Lagrangian & Hamiltonian Mechanics

Table of Contents

1. Applications

Quantum mechanics and statistical mechanics are based on Hamiltonian mechanics, and general relativity is based on Lagrangian mechanics. This two formalisms are not fully reconciled

2. Lagrangian

  • \[ L = T - V \] where \(T\) is the kinetic energy of the object, and \(V\) is the potential energy of the object.

3. Principle of Least Action

  • Principle of Stationary Action
  • The equation of motion can be derived from the Lagrangian \(L\) by the principle of the least action via the Euler-Lagrange equation .

The principle of least action does not apply for the polygenic system in which some forces do not come from a potential.

3.1. Maupertuis's Principle

The abbreviated action attains local minimum at the physical path within the space of paths with the same endpoints and same energy along a single path and between different path:

\[ S_0[\mathbf{q}(t)] := \int_{q_1}^{q_2} \mathbf{p}\cdot \mathrm{d}\mathbf{q}. \]

3.2. Hamilton's Principle

The energy conservation along a path still needs to hold.

\[ S[\mathbf{q}(t)] := \int_{t_1}^{t_2} L(\mathbf{q},\mathbf{\dot{q}}, t)\,\mathrm{d}t. \]

3.3. Holonomic Constraints

It is a constraints that can reduce the degree of freedom in the configuration space. Holonomic constraint must be expressible as a function: \[ f(x_1, x_2, \dots, x_n, t) = 0. \]

Among them, the constraint that only dependes on the configuration is called scleronomic, and one that depends on the configuration and the time is called rheonomic.

To give a non-example, the constraint on the particle that is allowed to fall off a sphere is given by: \[ r^2 - a^2 \ge 0 \]

4. Conservative Force

  • \[ \frac{\partial L}{\partial \mathbf{q}} - \frac{d}{dt}\left(\frac{\partial L}{\partial \mathbf{\dot{q}}}\right) = \mathbf{0}^\mathsf{T} \]
    • with \(q_i\) being the generalized coordinates.
    • \(n\) coordinates and \(m\) holonomic constrants yields \(n-m\) equations with \(n-m\) generalized coordinates.

4.1. Ignorable Coordinate

  • If the Lagrangian does not explicitly depends on \(q_k\) then the generalized momentum \(p_k\) is constant: \[ \dot{p}_k := \frac{\partial L}{\partial q_k} = 0. \]

5. Generalized Constraint Force

  • With constraints:
    • \[ \mathbf{f}(\mathbf{q}, t) = \mathbf{0} \]
    • Equivalently \[ \delta \mathbf{f} = \frac{\partial \mathbf{f}}{\partial \mathbf{q}}\delta \mathbf{q} = \mathbf{0} \]
  • and the variation:
    • \[ \left[\frac{\partial L}{\partial \mathbf{q}} - \frac{d}{dt}\left(\frac{\partial L}{\partial \mathbf{\dot{q}}}\right)\right]\delta \mathbf{q} = 0 \]
  • By the method of Lagrange multipliers, It follows that: \[ \frac{\partial L}{\partial \mathbf{q}} - \frac{d}{dt}\left(\frac{\partial L}{\partial \mathbf{\dot{q}}}\right) + \boldsymbol{\lambda}^\mathsf{T}\frac{\partial \mathbf{f}}{\partial \mathbf{q}} = \mathbf{0}^\mathsf{T}. \]
    • with \(\boldsymbol{\lambda}^\mathsf{T}\partial \mathbf{f}/\partial \mathbf{q}\) being the generalized constraint force.

6. Generalized Force

  • From the d'Alembert's principle on a system of particles: \[ \sum_i (\mathbf{F}_i -\mathbf{\dot{p}})\cdot \delta\mathbf{r}_i = 0 \]
  • The applied forces \(\mathbf{F}_i\) transforms into generalized forces \(\mathbf{Q}_i\)
    • \[ \mathbf{F}_i\cdot \delta \mathbf{r}_i = \mathbf{F}_i \cdot \frac{\partial \mathbf{r}_i}{\partial \mathbf{q}_i}\delta \mathbf{q}_i = \mathbf{Q}_i\cdot \delta \mathbf{q}_i \]
    • \[ \mathbf{Q}_i = \left(\frac{\partial \mathbf{r}_i}{\partial \mathbf{q}_i}\right)^\mathsf{T}\mathbf{F}_i \]
  • and the inertial force \(\mathbf{\dot{p}}_i\) becomes:
    • \[ \mathbf{\dot{p}}_i\cdot \delta \mathbf{r}_i = \left[\frac{d}{dt}\left(\frac{\partial T}{\partial \mathbf{\dot{q}}_i}\right)-\frac{\partial T}{\partial \mathbf{q}_i}\right]\delta \mathbf{q}_i \]
  • combining these two, and the D'Alembert's principle, yields:
    • \[ \left(\mathbf{Q}_i^\mathsf{T} - \left[\frac{d}{dt}\left(\frac{\partial T}{\partial \mathbf{\dot{q}}_i}\right)-\frac{\partial T}{\partial \mathbf{q}_i}\right]\right)\delta \mathbf{q}_i = \mathbf{0}^\mathsf{T} \]
  • Removing the conservative forces form the generalized force \(\mathbf{Q}_i\), gives:
    • \[ \left(\mathbf{Q'}_i^\mathsf{T} - \left[\frac{d}{dt}\left(\frac{\partial L}{\partial \mathbf{\dot{q}}_i}\right)-\frac{\partial L}{\partial \mathbf{q}_i}\right]\right)\delta \mathbf{q}_i = \mathbf{0}^\mathsf{T} \]
  • And lastly imposing the constraints \(\mathbf{f}_i(\mathbf{q}_i, t) = \mathbf{0}\), the general Lagrangian equation of motion is obtained:
    • \[ \mathbf{Q'}_i^\mathsf{T} - \left[\frac{d}{dt}\left(\frac{\partial L}{\partial \mathbf{\dot{q}}_i}\right)-\frac{\partial L}{\partial \mathbf{q}_i}\right] + \boldsymbol{\lambda}_i^\mathsf{T}\frac{\partial \mathbf{f}_i}{\partial \mathbf{q}_i} = \mathbf{0}^\mathsf{T}. \]

7. Hamilton's Canonical Equations

\[ \dot{q}_i = \frac{\partial H}{\partial p_i},\quad \dot{p}_i = -\frac{\partial H}{\partial q_i} \]

It is derived in a various ways, one of which is using the Legendre transformation.

It can be written succinctly as \[ \dot{\mathbf{x}} = \mathbf{J}\nabla H \] where \( J \) is the canonical symplectic matrix, and the gradient is with respect to each state variable.

8. Spaces

  • Configuration Space: the set of variable for each degree of freedom
  • Even Space: configuration space with additional time axis
  • Momentum Space: the set of conjugate momenta for each degree of freedom
  • State Space: configuration space and momentum space
  • State-Time Space: state space additional time axis

9. Virial Theorem

\begin{equation*} \left\langle T \right\rangle = -\frac{1}{2} \left\langle \sum_i \mathbf{F}_i \cdot \mathbf{r}_{i} \right\rangle. \end{equation*}

10. D'Alembert's Principle

10.1. Static System

  • Principle of Virtual Work

\[ \delta W = \sum_i \mathbf{F}_i \cdot \delta \mathbf{r}_i \] where \( \mathbf{F}_i \) is the applied force, excluding constraint forces, on the \( i \) th particle, and \( \delta\mathbf{r}_i \) is the virtual displacement of the \( i \) th particle, consistent with the constraints: \[ \sum_i \mathbf{C}_i \cdot \delta\mathbf{r}_i = 0. \]

10.2. Dynamic System

  • Lagrange-d'Alembert Principle

\[ \delta W = \sum_i (\mathbf{F}_i - \mathbf{\dot{p}}_i )\cdot \delta \mathbf{r}_i = 0 \] where \( \mathbf{\dot{p}}_i \) is the change in the momentum of \( i \) th particle.

It is the Newton's second law, minus the constraint forces, that vanishes with dot product.

10.2.1. Generalized Coordinates

\[ \delta W = \sum_i (Q_i + Q_i^*) \delta q_i = 0 \] where \( Q_i \) is the generalized applied force and \( Q_i^* \) is the generalized inertia force.

The arbitrariness of \( \delta q_i \) yields: \[ -Q_i^* = \frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial T}{\partial \dot{q}_i} -\frac{\partial T}{\partial q_i} = Q_i. \]

10.3. Generalized Hamilton's Principle

\[ \delta \int_I L(\mathbf{r}_1,\dots, \mathbf{r}_n, \mathbf{\dot{r}}_1,\dots, \mathbf{\dot{r}}_n, t)\,\mathrm{d}t + \sum_i \int_I \mathbf{F}_i\cdot \delta \mathbf{r}_i\,\mathrm{d}t = 0. \]

10.4. Thermodynamic System

For adiabatically closed system, \[ \delta \int_I L(\mathbf{r}_1,\dots, \mathbf{r}_n, \mathbf{\dot{r}}_1,\dots, \mathbf{\dot{r}}_n, S, t)\,\mathrm{d}t + \sum_i \int_I \mathbf{F}_i\cdot \delta \mathbf{r}_i\,\mathrm{d}t = 0. \] where the Lagrangian is given by: \[ L(\mathbf{r}_1,\dots, \mathbf{r}_n, \mathbf{\dot{r}}_1,\dots, \mathbf{\dot{r}}_n, S, t) = \sum_i\frac{1}{2} m_i\mathbf{\dot{r}}_i^2 - V(\mathbf{r}_1,\dots, \mathbf{r}_n, S), \] and with the generalized constraint: \[ \sum_i \mathbf{C}_i \cdot \delta \mathbf{r}_i + T\delta S = 0, \] \( T := \partial V /\partial S \) is the temperature of the system, \(\mathbf{F}_i\) are the external forces, \( \mathbf{C}_i \) are the internal dissipative forces.

11. Noether's Theorem

11.1. General Statement

The symmetry of the Lagrangian at the point of stationary action gives rise to a conserved quantity.

The invariant Lagrangian tells that the law of physics (the relations among physical quantities) does not change under that symmetry.

11.2. Details

Given that \[ \delta L = 0 \] with respect to \( \delta q \) with \( q \) satisfying \( \delta S = 0 \), \[ 0 = \delta L = \frac{\partial L}{\partial q}\delta q + \frac{\partial L}{\partial \dot{q}} \delta \dot{q} = \frac{d}{dt}\left( \frac{\partial L}{\partial \dot{q}}\delta q \right) + \left( \frac{\partial L}{\partial q} - \frac{d}{dt}\left( \frac{\partial L}{\partial \dot{q}} \right)\right) \delta q = \frac{d}{dt}\left( \frac{\partial L}{\partial \dot{q}}\delta q \right). \]

Therefore, the quantity \[ \frac{\partial L}{\partial \dot{q}} \delta q \] is conversed during the timespan of \( q \).

11.3. For Field

The Lagrangian density \( \mathcal{L} \) can be invariant under:

  • Internal Symmetry: the variation in the field \( \delta \phi \)
  • Spacetime Symmetry: the transformation of the spacetime coordinate

11.3.1. Internal Symmetry

  • The transformed coordinate have to be the same, and derivatives are not touched.
  • The continuity: \[ \partial_{\mu}\left( \frac{\partial \mathcal{L}}{\partial (\partial_{\mu}\phi )} \delta\phi \right) = 0 \] means a quantity is conserved over the entire field as a whole.

11.3.2. Spacetime Symmetry

The variation in the Lagrangian have to account for the change of variable \[ 0 = \delta \mathcal{L} = \mathcal{L}' - \mathcal{L} \] with \( 0 = \delta \phi(x) = \phi' (x') - \phi (x) \) since the field itself does not change (or simply moves around).

The coordinate transformation is given by the tangent vector field \( K^{\mu} \)

Accordingly, \( 0 = \phi'(x') - \phi'(x) + \phi'(x) - \phi(x) = \delta x^{\mu} \partial_{\mu}\phi' + \bar{\delta} \phi \) \[ \bar{\delta}\phi = -\delta x^{\mu} \partial_{\mu}\phi'. \]

And

\begin{align*} \partial_{\mu}\phi &= \frac{\partial x'^{\nu}}{\partial x^{\mu}}\frac{\partial \phi'}{\partial x'^{\nu}} \\ &= ( \delta^{\nu}_{\mu} + \partial_{\mu} \delta x^{\nu}) ( \partial_{\nu}\phi + \delta \partial_{\nu}\phi) \\ &= \partial'_{\mu}\phi' + (\partial_{\mu}\delta x^{\nu})\partial_{\nu}\phi \end{align*}

therefore, \[ \delta (\partial_{\mu}\phi) = - (\partial_{\mu}\partial x^{\nu}) \partial_{\nu}\phi. \]

To calculate the action the measure need to be adjusted:

\begin{align*} d^4x' &= d^4x \det \left( \frac{\partial x'^{\mu}}{\partial x^{\nu}} \right) \\ &= d^4x \det \left( \delta^{\mu}{}_{\nu} + \frac{\partial (\delta x)^{\mu}}{\partial x^{\nu}} \right) \\ &= d^4x (1 + \partial_{\mu} \delta x^{\mu}). \end{align*}

The conserved current turns out to be \[ \frac{\partial \mathcal{L}}{\partial(\partial_{\mu}\phi)}\bar{\delta}\phi + (\delta x^{\mu}) \mathcal{L}. \] And the energy-momentum tenror \( T^{\mu}{}_{\nu} \) that connects the transformation to the current \[ J^{\mu}(K) = T^{\mu}{}_{\nu}K^{\nu } \] is then given by: \[ T^{\mu}{}_{\nu} = -\frac{\partial \mathcal{L}}{\partial(\partial_{\mu}\phi)}\partial_{\nu}\phi + \delta^{\mu}{}_{\nu}\mathcal{L}. \]

12. References

Created: 2025-05-25 Sun 03:19