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Lagrangian and Hamiltonian Mechanics (Read 2251 times)
Gerrit-Jan Linker
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Lagrangian and Hamiltonian Mechanics
14.09.11 at 08:28:20
 
Newtonian, Lagrangian and Hamiltonian Mechanics
 

  • Newtonian mechanics
     
    Procedure
    • Write down Newton's second law. In one dimension this is:
       
      F(x)=ma=mx''
       
      where:
      F = Force
      m = mass
      a = acceleration
      x'' = d2x/dt2
    • Substitute F(x) (or more generally F(x,y,z) or F(r)) for the specific forces in the system.
    • Solve the resulting differential equation for x(t)

  • Lagrangian mechanics
    In Lagrangian mechanics one second order differential equation needs to be solved.
     
    Procedure
    • Define the Lagrangian as L = T - U, with T = kinetic energy and U = potential energy
      Kinetic energy is usually a function of v and the potential usually depends on x, we have:
      L(v,x) = T(v) - U(x)
    • Substitute T and U for an expression of the kinetic and potential energy for the system
    • Solve the Lagrangian equation. In one dimension this is:
      d/dt ( ∂L/∂v ) - ∂L/∂x = 0

     
  • Hamiltonian mechanics
    In Hamiltonian mechanics there are two coupled first order differential equations to be solved. One advantage of Hamiltonian mechanics is that it is similar to quantum mechanics.
     
    Procedure
    • The Hamiltonian is defined as the sum of kinetic and potential energy: H = K + U where K should be expressed in terms of momentum p: H(p,x) = K(p) + U(x)
    • Substitute K and U for an expression of the kinetic and potential energy for the system
    • Solve the Hamiltonian equations. In one dimension this is:
      dx/dt = ∂H/∂p
      dp/dt = -∂H/∂x


 
Source:
http://www.pgccphy.net/ref/advmech.pdf
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« Last Edit: 14.09.11 at 08:43:02 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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