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Using the Lagrangian
A Nexus mindmap
The Lagrangian is a scalar,
Newton's equations are vectorial
0
(Using "
generalised coordinates
")
0
Can easily take into account constraints
1
The form of the Euler-Lagrange
equations remains the same under
co-ordinate transformations
2
Why the Lagrangian
formulation is useful
0
Can eleminate some points
0
The trick to getting around
geometrical constraints is to
choose a new set of 'positions'
that
are
independent
1
If we have geometrical constraints
then not all the positions are
independent of each other
0
0
3
N
coordinates
0
0
3
N
velocities
1
(3N degrees of freedom)
0
6
N
variables
2
0
0
3
Unconstrained
system of
N
particles
2
0
0
0
1
1
0
2
0
2
0
0
3
0
0
1
4
Lagrangian
3
0
In practice they are chosen to be
displacements and angles naturaly
appearing in the problem
2
Distances over a sphere...
2
Co-ordinates
0
Angles
1
e.g.
0
In general these won't have
the dimensions of length
1
0
0
differentiating with time
gives generalised velocities
1
Generalised velocities
1
Need to invert
q(
x
)
0
1
0
To calculate the kinetic and potential energy, often it's
easier to determine particle positions
2
particle positions
and velocities
2
"degrees of freedom"
0
Though the generalised co-ordinates are not
unique there is a minimum number required to
determine the physical position of every particle
0
1
degrees of freedom
3
generalised
coordinates
1
Generalised
coordinates
1
0
0
1
2
0
0
0
1
1
[Morin,
Classical Mechanics
]
5
0
2
3
0
1
4
0
0
0
0
1
2
0
3
1
1
0
4
1
1
1
0
2
0
2
0
0
1
0
2
1
1
0
2
examples
2
Using the
Lagrangian