Classical Mechanics
Classical Mechanics
5th Edition
ISBN: 9781891389221
Author: John R. Taylor
Publisher: University Science Books
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Chapter 13, Problem 13.1P
To determine

The Hamilton’s equation for a free particle.

Expert Solution & Answer
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Answer to Problem 13.1P

The Hamilton’s equation for a free particle is x=v0t+x0 which represents a straight line.

Explanation of Solution

Write the expression for kinetic energy for a free particle moving along x direction.

  T=12mx˙2        (I)

Here, m is the mass..

Write the expression for potential energy for a free particle moving along x direction.

  U=0        (II)

Write the lagrangean by substituting expression (I) and (II),

  L=TUL=12mx˙2        (III)

Write the generalized momentum by substituting expression (III),

  p=Lx˙p=12m(2x˙)=mx˙x˙=pm        (IV)

Substitute expression (IV) in expression (I), re-write the expression for kinetic energy

  T=12m(pm)2        (V)

Write the Hamiltonian, by substituting expression (V) and (II),

  H=p22m+0        (VI)

Generalize the momentum

  p˙=Hxx˙=2p2mx˙=pmdpdt=0p=p0        (VII)

Conclusion:

Substitute expression (VII) in x˙=pm

  dxdt=p0mdx=p0mdt        (IX)

Integrating the expression (IX) on both sides

  dx=p0mdtx=p0mt+C        (X)

.

Consider the condition in with ar t=0,x=x0 in initial displacement.

  x=p0mt+x0x=v0t+x0        (XI)

This equation (XI) represents a straight line.

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