spatial translation

8/17/2019 Spatial Translation

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Conservation Laws and Symmetry

Dr. Kristel Torokoff

February 22, 2014


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Background

Suppose we have a particle of mass m in one spatial dimension. It has position x , which gives thedistance from some arbitrary origin, shown below.

Note - although the position of the origin is arbitrary, once it is chosen, it is fixed. The first timederivative of the position, dx

dt , gives the speed of the particle, and the second time derivative of the

position, d 2x dt 2

, gives the acceleration of the particle. Note: the derivative means the rate of change.

We can now define the linear momentum of the particle, p , as mass times speed

p = mdx

dt

Newton’s Second Law of Motion states that the net force F on the particle is equal to the rate of change of its linear momentum,

F = dp

dt .


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Background

Suppose F is a conservative force, this means that the work done by F in moving a particle frompoint A to point B is not dependent on the path taken.

Note: in particular, if the particle travels in a closed loop, the net work done by a conservative forceis zero. Here the work done by a conservative force F is the same independent of which of the abovepaths was taken between A and B. Examples of conservative forces include gravity and the springforce. Examples of non-conservative forces include friction. A conservative force depends only on theposition of the particle and it can be expressed as the negative rate of change of a potential u ,

F = −∂ u

∂ x

In general, the potential u can be a function of position x and time t , u = u (x , t ). The notation ∂∂x

means a partial derivative with respect to x , and ∂u ∂x

means that we only diff erentiate u (x , t ) withrespect to x , keeping any other variables such as t constant. Thus, for a particle of mass m in aconservative force field, we can write Newton’s Second Law of motion as

dp

dt = −

∂ u

∂ x

.


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Conservation Laws

Conservation of a particular measurable property of a physical system, such as linear momentum p ,

means that this property does not change as the system evolves, i.e. dp dt

= 0.

If something does not change, it means the rate of change is zero, and thus this something is aconstant, or an invariant.

From Newton’s Laws of motion, we can derive the following conservation laws:

• (i) the conservation of linear momentum p : dp dt

= 0

• (ii) the conservation of angular momentum L : dLdt

= 0

• (iii) the conservation of energy E : dE dt

= 0

These conservation laws, although derived from Newton’s laws, can be generalised to situations

beyond the reach of classical physics. Furthermore, in the following chapter we will connect theseconservation laws to the continuous symmetries of space and time.


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Symmetries 1 - Translations in Space

This symmetry is based on the idea that space is the same everywhere, sometimes called ‘thehomogeneity of space’. Since space is the same everywhere, we expect the laws of Physics to be thesame everywhere.

Consider a particle of mass m at position x from some origin O . In this frame it has linearmomentum p x = m

dx dt

.

The same particle considered in a diff erent frame has position x 0 from O 0 and has linear momentum

p 0

x = m dx 0

dt .The two frames are related by a linear translation x 0 = x − α, with α some constant, such that

p 0x = md (x −α)

dt = m dx

dt = p x , since

d αdt

= 0. So,

p 0

x = p x

Thus, the linear momentum is translation invariant. Furthermore, recalling Newton’s Second Law

of motion dp dt

= − ∂u ∂x

, we see that for an isolated system, that is, a system with no external forces

F = 0 or ∂u ∂x

= 0, the linear momentum is conserved/unchanged, dp dt

= 0, by any parallel

displacement of the entire system in space.


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Symmetries 1 - Translations in Space

The concept of an isolated system is useful for understanding a basic principle. Be we should alsoconsider what happens when we have an external potential u = u (x , t ). Under the above translationu → u (x 0, t ) = u (x − α, t ) and in general ∂u

∂x 6= 0. Thus for systems in an external field the

momentum is conserved only if the potential does not depend on x , i.e. the potential is a constant

or a function of time, and thus du dx = 0, which gives by Newton’s Second Law of motion

dp dt = 0, the

condition for linear momentum conservation.

In 3 dimensions, we can have 3 conserved quantities: p x , p y , p z provided that ∂u ∂x

= 0, ∂u ∂y

= 0 and∂u ∂z

= 0 respectively.

Let us illustrate this by the following example.


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Projectile in 2 dimensions: particle of mass m in a gravitational field

Initially we give the potential a horizontal velocity v , such that at t = 0 the horizontal component of speed is dx

dt and the vertical component of speed is 0. The particle has momentum in two directions

: horizontal p x and vertical p y .No changes in the horizontal direction, no external forces in x direction → symmetry, i.e.

horizontally the particle keeps doing what it started off doing, dp x dt = 0. In the vertical direction, wehave an external field - gravity. This breaks the symmetry, i.e. up is diff erent from down. No

symmetry → No conserved quantity. P y not conserved. dp y dt

= − ∂u ∂y 6= 0.


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Projectile in 2 dimensions: particle of mass m in a gravitational field

Consequently, in the vertical direction the particle’s momentum changes, such that vertically theparticle does not continue doing what it started off doing, and the resulting trajectory is shown inthe diagram.

spatial translation

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