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# The equations of Lagrange
## Principle of virtual work
> *Definition 1*: a virtual displacement is a displacement at a fixed moment in time that is consistent with the constraints at that moment.
The following principle addresses the problem that the constraint forces are generally unknown.
> *Principle 1*: let $\mathbf{\delta x}_i \in \mathbb{R}^m$ be a virtual displacement and let $\mathbf{F}_i: \mathbf{q} \mapsto \mathbf{F}_i(\mathbf{q})$ be the total force excluding the constraint forces. Then
>
> $$
> \sum_{i=1}^n \Big\langle \mathbf{F}_i(\mathbf{q}) - m_i \mathbf{x}_i''(\mathbf{q}), \mathbf{\delta x}_i \Big\rangle = 0,
> $$
>
> is true for sklerenomic constraints and all $t \in \mathbb{R}$.
Which implies that the constraint forces do not do any (net) virtual work.
## The equations of Lagrange
> *Theorem 1*: let $T: (\mathbf{q}, \mathbf{q}') \mapsto T(\mathbf{q}, \mathbf{q'})$ be the kinetic energy of the system. For holonomic constraints we have that
>
> $$
> d_t \Big(\partial_{q_j'} T(\mathbf{q},\mathbf{q}') \Big) - \partial_{q_j} T(\mathbf{q},\mathbf{q}') = Q_j(\mathbf{q}),
> $$
>
> for all $t \in \mathbb{R}$. With $Q_j: \mathbf{q} \mapsto Q_j(\mathbf{q})$ the generalized forces of type I given by
>
> $$
> Q_j(\mathbf{q}) = \sum_{i=1}^n \Big\langle \mathbf{F}_i(\mathbf{q}), \partial_j \mathbf{x}_i(\mathbf{q}) \Big\rangle,
> $$
>
> for all $t \in \mathbb{R}$ with $\mathbf{F}_i: \mathbf{q} \mapsto \mathbf{F}_i(\mathbf{q})$ the total force excluding the constraint forces.
??? note "*Proof*:"
Will be added later.
Obtaining the equations of Lagrange. Note that the position of each point mass $\mathbf{x}_i$ is defined in the [Lagrangian formalism](lagrangian-formalism.md#generalizations).
### Conservative systems
For conservative systems we may express the force $\mathbf{F}_i: \mathbf{q} \mapsto \mathbf{F}_i(\mathbf{q})$ in terms of a potential energy $V: X \mapsto V(X)$ by
$$
\mathbf{F}_i(\mathbf{q}) = -\nabla_i V(X),
$$
for $X: \mathbf{q} \mapsto X(\mathbf{q}) \overset{\mathrm{def}}= \{\mathbf{x}_i(\mathbf{q})\}_{i=1}^n$.
> *Lemma 1*: for a conservative holonomic system the generalized forces of type I $Q_j: \mathbf{q} \mapsto Q_j(\mathbf{q})$ may be expressed in terms of the potential energy $V: \mathbf{q} \mapsto V(\mathbf{q})$ by
>
> $$
> Q_j(\mathbf{q}) = -\partial_{q_j} V(\mathbf{q}),
> $$
>
> for all $t \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.
The equation of Lagrange may now be rewritten, which obtains the following lemma.
> *Lemma 2*: let $T: (\mathbf{q}, \mathbf{q}') \mapsto T(\mathbf{q}, \mathbf{q'})$ and $V: \mathbf{q} \mapsto V(\mathbf{q})$ be the kinetic and potential energy of the system. The Lagrange equations for conservative systems are given by
>
> $$
> d_t \Big(\partial_{q_j'} T(\mathbf{q},\mathbf{q}')\Big) - \partial_{q_j}T(\mathbf{q},\mathbf{q}') = - \partial_{q_j} V(\mathbf{q}),
> $$
>
> for all $t \in \mathbb{R}$
??? note "*Proof*:"
Will be added later.
> *Definition 2*: let $T: (\mathbf{q}, \mathbf{q}') \mapsto T(\mathbf{q}, \mathbf{q'})$ and $V: \mathbf{q} \mapsto V(\mathbf{q})$ be the kinetic and potential energy of the system. The Lagrangian $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ is defined as
>
> $$
> \mathcal{L}(\mathbf{q}, \mathbf{q'}) = T(\mathbf{q},\mathbf{q}') - V(\mathbf{q}),
> $$
>
> for all $t \in \mathbb{R}$.
With this definition we may write the Lagrange equations in a more formal way.
> *Theorem 2*: let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian, the equations of Lagrange for conservative holonomic systems are given by
>
> $$
> d_t \Big(\partial_{q_j'} \mathcal{L}(\mathbf{q}, \mathbf{q'}) \Big) - \partial_{q_j} \mathcal{L}(\mathbf{q}, \mathbf{q'}) = 0,
> $$
>
> for all $t \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.

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# Lagrange generalizations
## The generalized momentum and force
> *Definition 1*: let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian, the **generalized momentum** $p_j: (\mathbf{q}, \mathbf{q}') \mapsto p_j(\mathbf{q},\mathbf{q}')$ is defined as
>
> $$
> p_j(\mathbf{q},\mathbf{q}') = \partial_{q_j'} \mathcal{L}(\mathbf{q}, \mathbf{q'}),
> $$
>
> for all $t \in \mathbb{R}$.
The generalized momentum may also be referred to as the canonical or conjugated momentum. Recall that $j \in \mathbb{N}[j\leq f]$.
> *Definition 2*: let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian, the **generalized force of type II** $F_j: (\mathbf{q}, \mathbf{q}') \mapsto F_j(\mathbf{q},\mathbf{q}')$ is defined as
>
> $$
> F_j(\mathbf{q},\mathbf{q}') = \partial_{q_j} \mathcal{L}(\mathbf{q}, \mathbf{q'})
> $$
>
> for all $t \in \mathbb{R}$.
We may also write $\mathbf{p} = \{p_j\}_{j=1}^f$ and $\mathbf{F} = \{F_j\}_{j=1}^f$.
## The generalized energy
> *Theorem 1*: let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian, the generalized energy $h: (\mathbf{q}, \mathbf{q'},\mathbf{p}) \mapsto h(\mathbf{q}, \mathbf{q'},\mathbf{p})$ is given by
>
> $$
> h(\mathbf{q}, \mathbf{q'}, \mathbf{p}) = \sum_{j=1}^f \big(p_j q_j' \big) - \mathcal{L}(\mathbf{q}, \mathbf{q'}),
> $$
>
> for all $t \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.
A generalization of the concept of energy.
* If the Lagrangian $\mathcal{L}: (\mathbf{q}, \mathbf{q'},t) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'},t)$ is explicitly time-dependent $\partial_t \mathcal{L}(\mathbf{q}, \mathbf{q'},t) \neq 0$ and the generalized energy $h$ is not conserved.
* If the Lagrangian $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ is not explicitly time-dependent $\partial_t \mathcal{L}(\mathbf{q}, \mathbf{q'}) = 0$ and the generalized energy $h$ is conserved.
> *Theorem 2*: for autonomous systems with only conservative forces the generalized energy $h: (\mathbf{q}, \mathbf{q'}) \mapsto h(\mathbf{q}, \mathbf{q'})$ is conserved and is given by
>
> $$
> h(\mathbf{q}, \mathbf{q'}) = T(\mathbf{q},\mathbf{q}') + V(\mathbf{q}) \overset{\mathrm{def}}= E,
> $$
>
> for all $t \in \mathbb{R}$ with $T: (\mathbf{q}, \mathbf{q}') \mapsto T(\mathbf{q}, \mathbf{q'})$ and $V: \mathbf{q} \mapsto V(\mathbf{q})$ the kinetic and potential energy of the system and $E \in \mathbb{R}$ the total energy of the system.
??? note "*Proof*:"
Will be added later.
In this case the generalized energy $h$ is conserved and is equal to the total energy $E$ of the system.
## Conservation of generalized momentum
> *Definition 3*: let $\mathcal{L}: (\mathbf{q}, \mathbf{q'}) \mapsto \mathcal{L}(\mathbf{q}, \mathbf{q'})$ be the Lagrangian, a coordinate $q_j$ is **cyclic** if
>
> $$
> \partial_{q_j} \mathcal{L}(\mathbf{q}, \mathbf{q'}) = 0,
> $$
>
> for all $t \in \mathbb{R}$.
Therefore the Lagrangian is independent of a cyclic coordinate.
> *Proposition 1*: the generalized momentum $p_j$ corresponding to a cyclic coordinate $q_j$ is conserved.
??? note "*Proof*:"
Will be added later.
## Seperable systems
> *Proposition 2*: the Lagrangian is seperable if there exists two mutually independent subsystems.
??? note "*Proof*:"
Will be added later.
Obtaining a decoupled set of partial differential equations.
## Invariances
> *Proposition 3*: the Lagrangian is invariant for Gauge transformations and therefore **not unique**.
??? note "*Proof*:"
Will be added later.
There can exist multiple Lagrangians that may lead to the same equation of motion.
According to the theorem of Noether, the invariance of a closed system with respect to continuous transformations implies that corresponding conservation laws exist.

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# Lagrangian formalism of mechanics
The Lagrangian formalism of mechanics is based on the axioms, postulates and principles posed in the [Newtonian formalism](/en/physics/mechanics/newtonian-mechanics/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.
<br>
> *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 $\{\mathbf{x}_i\}_{i=1}^n$.
<br>
> *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.