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Unformatted text preview: p i = ∂ L /∂ ˙ x i . Show that if L = T − V , and T = ∑ i ∑ j T ij ˙ x i ˙ x j , where the T ij ’s are constants, then H = T + V . You may assume with no loss of generality that T ij = T ji . 1 6. Shankar, problem 2.7.2 (i). 7. OPTIONAL; NOT TO BE TURNED IN: Suppose we have a particle in one dimension, whose Lagrangian is L = − mc 2 r 1 − ˙ x 2 /c 2 − V ( x ) , (2) where m and c are positive constants. (Actually, m is the rest mass and c is the speed of light.) Assume  ˙ x  < c . (a). Find the Lagrange equation of motion for this particle. (b). Find the canonical momentum p = ∂ L /∂ ˙ x . (c). Find the corresponding Hamiltonian H . (d). What is H in the limit  ˙ x  ≪ c ? 2...
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 Fall '07
 STROUD
 Physics, Linear Algebra, mechanics, Fundamental physics concepts, Lagrangian mechanics, Shankar

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