A block of mass is attached to a spring of stiffness and slides on a frictionless…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Lagrangian mechanics

Action and Symmetry, Ch The action, the first variation and the boundary term

Problem

A block of mass

m = 1.0\,\mathrm{kg}

is attached to a spring of stiffness

k = 4.0\,\mathrm{N\,m^{-1}}

and slides on a frictionless horizontal table, so

L(q,\dot q) = \tfrac12 m\dot q^2 - \tfrac12 k q^2

with

q

the displacement from the spring's natural length. At

t_1 = 0

the block is at

q = 0

with velocity

\dot q(0) = v_0 = 2.0\,\mathrm{m\,s^{-1}}

. The final time

t_2 > 0

is fixed but the value of

q(t_2)

is left free. ((a)) Derive the natural boundary condition that stationarity of

S = \int_0^{t_2} L\,\mathrm{d}t

imposes at the free end, and use it to find the smallest

t_2 > 0

for which the given initial data are consistent with that condition, together with

q(t_2)

at that time. ((b)) Show that adding

f(q) = \alpha q^2

(for a constant

\alpha

) as

L' = L + \mathrm{d}f/\mathrm{d}t

leaves the equation of motion unchanged. ((c)) Since

L

has no explicit

t

, write down the Beltrami (Jacobi) first integral for this

L

and check numerically that it takes the same value at

t=0

and at the

t_2

found in part (a).