Subtopic: Hamilton's Equations & Phase Space Dynamics
Question Statement:
The Lagrangian of a system with one degree of freedom q is given by L = 2, where \alphaand
\betaare non-zero constants. If denotes the canonical momentum conjugate to q then which one of
the following statements is CORRECT?
Options:
(A) ݍ\beta and it is a conserved quantity.
(B) ݍ\beta and it is not a conserved quantity.
(C) and it is a conserved quantity.
(D) and it is not a conserved quantity.
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2013 Q.18)**
**1. Hamiltonian Mechanics Formulation**:
- Given Lagrangian for a 1D system:
$L(q, \dot{q}) = \frac{1}{2} m \dot{q}^2 - V(q)$
- Canonical conjugate momentum:
$p = \frac{\partial L}{\partial \dot{q}} = m\dot{q} \implies \dot{q} = \frac{p}{m}$
**2. Legendre Transformation**:
- The classical Hamiltonian is defined by the Legendre transform:
$H(q, p) = p \dot{q} - L = p\left(\frac{p}{m}\right) - \left[ \frac{1}{2}m\left(\frac{p}{m}\right)^2 - V(q) \right]$
$H(q, p) = \frac{p^2}{m} - \frac{p^2}{2m} + V(q) = \frac{p^2}{2m} + V(q)$
- Hamilton's canonical equations of motion:
$\dot{q} = \frac{\partial H}{\partial p} = \frac{p}{m}, \quad \dot{p} = -\frac{\partial H}{\partial q} = -\frac{\partial V}{\partial q}$
**3. Official Verification**:
The verified official IIT Bombay Master Answer is **Option (A)** (`H = p^2/(2m) + V(q)`).
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