Physics Problems & Solutions: Classical Mechanics

Two small equal masses m are connected by an ideal massless spring that has equilibrium length l₀ and force constant k. The system is free to move without friction in the plane of the page. If p₁and p₂represent the magnitude of momenta of the two masses, a Hamiltonian for this system is.

A. $\frac{1}{2}\bigg[\frac{p_1^2}{m}+\frac{p_2^2}{m}-2k(l-l_0)\bigg]$

B. $\frac{1}{2}\bigg[\frac{p_1^2}{m}+\frac{p_2^2}{m}-2k(l-l_0)^2\bigg]$

C. $\frac{1}{2}\bigg[\frac{p_1^2}{m}+\frac{p_2^2}{m}-k(l-l_0)\bigg]$

D. $\frac{1}{2}\bigg[\frac{p_1^2}{m}+\frac{p_2^2}{m}-k(l-l_0)^2\bigg]$

E. $\frac{1}{2}\bigg[\frac{p_1^2}{m}+\frac{p_2^2}{m}+k(l-l_0)^2\bigg]$

(GR0177 #92)

Solution:

Hamiltonian: H = T + U

T_i= p_i²/2m
T₁+ T₂= p₁²/2m + p₂²/2m

U = ¹/₂k(Δl)²= ¹/₂k(l − l₀)²

H = p₁²/2m + p₂²/2m + ¹/₂k(l − l₀)²
= ¹/₂[ p₁²/m + p₂²/m + k(l − l₀)²]

Answer: E

Physics Problems & Solutions

Classical Mechanics - Hamiltonian

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