Sometimes it is not all that easy to find the equations of motion and there is an alternative approach known as lagrangian mechanics which enables us to find the equations of motion when the newtonian method is proving difficult. In lagrangian mechanics we start, as usual, by drawing a large, clear diagram of the system, using a ruler and a compass. But, rather than drawing the forces and accelerations with red and green arrows, we draw the velocity vectors (including angular velocities) with blue arrows, and, from these we write down the kinetic energy of the system. If the forces are conservative forces (gravity, springs and stretched strings), we write down also the potential energy. That done, the next step is to write down the lagrangian equations of motion for each coordinate. These equations involve the kinetic and potential energies, and are a little bit more involved than \(F=ma\), though they do arrive at the same results.
- 13.1: Introduction to Lagrangian Mechanics
- I shall derive the lagrangian equations of motion, and while I am doing so, you will think that the going is very heavy, and you will be discouraged. At the end of the derivation you will see that the lagrangian equations of motion are indeed rather more involved than F=ma , and you will begin to despair – but do not do so! In a very short time after that you will be able to solve difficult problems in mechanics that you would not be able to start using the familiar newtonian methods.
- 13.2: Generalized Coordinates and Generalized Forces
- A state of a molecule may described by a number of parameters, e.g., bond lengths and the angles). These bonds lengths and bond angles constitute a set of coordinates which describe the molecule. We are not going to think about any particular sort of coordinate system or set of coordinates. Rather, we are going to think about generalized coordinates, which may be lengths or angles or various combinations of them.
- 13.3: Holonomic Constraints
- The state of the system at any time can be represented by a single point in 3N -dimensional space. However, in many systems, the particles may not be free to wander anywhere at will; they may be subject to various constraints. A constraint that can be described by an equation relating the coordinates (and perhaps also the time) is called a holonomic constraint, and the equation that describes the constraint is a holonomic equation.
- 13.4: The Lagrangian Equations of Motion
- So, we have now derived Lagrange’s equation of motion. It was a hard struggle, and in the end we obtained three versions of an equation which at present look quite useless. But from this point, things become easier and we rapidly see how to use the equations and find that they are indeed very useful.
- 13.5: Acceleration Components
- The radial and transverse components of velocity and acceleration in two-dimensional coordinates are derived using Lagrange’s equation of motion.
- 13.6: Slithering Soap in Conical Basin
- We imagine a slippery (no friction) bar of soap slithering around in a conical basin.
- 13.7: Slithering Soap in Hemispherical Basin
- Suppose that the basin is of radius a and the soap is subject to the holonomic constraint r=a - i.e. that it remains in contact with the basin at all times. Note also that this is just the same constraint of a pendulum free to swing in three-dimensional space except that it is subject to the holonomic constraint that the string be taut at all times. Thus any conclusions that we reach about our soap will also be valid for a pendulum.
- 13.8: More Lagrangian Mechanics Examples
- More examples of using Lagrangian Mechanics to solve problems.
- 13.9: Hamilton's Variational Principle
- Hamilton’s variational principle in dynamics is slightly reminiscent of the principle of virtual work in statics. When using the principle of virtual work in statics we imagine starting from an equilibrium position, and then increasing one of the coordinates infinitesimally. We calculate the virtual work done and set it to zero. I am slightly reminded of this when discussing Hamilton’s principle in dynamics
Thumbnail: Simple pendulum. Since the rod is rigid, the position of the bob is constrained according to the equation f(x,y)=0, the constraint force is C, and the one degree of freedom can be described by one generalized coordinate (here the angle theta). Image used with permission (Public Domain;).