3.2 KiB
Lagrangian formalism of mechanics
The Lagrangian formalism of mechanics is based on the axioms, postulates and principles posed in the Newtonian formalism.
Configuration of a system
Considering a system of n \in \mathbb{R}
point masses m_i \in \mathbb{R}
with positions \mathbf{x}_i \in \mathbb{R}^m
in dimension m \in \mathbb{N}
, for i \in \mathbb{N}[i \leq n]
.
Definition 1: the set of positions
\{\mathbf{x}_i\}_{i=1}^n
is defined as the configuration of the system.
Obtaining a n m
dimensional configuration space of the system.
Definition 2: let
N = nm
, the set of time dependent coordinates\{q_i: t \mapsto q_i(t)\}_{i=1}^N
at a timet \in \mathbb{R}
is a point in theN
dimensional configuration space of the system.
Definition 3: let the generalized coordinates be a minimal set of coordinates which are sufficient to specify the configuration of a system completely and uniquely.
The minimum required number of generalized coordinates is called the number of degrees of freedom of the system.
Classification of constraints
Definition 4: geometric constraints define the range of the positions
\{\mathbf{x}_i\}_{i=1}^n
.
Definition 5: holonomic constraints are defined as constraints that can be formulated as an equation of generalized coordinates and time.
Let g: (q_1, \dots, q_N, t) \mapsto g(q_1, \dots, q_N, t) = 0
is an example of a holonomic constraint.
Definition 6: a constraint that depends on velocities is defined as a kinematic constraint.
If the kinematic constrain is integrable and can be formulated as a holonomic constraint it is referred to as a integrable kinematic constraint.
Definition 7: a constraint that explicitly depends on time is defined as a rheonomic constraint. Otherwise the constraint is defined as a sklerenomic constraint.
If a system of n
point masses is subject to k
indepent holonomic constraints, then these k
equations can be used to eliminate k
of the N
coordinates. Therefore there remain f \overset{\mathrm{def}}= N - k
"independent" generalized coordinates.
Generalizations
Definition 8: the set of generalized velocities
\{q_i'\}_{i=1}^N
at a timet \in \mathbb{R}
is the velocity at a point along its trajectory through configuration space.
The position of each point mass may be given by
\mathbf{x}_i: \mathbf{q} \mapsto \mathbf{x}_i(\mathbf{q}),
with \mathbf{q} = \{q_i\}_{i=1}^f
generalized coordinates.
Therefore the velocity of each point mass is given by
\mathbf{x}i'(\mathbf{q}) = \sum{r=1}^f \partial_r \mathbf{x}_i(\mathbf{q}) q_r',
for all t \in \mathbb{R}
(inexplicitly).
Theorem 1: the total kinetic energy
T: (\mathbf{q}, \mathbf{q}') \mapsto T(\mathbf{q}, \mathbf{q}')
of the system is given by
T(\mathbf{q}, \mathbf{q}') = \sum_{r,s=1}^f a_{rs}(\mathbf{q}) q_r' q_s',
with
a_{rs}(\mathbf{q}) = \sum_{i=1}^n \frac{1}{2} m_i \Big\langle \partial_r \mathbf{x}_i(\mathbf{q}), \partial_s \mathbf{x}_i(\mathbf{q}) \Big\rangle,
for all
t \in \mathbb{R}
.
??? note "Proof:"
Will be added later.