Physics Problems & Solutions: Classical Mechanics

Question 34-36

The potential energy of a body constrained to move on a straight line is kx⁴ where k is a constant. The position of the body is x, its speed v, its linear momentum p, and its mass m.

The body moves from x₁ at time t₁ to x₂ at time t₂. Which of the following quantities is an extremum for the x-t curve corresponding to this motion, if end points are fixed?

A. $\int_{t_1}^{t_2}\bigg(\frac{1}{2}mv^2-kx^4\bigg)dt$
B. $\int_{t_1}^{t_2}\frac{1}{2}mv^2dt$
C. $\int_{t_1}^{t_2}mxv$ $dt$
D. $\int_{x_1}^{x_2}\bigg(\frac{1}{2}mv^2+kx^4\bigg)dx$
E. $\int_{x_1}^{x_2}mv$ $dx$

(GR8677 #36)

Solution:

$\\$Lagrangian: $L=T-U \\$Hamiltonian: $ H=\int_{t_1}^{t_2} L dt \\U=kx^4 \\\\T=\frac{1}{2} mv^2 \\\rightarrow H=\int_{t_1}^{t_2}\bigg(\frac{1}{2} mv^2-kx^4\bigg)dt$

Answer: A

Physics Problems & Solutions

Classical Mechanics - Lagrangian and Hamiltonian

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