Mechanics

by Nikolai V. Shokhirev

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Introduction

All the presentation below are equivalent. The choice depends on the convenience and the simplicity of the equations of motion.

Lagrangian mechanics

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 pi corresponding to a generalized coordinate qi is

     (5)

The generalized forces

     (6)

The equations of motion in Lagrangian mechanics are Lagrange's equations:
 

     (7a)

or

     (7b)

Hamiltonian mechanics

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)

Energy conservation

If the potential U does not directly depend on time

     (10)

This conservation law is the advantage of the Hamiltonian presentation.

Cartesian coordinates

In the Cartesian coordinates both (7) and (9) reduce to the well-known equation of motion

     (11)

Let us consider the example.

Spring pendulum

This is a spring-mass system in a uniform gravitational field along the z-axis

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.

References

  1. Lagrangian mechanics. http://en.wikipedia.org/wiki/Lagrangian_mechanics
  2. Hamiltonian mechanics. http://en.wikipedia.org/wiki/Hamiltonian_mechanics
     

 

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