=constant; precise determination p=1/12 with error term -(h⁵/720)f⁽⁴⁾(ξ); accurate spherical Hamiltonian with all six correct equations of motion; correct coupled oscillator identification with proper canonical transformation | Minor computational slips: sign errors in characteristics, incorrect coefficient in error term (e.g., 1/6 instead of 1/12), one or two incorrect Hamilton equations, or algebra errors in Legendre transform | Major computational failures: wrong characteristic curves, incorrect error term order or coefficient, fundamentally wrong Hamiltonian structure, or inability to solve for momenta in (c)(ii) |
| Step justification | 20% | 10 | Every nontrivial step justified: explains why dp=0, dq=0 from Charpit equations; shows Taylor expansion to sufficient order for coefficient matching; justifies spherical coordinate conjugate momenta; explains why standard p=mẋ fails for (c)(ii) and how invertibility is maintained | Most steps shown but with gaps in justification; assumes results without derivation (e.g., stating error term without derivation, quoting Hamilton equations without showing partial derivatives) | Missing critical justifications: no explanation of characteristic method, unjustified Taylor truncation, assertion of Hamiltonian without derivation of momenta, or failure to explain the unusual canonical structure in (c)(ii) |
| Final answer & units | 20% | 10 | All four sub-parts with complete, boxed final answers: explicit integral surface z = y + √(2-a²)x or equivalent; p = 1/12 with error term and composite rule formula; six Hamilton equations for (c)(i); H = pₓpᵧ/m + mω₀²xy with four equations and identification as 'rotated harmonic oscillator' or '2D isotropic oscillator in skewed coordinates'; dimensional analysis verified | Most answers present but incomplete: missing composite rule, one or two Hamilton equations omitted, or system identification vague ('oscillator' without specification) | Missing or wrong final answers: no integral surface determination, incorrect constant p, incomplete Hamilton equations, or failure to identify the physical system in (c)(ii) |
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