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4 Lagrange’s Method

Lagrange’s method is an energy method used to derive the equations of motion for a discrete system. For large systems with multiple bodies, it is often simpler to apply than Newton’s method because forces and moments acting between bodies do not need to be accounted for explicitly. The steps needed to derive equations of motion are as follows.

4.1 Compute the system kinetic energy.

The system kinetic energy is the sum of the kinetic energies of all the bodies in the system, or
(7) KE = ⎲⎳⎛⎝(1)/(2)mv⋅v + (1)/(2) ω⋅I⋅ω⎞⎠
Computing the system kinetic energy requires expressions for the translational and angular velocity, which is often the most challenging part of applying Lagrange’s method.

4.2 Compute the system potential energy.

The system potential energy, PE, is the sum of the potential energies of all bodies in the system. As there are many different types of potential energy, a single expression will not be written. The most common types in aerospace engineering problems are gravitational potential energy and elastic potential energy. Note that there is some flexibility in that the forces from potential energy can be accounted for in the generalized force expressions.

4.3 Compute the Lagrangian.

The Lagrangian is simply the difference between the system kinetic and potential energy, or
(8) ℒ = KE − PE
The Lagrangian should be written in terms of the generalized coordinates, qk, where k = 1, ..., N. These may be the actual coordinate directions, but often it is more convenient to use other variables instead.

4.4 Determine the generalized forces.

The generalized forces are the effects of nonconservative forces applied to the system and any conservative forces not included in the system potential energy. The generalized forces are only those applied to the system externally, which does not include any forces acting between bodies in the system. The generalized forces, Qk, are commonly determined using the concept of virtual work. They can also be determined from the applied forces and discrete body velocities
(9) Qk = n⎲⎳i = 1 Fi⋅(∂vi)/(∂qk̇),   k = 1, ..., N
where n is the number of discrete bodies in the system, Fi is the sum of external forces acting on body i, and vi is the velocity of particle i.

4.5 Write Lagrange’s equation for each generalized coordinate.

Lagrange’s equation is
(10) (d)/(dt)⎛⎝(∂ℒ)/(∂qk̇)⎞⎠ − (∂ℒ)/(∂qk) = Qk,   k = 1, ..., N
Implicit in this step is the evaluation of the partial derivatives (∂ℒ)/(∂qk̇) and (∂ℒ)/(∂qk) and the derivative (d)/(dt)⎛⎝(∂ℒ)/(∂qk̇)⎞⎠. At this point the system of equations governing the motion are written for each generalized coordinate. Lagrange’s method for deriving equations of motion gives an expression equivalent to Newton’s method, provided both are applied correctly.
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