# Lagrangian Mechanics

### From TorontoMathWiki

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 obeys Lagrange's equation

- for all

where is the Lagrangian of the system.

We may write in place of .

# Generalized coordinates

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

# Conservation laws

Consider a system of $n$ particles, and let be the $i^{th}$ particle’s position in space, and let be its velocity. We define the generalized momentum of the $i^{th}$ particle as

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

In this notation, Lagrange's equations are written

We now discuss the *symmetries* and *conservation laws* implied by Lagrange's equation. A *symmetry* refers to a family of transformations 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 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 is invariant under the transformations . 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

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

it follows that

Therefore, is conserved. When , where

- is the kinetic energy and

- is the potential energy,

- ( )

then and

- .

In this case, the total energy is conserved.

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

Proof: Since the Lagrangian does not depend on , then

Thus, is conserved.

**Corollary 1**: If the Lagrangian is invariant under the transformation , for all $i \in {1,...,n} $, then the total momentum is conserved.

This follows immediately from Proposition 2.

**Corollary 2**: If the Lagrangian is invariant under the transformation , then the z-component of the generalized momentum, i.e. , 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 represent the infinitesimal rotation, whose magnitude is the angle of rotation , and whose direction is that of the axis of rotation, so the direction of rotation is that of a right-handed screw driven along . Let denote the resulting change in the position of the $i^{th}$ particle in the system undergoing rotation. Let be the angle that makes with . Then

- .

The direction of is perpendicular to the plane spanned by and . Hence

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

Since the Lagrangian is unchanged by the rotation,

Permuting factors and taking outside the sum,

does not change with time, so

Since the angular momentum of the system is the vector

this implies that the component of angular momentum along , 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 is conserved.

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

for arbitrary . Therefore,

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 placed inside a Newtonian system. Then, the Lagrangian of the particle in Cartesian co-ordinates is given by

where

- is the kinetic energy of the particle, and
- is the potential energy.

Hence,

- for all

and

- for all

Substituting into Lagrange's equation, we obtain

- for all

or

This qualifies Lagrangian mechanics as a reformulation of Newtonian mechanics.

# Hamilton's principle

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

with conditions and .

Hamilton's principle states that the evolution of the system from to is an extremum of . In other words, δ*S* = 0.

Consequently, from the Euler-Lagrange equation, we obtain

- for all

as expected.

This principle is also called the principle of least action.