by Nikolai V. Shokhirev
Up: Physics
 Mechanics
 Hardsphere dynamics
 MD Simulation
 Kinetics

All the presentation below are equivalent. The choice depends on the convenience and the simplicity of the equations of motion.
The definition of the Lagrangian is
(1) 
Here K is the kinetic energy and U is the potential energy. In the Cartesian coordinates
(2) 
The dot denotes the time derivative.
Using the following transformation
(3) 
the Lagrangian can be expressed in terms of the generalized coordinates q and their derivatives:
(4) 
The generalized momentum p_{i} corresponding to a generalized coordinate q_{i} is
(5) 
The generalized forces
(6) 
The equations of motion in Lagrangian mechanics are Lagrange's equations:
(7a) 
or
(7b) 
The Hamiltonian is defined as
(8) 
The Hamiltonian is the otal energy of the system as a function of the
generalized coordinates and momentums (5). The Hamilton equations of
motion are written as follows:
(9) 
If the potential U does not directly depend on time
(10) 
This conservation law is the advantage of the Hamiltonian presentation.
In the Cartesian coordinates both (7) and (9) reduce to the wellknown equation of motion
(11) 
Let us consider the example.
This is a springmass system in a uniform gravitational field along the zaxis
1. Cartesian coordinates
(12) 
The kinetic and potential energies are
(13) 
here g is the acceleration constant, k is the spring constant. The momentums (5) are simple
(14) 
However, the forces are complex and inconvenient for calculations
(15) 
2. Polar coordinates
(16) 
The coordinates and velocities are defined as
(17a)  
(17b) 
Now the kinetic and potential energies are
(18) 
The forces are much simpler now
(19) 
The momentums are now more complex
(20) 
3. Discussion
Despite of the complexity of Eq. (20) it is more suitable for the analysis of limiting cases. In particular, for a stiff spring,
(21) 
Only the equation for the angle is necessary, which can be obtained from the second components of (19, 20)
(22) 
The analysis of small angles is also simpler in this coordinates.
Up: Physics
 Mechanics
 Hardsphere dynamics
 MD Simulation
 Kinetics

Please email me at nikolai@shokhirev.com 
ŠNikolai V. Shokhirev, 20012009