1 Introduction to Lagrange Equations of Motion for NonConservative Forces
The Lagrangian description of a mechanical system is different. For Lagrangian mechanics it is assumed that the position of the system at 2 instances of time (t1 and t2) are known (or knowable) and fixed. The system must move/behave between times t1 and t2 such that the system has the least "action". The "action" being defined by
- is called the Lagrange function or Lagrangian
- is a generalized coordinate which can represent the x, y, z of Cartesian coordinates or the θ, R of Polar coordinates, etc.
2 Lagrange Equation of Motion for Conversative Forces
The Lagrange Equation of Motion for Conservative Forces is below
This results is based on D'Alembert's equation which starts by calculating differential work work done on a particle by calculating the change kinetic energy:
This is work done by a force for a displacement . In terms of generalized coordinates we have
A generalized force is of such nature that the product is the work done by driving forces when alone is changed by . We mention that a generalized force doe snot have to be a force in the usual sense. For example, if is an angle, the must be a torque in order that be work.
3 Derivation of Lagrange Equation of Motion for NonConversative Forces
The Lagrange equation above was derived for conservative forces only. A more general form would need to include nonconservative forces, such as friction, and that is what is derived in this section.
The Lagrange equation of motion for the q1 coordinate if some of the forces are nonconservative (such as friction) is The final result of this derivation is
|.||Lagrange Equation of Motion for All Forces|
- is the Lagrangian function,
- is the generalized coordinate, and
- are the conservative forces.
The kinetic energy of a particle is its ability to do work by virute of its motion. All forces, both conservative and nonconservative, acting on the particle contribute to the particle's motion. If we add a single nonconservative force, friction, to the conservative forces acting on the particle then
- is the total work done by all conservative forces,
- is the work done by friction, and
- is the change in kinetic (mechanical) energy.
As a result of the nonconservative force, friction, the total kinetic energy is not constant. The change is equal to the amount of work done by the nonconservative force. The mechanical (kinetic) energy of the particle can be described as follows:
- is the final mechanical energy of the particle and
- is the particle initial mechanical energy.
Since friction always removes kinetic energy (it is always negative) the final energy of the particle () is less than the intial energy (). The difference between the final and initial energies captured as internal energy.
- is the internal energy or "lost" mechanical energy.
Further generalizing we get an equation stating the principle of conservation of energy. In other words the total energy does not change, just the form of the energy from kinetic to other such as internal or heat.
Earlier we described a general force
If all the forces are conservative then
- is potential energy and
- is the change in potential energy.
Then we have
Therefore, the Lagrange equations of motion for a single particle for conservative forces can be represented as
Rearranging the equation so that all non-zero components are on the left side of the equation we get
Define the Lagrangian function as
The partial derivative with respect to is
Since potential energy () is not a function of velocity () then
Substituting Eqn. (1) and (2) into (3), the Lagrange equations of motion for conservative forces becomes
When there are nonconservative forces present the Lagrange equations of motion become
- are the nonconservative forces acting on the particle.
- Zak, pp. 15-17