Lagrangian Mechanics

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Lagrangian mechanics is a reformulation of classical mechanics introduced by Joseph-Louis Lagrange in 1788. Here we discuss the use of generalized co-ordinates and the connection to Newtonian mechanics and Hamilton's principle.

Contents

Statement

A system described by generalized coordinates \ q_1, q_2, ..., q_n obeys Lagrange's equation

{\partial{L}\over \partial q_i} - {\mathrm{d} \over \mathrm{d}t}\left({\partial{L}\over \partial{\dot{q_i}}}\right) = 0. for all \ i = 1, 2, ..., n

where \ L = L (t, q_1, q_2, ..., q_n, \dot{q_1}, \dot{q_2}, ..., \dot{q_n}) is the Lagrangian of the system.

We may write \ (t, q, \dot q) in place of \ (t, q_1, q_2, ..., q_n, \dot{q_1}, \dot{q_2}, ..., \dot{q_n}).

Generalized coordinates

The state of a system of \ N particles in 3D space can be uniquely defined using no more than \ 3N independent quantities. The number of such quantities required is called the number of degrees of freedom. Any \ n such quantities \ q_1, q_2, ..., q_n are called generalized co-ordinates, and \dot{q_1}, \dot{q_2}, ..., \dot{q_n} generalized velocities. These quantities are not necessarily Cartesian.

Conservation laws

Consider a system of $n$ particles, and let \ \vec q_i be the $i^{th}$ particle’s position in space, and let \ \dot{\vec q_i} = \frac{d \vec q_i}{dt} be its velocity. We define the generalized momentum of the $i^{th}$ particle as

\ \vec p_i = {\partial{L}\over \partial{\dot{\vec q_i}}} 
= \left( \frac{\partial L}{\partial \dot{q_{ix}}}, \frac{\partial L}{\partial \dot{q_{iy}}}, \frac{\partial L}{\partial \dot{q_{iz}}} \right)

and the generalized force acting on the $i^{th}$ particle as

\ \vec F_i = {\partial L\over \partial \vec q_i} = \left( \frac{\partial L}{\partial q_{ix}}, \frac{\partial L}{\partial q_{iy}}, \frac{\partial L}{\partial q_{iz}} \right)

In this notation, Lagrange's equations are written

\dot{\vec p_i} = \vec F_i, \quad i = 1,...,n

We now discuss the symmetries and conservation laws implied by Lagrange's equation. A symmetry refers to a family of transformations \ t \longmapsto t ^\prime, \vec q_i \longmapsto {\vec{q_i}}^{\prime} that leads to zero first-order changes in the Lagrangian. Three standard symmetries in classical mechanics are

1. Invariance under translations of time $\Rightarrow$ the Lagrangian has no explicit time dependence

2. Invariance under translations of space $\Rightarrow$ the Lagrangian does not depend on \ \vec q_i for all $i=1,...,n$.

3. Invariance under rotations about an axis

Each one implies a conservation law: a conserved quantity or an integral of Lagrange’s equations can be identified.

Proposition 1 (Conservation of energy): Suppose the Lagrangian \ L(t,\vec q_1,...,\vec q_n,\dot{\vec q_1},...,\dot{\vec q_n}) is invariant under the transformations \ t \longmapsto t + \epsilon. Then the Hamiltonian of the system is conserved.

Proof: Since the Lagrangian is invariant under translations of time, it has no explicit dependence on time. So \ \frac{\partial L}{\partial t} = 0

Since the Hamiltonian, or the energy of the system, is defined by

 H = \sum_{i=1}^n \dot{\vec q_i} \cdot \frac{\partial L}{\partial \dot{\vec q_i}} - L

it follows that

 \begin{align} \frac{\partial H}{\partial t}
& = \sum_{i=1}^n \ddot{\vec{q_i}} \cdot \frac{\partial L}{\partial \dot{\vec{q_i}}} + \sum_{i=1}^n \dot{\vec{q_i}} \cdot \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\vec{q_i}}} \right) - \frac{\partial L}{\partial t} \\
& = \sum_{i=1}^n \ddot{\vec{q_i}} \cdot \frac{\partial L}{\partial \dot{\vec{q_i}}} + \sum_{i=1}^n \dot{\vec{q_i}} \cdot \frac{\partial L}{\partial \vec{q_i}} - \frac{\partial L}{\partial t} \\
& = \frac{\partial L}{\partial t} - \frac{\partial L}{\partial t} \\
& = 0-0 \\
& = 0 \\
\end{align}

Therefore, \ H is conserved. When \ L = T - V, where

\ T = \sum_{i=1}^n \frac{1}{2}m_i \dot{\vec q_i}^2 is the kinetic energy and
\ V = V(\vec q_1,...,\vec q_n) is the potential energy,
( \dot{\vec q_i}^2 = \dot{\vec q_i} \cdot \dot{\vec q_i} )

then \ \vec p_i = m_i \dot{\vec q_i} and

\ H = \sum_{i=1}^n m_i \dot{\vec q_i}^2 - L = 2T - (T - V) = T + V.

In this case, the total energy is conserved.

Proposition 2 (Conservation of Momentum): Suppose the Lagrangian is invariant under the transformations \ \vec{q_i} \longmapsto \vec{q_i} + \vec a, where $i \in {1,...,n} $. Then the generalized momentum \ \vec p_i is conserved.

Proof: Since the Lagrangian does not depend on \ \vec q_i, then

\ \vec 0 = \frac{\partial L}{\partial \vec{q_i}} = \dot{\vec{p_i}}

Thus, \ p_i is conserved.

Corollary 1: If the Lagrangian is invariant under the transformation \ \vec{q_i} \longmapsto \vec{q_i} + \vec a, for all $i \in {1,...,n} $, then the total momentum \ \vec{P} = \textstyle \sum_{i=1}^n \vec{p_i} is conserved.

This follows immediately from Proposition 2.

Corollary 2: If the Lagrangian is invariant under the transformation \ q_{iz} \longmapsto q_{iz} + a, then the z-component of the generalized momentum, i.e. \ p_{iz} , is conserved.

The proof is the same as that of Proposition 2, with the subscript $i$ replaced by $iz$, so we obtain the result from Lagrange's equation for the $z$-coordinate. The statement in Corollary 2 also holds if we replace $z$ with $x$ or $y$ instead. A coordinate on which the Lagrangian has no explicit dependence is called cyclic. In general, if a coordinate is cyclic, then the generalized momentum corresponding to that coordinate is conserved.

Proposition 3 (Conservation of Angular Momentum): Suppose that the Lagrangian is invariant under an infinitesimal rotation of the system. Then the component of angular momentum along the axis of rotation is conserved.

Proof: Let the vector \ \delta \vec{\phi} represent the infinitesimal rotation, whose magnitude is the angle of rotation \ \delta \phi , and whose direction is that of the axis of rotation, so the direction of rotation is that of a right-handed screw driven along \ \delta \vec{\phi} . Let \ \delta \vec{q_i} denote the resulting change in the position \ \vec q_i of the $i^{th}$ particle in the system undergoing rotation. Let \ \theta be the angle that \ \vec{q_i} makes with \ \delta \vec{\phi} . Then


\ | \delta \vec{r} | = \delta \phi | \vec{q_i} | \sin \theta  .


The direction of \ \delta \vec{q_i} is perpendicular to the plane spanned by \ \vec{q_i} and \ \delta \vec{\phi} . Hence

\ \delta \vec{q_i} = \delta \vec{\phi} \times \vec{q_i}

When the system rotates, the velocities of the particles also change direction. The velocity increment relative to a fixed system of coordinates is

\ \delta \dot{\vec{q_i}} = \delta \vec{\phi} \times \dot{\vec{q_i}}

Since the Lagrangian is unchanged by the rotation,

\ \begin{align} 0 & = \delta L \\ & = \sum_{i=1}^n \left( \frac{\partial L}{\partial \vec{q_i}} \cdot \delta \vec{q_i} + \frac{\partial L}{\partial \dot{\vec{q_i} } } \cdot \delta \dot{\vec{q_i}} \right) \\
& = \sum_{i=1}^n \left( \dot{\vec{p_i}} \cdot \left( \delta \vec{\phi} \times \vec{q_i} \right) 
+ \vec{p_i} \cdot \left( \delta \vec{\phi} \times \dot{\vec{q_i}} \right) \right) \\  \end{align}

Permuting factors and taking \ \delta \vec{\phi} outside the sum,

 \begin{align} 0 & = \delta \vec{\phi} \cdot \sum_{i=1}^n \left( \vec{q_i} \times \dot{\vec{p_i}} + \dot{\vec{q_i}} \times \vec{p_i} \right) \\
& = \delta \vec{\phi} \cdot \frac{d}{dt} \left( \sum_{i=1}^n \vec{q_i} \times \vec{p_i} \right)
\end{align}

\ \delta \vec{\phi} does not change with time, so

\ \frac{d}{dt} \left( \delta \vec{\phi} \cdot \sum_{i=1}^n \vec{q_i} \times \vec{p_i} \right) = 0

Since the angular momentum of the system is the vector

\ \vec M = \sum_{i=1}^n \vec{q_i} \times \vec{p_i}

this implies that the component of angular momentum along \ \delta \vec{\phi} , i.e. the axis of rotation, is conserved.

Corollary 3: If the Lagrangian is invariant under any rotation of the system, then the angular momentum \ \vec M is conserved.

Proof: As shown in the proof of Proposition 3, we have

\ \delta \vec{\phi} \cdot \frac{d \vec M}{dt} = 0

for arbitrary \ \delta \vec{\phi} . Therefore,

\ \frac{d \vec M}{dt} = \vec 0

Remark: Propositions 2 and 3 are actually consequences of a more general result called Noether's Theorem. Thus, we can describe conservation of momentum and of angular momentum as being consequences of the invariance of the action functional under a family of spatial translations and a family of rotations, respectively. The idea of Noether's Theorem is to find a conserved quantity given any symmetry. Thus, once we know the symmetries of a given problem, we can easily find conserved quantities.


References:

  • Calculus of Variations by I.M. Gelfand & S.V. Fomin (translated and edited by Richard A. Silverman), 1991.
  • Introductory Classical Mechanics with Problems and Solutions, by David Morin, 2004.
  • Course of Theoretical Physics Vol. 1: Mechanics, 3rd ed., by Landau and Lifshitz; first published in 1960.

To be completed by Alex.Baiyun 23:19, 21 May 2010 (UTC)

The Lagrangian for a test particle in a Newtonian system

Consider a test particle of mass \ m placed inside a Newtonian system. Then, the Lagrangian of the particle in Cartesian co-ordinates \vec x = (x_1, x_2, x_3) is given by

\ L = T - V

where

\ T = \frac 12 m \dot{\vec{x}} \cdot \dot{\vec{x}} is the kinetic energy of the particle, and
\ V = V(\vec x) is the potential energy.

Hence,

{\partial{L}\over \partial x_i} = - {\partial V\over \partial x_i} for all \ i=1,2,3

and

{\partial{L}\over \partial{\dot{x_i}}} = m \dot x_i for all \ i=1,2,3

Substituting into Lagrange's equation, we obtain

\ m \ddot x_i + {\partial V \over \partial x_i} = 0 for all \ i=1,2,3

or

\ m \ddot{\vec x} + \vec{\nabla} V = 0

This qualifies Lagrangian mechanics as a reformulation of Newtonian mechanics.

Hamilton's principle

For a given system, we define the action from \ t_1 to \ t_2 as

\ S = \int_{t_1}^{t_2} L (t, q, \dot q) dt

with conditions \ q(t_1)=q^{(1)} and \ q(t_2)=q^{(2)}.

Hamilton's principle states that the evolution \ q of the system from \ t_1 to \ t_2 is an extremum of \ S. In other words, δS = 0.

Consequently, from the Euler-Lagrange equation, we obtain

{\partial{L}\over \partial q_i} - {\mathrm{d} \over \mathrm{d}t}{\partial{L}\over \partial{\dot{q_i}}} = 0. for all \ i = 1, 2, ..., n

as expected.

This principle is also called the principle of least action.

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