Action, Lagrangian and Hamiltonian Mechanics

More on the origins of physics

The view of physics taught at school is quite different from modern physics. Elementary School Physics concentrates on lumps of matter undergoing accelerations, collisions, extensions and motions in set directions. In real physics the world is understood as a collection of events occurring in a 'manifold' that dictates the freedom for motion. Each event or phenomenon is either directed and has the properties of a vector or not directed and has the properties of a scalar. Directed events have a magnitude that is a property of the event or phenomenon itself. The interaction of events with the world depends upon the angle between the direction of the event and the thing with which it interacts. So, in physics phenomena have an intrinsic magnitude and this can cause effects on other things according to the way they interact in space and time. In real physics all interactions depend on both magnitude and the spatio-temporal relations between things.

The next big departure between physics and School Physics is the conservation laws. In physics it is understood that space and time are symmetrical and that the freedom for things to move evenly in all spatial directions results in the conservation of linear momentum, the evenness of time results in the conservation of energy etc. The discovery of the role of symmetry in the conservation laws arose out of the Lagrangian form of analysis and will be discussed below. The application of the Lagrangian method in quantum mechanics demonstrates why large objects take particular paths in the real world and can be used to derive and explain Newtonian mechanics.

It is the deep attachment of biologists and AI researchers to School Physics that is probably the chief obstacle to making progress in areas such as consciousness studies. The fact that the world can, to some extent, be described by processes should not blind us to the fact that the classical world of our observation is actually arrangements of things governed by metrical geometry and quantum physics.

Action

Most of those people who have been taught physics at school learnt Newton's original approach. This seventeenth century approach to physics has been superseded. It will come as a shock to many students to know that it was first superseded in the eighteenth century. What you were taught in school physics is an approach that is two hundred years out of date.

Fortunately the approach discovered over two centuries ago is still widely applicable, even in modern physics, so it is easy to catch up. The eighteenth century approach is known as Lagrangian mechanics (devised by Joseph Lagrange between 1772 and 1788). Lagrangian mechanics concentrates on the energy exchanges during motion rather than on the forces involved. Lagrangian mechanics gave rise to Hamiltonian mechanics (devised by William Hamilton 1833).

Consider a toy train running along friction free tracks. We want to work out the path it takes to get from the start to the finish of the track using our intuitions about energy and motion. If the train is run freely along the track it is found to take a particular time to get to the finish. If, on a second run, the train is reversed then set back in motion in the original direction to finish at the same time any amount of energy could be used to get the train from the start to the finish. It is evident that if the train is to get from the start to the finish in a given time then the least amount of energy used over the period occurs when there are no interventions.

We could measure all the forms of energy used to slow down or speed up the train to see if an intervention has occurred but it turns out that only the kinetic energy of the train needs to be measured. If the train is slowed down, subtracting kinetic energy, then for the train to get to the end of the track at the proper time even more kinetic energy must be added when pushing it forward again for it to arrive at the finish on time. This means that we can account for the energy expenditure that affects the motion of the train by simply measuring the kinetic energy at intervals. The minimum amount of kinetic energy over the whole period of the trip corresponds to no interventions.

No interventions occur when the sum of all the kinetic energy measurements are zero. The sum of the kinetic energy measurements in the toy train system is known as the action and has the symbol S.

The action can be more complicated than a simple sum of kinetic energies, for instance when a ball is thrown into the air the kinetic energy can be converted into potential energy and vice versa. If a ball is thrown into the air and hits the ground after a definite time then the minimum interventions occur when the sum of the measurements of the difference between the kinetic and potential energy over the interval is a minimum. In this case the 'action' is the sum of the measurements of the difference between the kinetic and potential energies.

Pierre Louis Moreau de Maupertuis discovered the idea of least action in 1746. He defined the action as the product of the time over which a movement occurs and twice the kinetic energy of the moving object. He found that this product tends to a minimum and this idea became called the Principle of Least Action.

The work of Euler, Lagrange and Hamilton has led to the concepts in the principle of least action being applied to the whole of physics. This wider and modified principle of least action is now called the Principle of Stationary Action.

In mathematical terms the action, S, is given by:

$S=\int _{t_{1}}^{t_{2}}\ (T-U)\,dt.$

where T is the kinetic energy and U is the potential energy.

The quantity (T - U) is known as the Langrangian function so if:

$L=T-U$

The Lagrangian depends upon the position and the derivative of the position with respect to time $(x,{\dot {x}})$ . The action is:

S=\int _{t_{1}}^{t_{2}}\;L(x,{\dot {x}})\,dt.

The problem is to determine how the Lagrangian, $(T-U)$ , can vary with distance, $x$ , so that the action, $S$ is minimised. In other words, given relationships between $T,U$ and $x$ , what curve of $L$ against $t$ will contain the minimum area. (This process is known as finding the minimising extremal curve for the integral).

The starting point for calculating the least action in this way is Euler's calculation of variations method (see Hanc 2005). This results in the Euler-Lagrange equation:

{\partial L \over \partial x_{a}}-{d \over dt}{\partial L \over \partial {\dot {x}}_{a}}=0

which is a complicated formula for finding the extremal curve.

The Lagrangian

The Langrangian $(T-U)$ finds immediate applications in simple mechanics. In simple mechanics the kinetic energy of a moving object is given by:

T={\frac {1}{2}}mv^{2}

which, as $v={\dot {x}}$ (the time derivative of distance), equals:

T={\frac {1}{2}}m{\dot {x}}^{2}

and the potential energy is usually directly proportional to distance:

U=kx

or

U=mgh

etc..

The Lagrangian is then:

L={\frac {1}{2}}m{\dot {x}}^{2}-U(x)

Differentiating the Lagrangian with respect to $x$ :

{\partial L \over \partial x}=-{dU \over dx}

but Newtonian force is the change in potential energy with distance so:

{\partial L \over \partial x}=force

Differentiating Lagrangian with respect to ${\dot {x}}$ :

{\partial L \over \partial {\dot {x}}}=m{\dot {x}}

and $m{\dot {x}}$ is Newtonian momentum.

So:

{d \over dt}{\partial L \over \partial {\dot {x}}}=m{\dot {\dot {x}}}

which is Newtonian force. Hence:

{\partial L \over \partial x}={d \over dt}{\partial L \over \partial {\dot {x}}}

which is the Lagrangian equivalent of $f=ma$

Hamiltonian mechanics

Hamiltonian mechanics starts from the idea of expressing the total energy of a system:

H=T+U

where T is the kinetic energy and U is the potential energy. The Hamiltonian can be expressed in terms of the momentum, $p$ and the Lagrangian:

H_{(p,{\dot {x}})}=p{\dot {x}}-L_{(x,{\dot {x}})}

Differentiating the Hamiltonian with respect to momentum velocity is given by:

{\partial H \over \partial p}={\dot {x}}

Differentiating the Hamiltonian with respect to $x$ we can derive the Hamiltonian expression for force:

{\partial H \over \partial x}=-{\partial L \over \partial x}

and

-{\partial H \over \partial x}={\dot {p}}=force

.

Lagrangian analysis and conservation laws

The Euler-Lagrange equation can be re-organised as:

{\partial L \over \partial x}={d \over dt}{\partial L \over \partial {\dot {x}}}

If one side of this equation is zero then the other side is also zero. This means that, for instance, if there is no change in kinetic-potential energy with distance then $\partial L \over \partial {\dot {x}}$ is constant or conserved.

In the discussion of action above it was shown that changes in the kinetic and potential energy are due to perturbations in the course or progress of an object. In other words changes in the Lagrangian will occur in Euclidean space if an object is perturbed in its motion and $\partial L \over \partial x$ will be zero if the path is unperturbed.

In the case of a freely moving particle:

L={\frac {1}{2}}m{\dot {x}}^{2}

and

{\partial L \over \partial {\dot {x}}}=m{\dot {x}}

If

{\partial L \over \partial x}=0

Then

m{\dot {x}}

the momentum, is conserved.

Emmy Noether systematically investigated the relationship between conservation laws, symmetries and invariant quantities. The following symmetries are shown with their corresponding conservation laws:

Translation in space: conservation of momentum.

Translation in time: conservation of energy

Spatial rotation: conservation of angular momentum

Hyperbolic rotation (Lorentz boost): conservation of energy-momentum 4 vector

See http://www.eftaylor.com/pub/Symmetry0104.pdf

http://www.mathpages.com/home/kmath564/kmath564.htm

This is a stub

The role of quantum mechanics

See Quantum physics explains Newton’s laws of motion http://www.eftaylor.com/pub/OgbornTaylor.pdf

Special relativity for beginners http://en.wikipedia.org/wiki/Special_relativity_for_beginners

This is a stub

What is it like to be physical?

Space-time vector or QM field? This is a stub

References

Hanc, J. (2005). The original Euler’s calculus-of-variations method:Key to Lagrangian mechanics for beginners.Submitted to Eur. J. Phys. http://www.eftaylor.com/pub/HancEulerEJP.pdf

Norbury, J.W. Lagrangians and Hamiltonians for High School Students. http://arxiv.org/PS_cache/physics/pdf/0004/0004029.pdf

Calvert, J.B. The beautiful theory. http://www.du.edu/~jcalvert/math/lagrange.htm see also http://www.du.edu/~jcalvert/