notes/docs/physics/classical-mechanics/lagrangian-mechanics/lagrangian-formalism.md

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

\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.