Problem 2 The relativistic Lagrangian for a particle of rest mass m moving along the x-axis in a potential V(x) is given by L= -mc² V(x) c2 (a) Derive the Euler-Lagrange equation of motion. (b) Show that it reduces to Newton's equation in the limit || <<< c. (c) Compute the Hamiltonian H of the system. Eliminate & from the Hamiltonian by using the equation aL p = Ox and write H = H(p, x) as a function of x and p only.
Problem 2 The relativistic Lagrangian for a particle of rest mass m moving along the x-axis in a potential V(x) is given by L= -mc² V(x) c2 (a) Derive the Euler-Lagrange equation of motion. (b) Show that it reduces to Newton's equation in the limit || <<< c. (c) Compute the Hamiltonian H of the system. Eliminate & from the Hamiltonian by using the equation aL p = Ox and write H = H(p, x) as a function of x and p only.
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Transcribed Image Text:Problem 2
The relativistic Lagrangian for a particle of rest mass m moving along the x-axis in a potential V(x) is given
by
2
L = -mc² 1
V(x)
c2
(a) Derive the Euler-Lagrange equation of motion.
(b) Show that it reduces to Newton's equation in the limit |*| << c.
(c) Compute the Hamiltonian H of the system. Eliminate ȧ from the Hamiltonian by using the equation
ƏL
p =
ax
and write H = H(p, x) as a function of x and p only.
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