Added first part of Lagrangian mechanics.
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- 'Energy': physics/mechanics/newtonian-mechanics/energy.md
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- 'Rotation': physics/mechanics/newtonian-mechanics/rotation.md
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- 'Particle systems': physics/mechanics/newtonian-mechanics/particle-systems.md
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# - 'Lagrangian mechanics':
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- 'Lagrangian mechanics':
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- 'Lagrangian formalism': physics/mechanics/lagrangian-mechanics/lagrangian-formalism.md
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- 'Lagrange equations': physics/mechanics/lagrangian-mechanics/lagrange-equations.md
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- 'Lagrange generalizations': physics/mechanics/lagrangian-mechanics/lagrange-generalizations.md
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# - 'Hamiltonian mechanics':
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# - 'Relativistic mechanics':
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# - 'Quantum mechanics':
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# The equations of Lagrange
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## Principle of virtual work
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> *Definition 1*: a virtual displacement is a displacement at a fixed moment in time that is consistent with the constraints at that moment.
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The following principle addresses the problem that the constraint forces are generally unknown.
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> *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
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>
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> $$
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> \sum_{i=1}^n \Big\langle \mathbf{F}_i(\mathbf{q}) - m_i \mathbf{x}_i''(\mathbf{q}), \mathbf{\delta x}_i \Big\rangle = 0,
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> $$
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>
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> is true for sklerenomic constraints and all $t \in \mathbb{R}$.
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Which implies that the constraint forces do not do any (net) virtual work.
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## The equations of Lagrange
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> *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
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>
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> $$
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> d_t \Big(\partial_{q_j'} T(\mathbf{q},\mathbf{q}') \Big) - \partial_{q_j} T(\mathbf{q},\mathbf{q}') = Q_j(\mathbf{q}),
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> $$
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>
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> for all $t \in \mathbb{R}$. With $Q_j: \mathbf{q} \mapsto Q_j(\mathbf{q})$ the generalized forces of type I given by
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>
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> $$
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> Q_j(\mathbf{q}) = \sum_{i=1}^n \Big\langle \mathbf{F}_i(\mathbf{q}), \partial_j \mathbf{x}_i(\mathbf{q}) \Big\rangle,
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> $$
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>
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> 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.
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??? note "*Proof*:"
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Will be added later.
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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).
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### Conservative systems
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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
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$$
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\mathbf{F}_i(\mathbf{q}) = -\nabla_i V(X),
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$$
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for $X: \mathbf{q} \mapsto X(\mathbf{q}) \overset{\mathrm{def}}= \{\mathbf{x}_i(\mathbf{q})\}_{i=1}^n$.
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> *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
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>
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> $$
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> Q_j(\mathbf{q}) = -\partial_{q_j} V(\mathbf{q}),
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> $$
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>
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> for all $t \in \mathbb{R}$.
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??? note "*Proof*:"
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Will be added later.
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The equation of Lagrange may now be rewritten, which obtains the following lemma.
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> *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
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>
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> $$
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> d_t \Big(\partial_{q_j'} T(\mathbf{q},\mathbf{q}')\Big) - \partial_{q_j}T(\mathbf{q},\mathbf{q}') = - \partial_{q_j} V(\mathbf{q}),
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> $$
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>
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> for all $t \in \mathbb{R}$
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??? note "*Proof*:"
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Will be added later.
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> *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
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>
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> $$
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> \mathcal{L}(\mathbf{q}, \mathbf{q'}) = T(\mathbf{q},\mathbf{q}') - V(\mathbf{q}),
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> $$
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>
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> for all $t \in \mathbb{R}$.
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With this definition we may write the Lagrange equations in a more formal way.
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> *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
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>
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> $$
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> d_t \Big(\partial_{q_j'} \mathcal{L}(\mathbf{q}, \mathbf{q'}) \Big) - \partial_{q_j} \mathcal{L}(\mathbf{q}, \mathbf{q'}) = 0,
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> $$
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>
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> for all $t \in \mathbb{R}$.
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??? note "*Proof*:"
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Will be added later.
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# Lagrange generalizations
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# Lagrangian formalism
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The Lagrangian formalism of mechanics is based on the axioms, postulates and principles posed in the [Newtonian formalism](../newtonian-mechanics/newtonian-formalism).
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## Configuration of a system
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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 $i \in \mathbb{N}[i \leq n]$.
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> *Definition 1*: the set of positions $\{\mathbf{x}_i\}_{i=1}^n$ is defined as the configuration of the system.
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Obtaining a $n m$ dimensional configuration space of the system.
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> *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.
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<br>
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> *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.
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The minimum required number of generalized coordinates is called the number of degrees of freedom of the system.
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## Classification of constraints
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> *Definition 4*: geometric constraints define the range of the positions $\{\mathbf{x}_i\}_{i=1}^n$.
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<br>
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> *Definition 5*: holonomic constraints are defined as constraints that can be formulated as an equation of generalized coordinates and time.
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Let $g: (q_1, \dots, q_N, t) \mapsto g(q_1, \dots, q_N, t) = 0$ is an example of a holonomic constraint.
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> *Definition 6*: a constraint that depends on velocities is defined as a kinematic constraint.
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If the kinematic constrain is integrable and can be formulated as a holonomic constraint it is referred to as a integrable kinematic constraint.
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> *Definition 7*: a constraint that explicitly depends on time is defined as a rheonomic constraint. Otherwise the constraint is defined as a sklerenomic constraint.
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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.
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## Generalizations
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> *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.
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The position of each point mass may be given by
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$$
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\mathbf{x}_i: \mathbf{q} \mapsto \mathbf{x}_i(\mathbf{q}),
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$$
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with $\mathbf{q} = \{q_i\}_{i=1}^f$ generalized coordinates.
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Therefore the velocity of each point mass is given by
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$$
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\mathbf{x}_i'(\mathbf{q}) = \sum_{r=1}^f \partial_r \mathbf{x}_i(\mathbf{q}) q_r',
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$$
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for all $t \in \mathbb{R}$ (inexplicitly).
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> *Theorem 1*: the total kinetic energy $T: (\mathbf{q}, \mathbf{q}') \mapsto T(\mathbf{q}, \mathbf{q}')$ of the system is given by
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>
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> $$
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> T(\mathbf{q}, \mathbf{q}') = \sum_{r,s=1}^f a_{rs}(\mathbf{q}) q_r' q_s',
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> $$
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>
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> with
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>
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> $$
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> a_{rs}(\mathbf{q}) = \sum_{i=1}^n \frac{1}{2} m_i \langle \partial_r \mathbf{x}_i(\mathbf{q}), \partial_s \mathbf{x}_i(\mathbf{q}),
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> $$
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>
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> for all $t \in \mathbb{R}$.
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??? note "*Proof*:"
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Will be added later.
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