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 time \(t \in \mathbb{R}\) is a point in the \(N\) 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 time \(t \in \mathbb{R}\) is the velocity at a point along its trajectory through configuration space.
The position of each point mass may be given by
with \(\mathbf{q} = \{q_i\}_{i=1}^f\) generalized coordinates.
Therefore the velocity of each point mass is given by
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 \(t \in \mathbb{R}\).
Proof:
Will be added later.