ModelAdmin with a has_many relationship

I'm extending ModelAdmin to manage a DataObject and I have a $has_many relationship that is managed by another DataObject. I'd like to manage this object on another tab but am lost as to how I'd add it. My basic code:

ApplicationAdmin.php:

class Applications extends ModelAdmin {
    private static $menu_title = 'Applications';

    private static $url_segment = 'apps';

    private static $managed_models = array (
        'Application'
    );
}

Application.php

class Application extends DataObject {
    private static $db = array(
        'Name' => "varchar"
    );

    private static $has_many = array(
        'Documents' => 'Document',
    );

    public function getCMSFields() {
        $fields = FieldList::create(
            TextField::create('Name'),
        );

        return $fields;
    }
}

Document.php

class Document extends DataObject {
    private static $db = array(
        'Title' => "varchar",
    );

    private static $has_one = array(
        "Document" => "File",
        "Application" => "Application"
    );

    public function getCMSFields () {
        $fields = FieldList::create(
            TextField::create('Title'),
            $doc = new UploadField('Document')
        );
        $doc->getValidator()->setAllowedExtensions(array('pdf', 'docx'));

        return $fields;
    }   
}

Basically I'd like to manage the documents for this entry under Root.Documents tab.

You can use a GridField to handle the relation between Application and Document, and place that field on its own tab if you wish. Example:

# Application.php
public function getCMSFields() {

    $fields         = parent::getCMSFields();
    $nameField      = TextField::create('Name');
    $documentsField = GridField::create(
        'Documents',
        'Documents',
        $this->Documents(),
        GridFieldConfig_RelationEditor::create()
    );

    $fields->addFieldToTab(
        'Root.Main',
        $nameField
    );

    $fields->addFieldToTab(
        'Root.Documents',
        $documentsField
    );

    return $fields;

}

Your code contains a some typos in the Document class (the classname has .php in it, getValidator should be getValidator()), and that class also needs a $has_one = ['Application' => 'Application']; for the relation to work.