Newtonian Mechanics 05

— 4. Hamilton's Principle — Minimum principles have a long history in the history of Physics:

• Heron explained the law of light reflection using a principle of minimum time.
• Fermat corrected Heron's Principle by stating that light travels between two points in the shortest time available.
• Maupertuis stated his minimum action principle that postulated that a particle's dynamics always minimized the action.
• Gauss postulated his principle of least constraint.
• Hertz postulaed his principle of minimum curvature.

In modern Physics one uses a more general extremum principle and the focus of this section will be to state this principle and flesh out its consequences.

Definition 2 The lagrangian (sometimes called the lagrangian function), $ {L}$, of a particle is the difference between its kinetic and potential energies. $ \displaystyle L=K-U \ \ \ \ \ (1)$

Definition 3 The action, $ {S}$, of a particle's movement (be it a real or virtual one) is: $ \displaystyle \int_{t_1}^{t_2}(K-U)dt \ \ \ \ \ (2)$

Axiom 1 Given a collection of paths that a particle can take between points $ {x_1}$ and $ {x_2}$ in the the time interval $ {\Delta t= t_2-t_1}$ the actual path that the particle takes is the one that makes the action stationary $ \displaystyle \delta S=\delta \int_{t_1}^{t_2}(K-U)dt=0 \ \ \ \ \ (3)$

For rectangular coordinates it is $ {T=T(x_i)}$, $ {U=U(x_i)}$, so $ {L=T-U=L(x_i,\dot{x}_i)}$ (where $ {\dot{x}_i=\dfrac{dx_i}{dt}}$ is called Newton's notation).

The function $ {L}$ can be identified with the function $ {f}$ that we saw on Newtonian Mechanics 04 if one makes the obvious analogies

• $ {x \rightarrow t}$
• $ {y_i(x) \rightarrow x_i(t)}$
• $ {y\prime_i(x) \rightarrow x\prime_i(t)}$
• $ {f(y_i(x),y\prime_i (x),x) \rightarrow L(x_i,\dot{x}_i,t)}$

In this case the Euler equations are called the Euler-Lagrange equations and it is

$ \displaystyle \frac{\partial L}{\partial x_i}-\frac{d}{dt}\frac{\partial L}{\partial \dot{x}_i}=0 $

Example 3 Let us study the harmonic oscillator under Langrangian formalism $ \displaystyle L=T-U=1/2m\dot{x}^2-1/2kx^2 \ \ \ \ \ (4)$ First it is $ {\dfrac{\partial L}{\partial x_i}=-kx}$. Then we have $ {\dfrac{d}{dt}\dfrac{\partial L}{\partial \dot{x}_i}=\dfrac{d}{dt}m\dot{x}=m\ddot{x}}$. Hence it is $ {\dfrac{\partial L}{\partial x_i}-\dfrac{d}{dt}\dfrac{\partial L}{\partial \dot{x}_i}=0 \Rightarrow m\ddot{x}+kx=0 \Rightarrow m\ddot{x}=-kx}$ which is just the harmonic oscillator dynamic equation that we already know.

Example 4 Consider a planar pendulumplanar pendulum write its Lagrangian and derive its equation of motion. The Lagrangian for the planar pendulum is $ \displaystyle L=1/2ml^2\dot{\theta}^2-mgl(1-\cos \theta) \ \ \ \ \ (5)$ If we consider $ {\theta}$ to be a rectangular coordinate (which it isn't!) it follows that the equation of motion is: $ \displaystyle \ddot{\theta}+g/l\sin \theta=0 $ This is precisely the equation of motion of a planar pendulum and this result is apparently unexpected since we only analyzed the Lagrangian for rectangular coordinates.

— 5. Generalized coordinates —

Consider a mechanical system constituted by $ {n}$ particles. In this case one would need $ {3n}$ quantities to describe the position of all particles (since we have 3 degrees of freedom). In the case of having any kind of restraints on the motion of the particles the number of quantities needed to describe the motion of particle is less than $ {3n}$. Suppose that one has $ {m}$ restrictions than the degrees of freedom are $ {3n-m}$.

Let $ {s=3n-m}$. These $ {s}$ coordinates don't need to be rectangular, polar, cylindrical nor spherical. 

These coordinates can be of any kind provided that they completely specify the mechanical state of the system.

Definition 4 The set of $ {s}$ coordinates that totally specify the mechanical state of $ {n}$ particles is defined to be the set of generalized coordinates. The generalized coordinates are represented by $ \displaystyle q_1,q_2,\cdots,q_s$

Since we defined the generalized coordinates of a system of particles one can also define its set of generalized velocities.

Definition 5 The set of $ {s}$ velocities of a system of $ {n}$ particles described by a set of generalized coordiinattes is defined to be the set of generalized velocities. The generalized velocities are represented by $ \displaystyle \dot{q_1},\dot{q_2},\cdots,\dot{q_s}$

Let $ {\alpha}$ denote the particle, $ {\alpha=1,2,\cdots,n}$, $ {i}$ represent the degrees of freedom $ {i}$, $ {i=1,2,3}$ and $ {j}$ the number generalized coordinates $ {j=1,2,\cdots,s}$.

$ \displaystyle x_{\alpha,i}=x_{\alpha,i}(q_1,q_2,\cdots,q_s,t)=x_{\alpha,i}(q_j,t) \ \ \ \ \ (6)$

For the generalized velocities it is

$ \displaystyle \dot{x}_{\alpha,i}=\dot{x}_{\alpha,i}(q_j,t) \ \ \ \ \ (7)$

The inverse transformations are

$ \displaystyle q_j=q_j(x_{\alpha,i},t) \ \ \ \ \ (8)$

and

$ \displaystyle \dot{q_j}=\dot{q}_j(x_{\alpha,i},t) \ \ \ \ \ (9)$

Finally let us note that we also need $ {m=3n-s}$ equations of constraint

$ \displaystyle f_k=f_k(x_{\alpha,i},t) \ \ \ \ \ (10)$

with $ {k=1,2,\cdots,m}$.

Example 5 Consider a point particle that moves along the surface of a semi-sphere of radius $ {R}$ whose center is the origin of the coordinate system. The relevant equations are $ {x^2+y^2+z^2-R^2\geq 0}$ and $ {z\geq 0}$. Let $ {q_1=x/R}$, $ {q_2=y/R}$ and $ {q_3=z/R}$ be our generalized coordinates. Furthermore we have the condition $ {q_1^2+q_2^2+q_3^2=1}$ as a constraint equation. Hence $ {q_3=\sqrt{1-(q_1^2+q_2^2)}}$

The time evolution of a mechanical system can be represented as a curve in the configuration space.

— 6. Euler-Lagrange Equations in generalized coordinates —

Since $ {K}$ and $ {U}$ are scalar functions $ {L}$ is also a scalar function. Therefore $ {L}$ is an invariant for coordinate transformations.

Hence it is

$ \displaystyle L=K(\dot{x}_{\alpha,i})- U(x_{\alpha,i})=T(q_j,\dot{q}_j,t)-U(q_j,t) \ \ \ \ \ (11)$

and $ {L=L(q_j,\dot{q}_j,t)}$.

Hence we can write Hamilton's Principle (Section 4) in the form

$ \displaystyle \delta \int_{t_1}^{t_2} L(q_j,\dot{q}_j,t) dt=0 \ \ \ \ \ (12)$

That is

• $ {x \rightarrow t}$
• $ {y_i(x) \rightarrow q_j(t)}$
• $ {y\prime_i(x) \rightarrow q\prime_j(t)}$
• $ {f(y_i(x),y\prime_i (x),x) \rightarrow L(q_j,\dot{q}_j,t)}$

are the analogies to be made now.

Finally the Euler-Lagrange Equations are

$ \displaystyle \frac{\partial L}{\partial q_j}-\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_j}=0 \ \ \ \ \ (13)$

for $ {j=1,2,\cdots,s}$

To finalize this section let us note the conditions of validity for the Euler-Lagrange equations:

• The system is conservative.
• The equations of constraint have to be functions between the coordinates of the particles and can also be a function of time.

Definition 6 Configuration space is the vector space defined by the generalized coordinates

Example 6 Consider the motion of a particle of mass $ {m}$ along the surface of a half-angle cone under the action of the force of gravity . The equations are $ {z=r\cot\alpha}$ and $ {v^2=\dot{r}^2\csc^2\alpha+r^2\dot{\theta}^2}$ Now for the potential energy it is $ {U=mgz=mgr\cot\alpha}$. Thus the lagrangian is $ \displaystyle L=1/2m(\dot{r}^2\csc^2\alpha+r^2\dot{\theta}^2)-mgr\cot\alpha$ Since $ {\dfrac{\partial L}{\partial \theta}=0}$ it is $ {\dfrac{d}{dt}\dfrac{\partial L}{\partial \dot{\theta}}=0}$. Hence it is $ {\dfrac{\partial L}{\partial \dot{\theta}}=mr^2\dot{\theta}=\mathrm{const}}$. The angular momentum about the $ {z}$ axis is $ {mr^2\dot{\theta}=mr^2\omega}$. Thus $ {mr^2\omega=\mathrm{const}}$ expresses the conservation of angular momentum about the axis of symmetry of the system. It is left as an exercise for the reader to find the Euler-Lagrange equation for $ {r}$.